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href="/nlab/show/arithmetic+topology">arithmetic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+arithmetic+geometry">higher arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+arithmetic+geometry">E-∞ arithmetic geometry</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/number">number</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>, <a class="existingWikiWord" href="/nlab/show/integer+number">integer number</a>, <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a>, <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, <a class="existingWikiWord" href="/nlab/show/irrational+number">irrational number</a>, <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a>, <a class="existingWikiWord" href="/nlab/show/quaternion">quaternion</a>, <a class="existingWikiWord" href="/nlab/show/octonion">octonion</a>, <a class="existingWikiWord" href="/nlab/show/adic+number">adic number</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+number">cardinal number</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+number">ordinal number</a>, <a class="existingWikiWord" href="/nlab/show/surreal+number">surreal number</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+arithmetic">Peano arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/second-order+arithmetic">second-order arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transfinite+arithmetic">transfinite arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+arithmetic">cardinal arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+arithmetic">ordinal arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+field">prime field</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+integer">p-adic integer</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+rational+number">p-adic rational number</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+complex+number">p-adic complex number</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></strong>, <a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+scheme">arithmetic scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+curve">arithmetic curve</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+genus">arithmetic genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chern-Simons+theory">arithmetic Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chow+group">arithmetic Chow group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil-%C3%A9tale+topology+for+arithmetic+schemes">Weil-étale topology for arithmetic schemes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absolute+cohomology">absolute cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+conjecture+on+Tamagawa+numbers">Weil conjecture on Tamagawa numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borger%27s+absolute+geometry">Borger's absolute geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Iwasawa-Tate+theory">Iwasawa-Tate theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic jet space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adelic+integration">adelic integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shtuka">shtuka</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenioid">Frobenioid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Arakelov+geometry">Arakelov geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Riemann-Roch+theorem">arithmetic Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+algebraic+K-theory">differential algebraic K-theory</a></p> </li> </ul> </div></div> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="rational_numbers">Rational numbers</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#as_a_field'>As a field</a></li> <li><a href='#as_a_commutative_ring'>As a commutative ring</a></li> <li><a href='#as_an_sequential_colimit_in_cring'>As an sequential colimit in CRing</a></li> <li><a href='#as_an_abelian_group'>As an abelian group</a></li> <li><a href='#as_an_initial_object_in_a_category'>As an initial object in a category</a></li> <li><a href='#in_dependent_type_theory'> In dependent type theory</a></li> <ul> <li><a href='#as_a_dependent_sum_type'>As a dependent sum type</a></li> <li><a href='#as_a_higher_inductive_type'>As a higher inductive type</a></li> </ul> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#commutative_ring_structure_on_the_rational_numbers'>Commutative ring structure on the rational numbers</a></li> <li><a href='#order_structure_on_the_rational_numbers'>Order structure on the rational numbers</a></li> <li><a href='#pseudolattice_structure_on_the_rational_numbers'>Pseudolattice structure on the rational numbers</a></li> <li><a href='#uniform_structure_on_the_rational_numbers'>Uniform structure on the rational numbers</a></li> <li><a href='#algebraic_closure'>Algebraic closure</a></li> <li><a href='#topologies'>Topologies</a></li> <li><a href='#analysis'>Analysis</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>A <em>rational number</em> is a <a class="existingWikiWord" href="/nlab/show/fraction">fraction</a> of two <a class="existingWikiWord" href="/nlab/show/integer">integer</a> <a class="existingWikiWord" href="/nlab/show/numbers">numbers</a>.</p> <h3 id="as_a_field">As a field</h3> <p>The <a class="existingWikiWord" href="/nlab/show/field">field</a> of <strong>rational numbers</strong>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>, is the <a class="existingWikiWord" href="/nlab/show/field+of+fractions">field of fractions</a> of the <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>, hence the field consisting of formal <a class="existingWikiWord" href="/nlab/show/fractions">fractions</a> (“<a class="existingWikiWord" href="/nlab/show/ratios">ratios</a>”) of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>.</p> <h3 id="as_a_commutative_ring">As a commutative ring</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>,</mo><mn>1</mn><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>,</mo><mi>s</mi><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>→</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+)</annotation></semantics></math> be the set of positive integers. The positive integers are embedded into every <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>: there is an injection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inj</mi><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>→</mo><mo lspace="0em" rspace="thinmathspace">R</mo></mrow><annotation encoding="application/x-tex">inj:\mathbb{N}^+\to\R</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inj</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">inj(1) = 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inj</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>inj</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">inj(s(n)) = inj(n) + 1</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">n:\mathbb{N}^+</annotation></semantics></math>.</p> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> has an injection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inv</mi><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>→</mo><mo lspace="0em" rspace="thinmathspace">R</mo></mrow><annotation encoding="application/x-tex">inv:\mathbb{N}^+\to\R</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inj</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>inv</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">inj(n) \cdot inv(n) = 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">inv</mo><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo lspace="0em" rspace="thinmathspace">inj</mo><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\inv(n) \cdot \inj(n) = 1</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">n:\mathbb{N}^+</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/commutative+algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>ℚ</mi> </mrow> <annotation encoding="application/x-tex">\mathbb{Q}</annotation> </semantics> </math>-algebra</a></strong>, and the commutative ring of <strong>rational numbers</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>-algebra.</p> <p>It can then be proven from the ring axioms and the properties of the integers that every rational number apart from zero and has a multiplicative inverse, making <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> a field.</p> <h3 id="as_an_sequential_colimit_in_cring">As an sequential colimit in CRing</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mi>n</mi><mo>!</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[1/n!]</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/localization+of+a+commutative+ring">localization</a> of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/localisation+of+a+commutative+ring+away+from+an+element">away from</a> the <a class="existingWikiWord" href="/nlab/show/factorial">factorial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>!</mo></mrow><annotation encoding="application/x-tex">n!</annotation></semantics></math>, and for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">i \in \mathbb{N}</annotation></semantics></math>, there is a unique commutative <a class="existingWikiWord" href="/nlab/show/ring+homomorphism">ring homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>i</mi></msub><mo>:</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mi>i</mi><mo>!</mo><mo stretchy="false">]</mo><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">h_{i}:\mathbb{Z}[1/i!]\to\mathbb{Z}[1/(i + 1)!]</annotation></semantics></math> defined by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of localisation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mi>i</mi><mo>!</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[1/i!]</annotation></semantics></math> away from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i + 1</annotation></semantics></math>. Then the commutative ring of <strong>rational numbers</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/sequential+colimit">sequential colimit</a> of the <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mn>0</mn><mo>!</mo><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><msub><mi>h</mi> <mn>0</mn></msub></mrow></mover><mi>ℤ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mn>1</mn><mo>!</mo><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><msub><mi>h</mi> <mn>1</mn></msub></mrow></mover><mi>ℤ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo>!</mo><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><msub><mi>h</mi> <mn>2</mn></msub></mrow></mover><mi>…</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}[1/0!] \overset{h_0}\to \mathbb{Z}[1/1!] \overset{h_1}\to \mathbb{Z}[1/2!] \overset{h_2}\to \ldots</annotation></semantics></math></div> <h3 id="as_an_abelian_group">As an abelian group</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>,</mo><mn>1</mn><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>,</mo><mi>s</mi><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>→</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{N}^+,1:\mathbb{N}^+,s:\mathbb{N}^+\to \mathbb{N}^+)</annotation></semantics></math> be the set of positive integers and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>0</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{Z},0,+,-,1)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> on the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">{1}</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> containing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> as an abelian subgroup. The positive integers are embedded into the <a class="existingWikiWord" href="/nlab/show/function+algebra">function abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A \to A</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>A</mi></msub><mo>:</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">id_A:A \to A</annotation></semantics></math> being the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>; i.e. there is an injection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inj</mi><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>→</mo><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">inj:\mathbb{N}^+\to (A \to A)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inj</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">inj(1) = id_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inj</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>inj</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">inj(s(n)) = inj(n) + id_A</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">n:\mathbb{N}^+</annotation></semantics></math>.</p> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> has an injection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inv</mi><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>→</mo><mo stretchy="false">(</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">inv:\mathbb{N}^+\to (A \to A)</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>:</mo><msup><mi>ℕ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">n:\mathbb{N}^+</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inj</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>inv</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">inj(n) \circ inv(n) = id_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>inv</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>inj</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">inv(n) \circ inj(n) = id_A</annotation></semantics></math>. Then the abelian group of <strong>rational numbers</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> such abelian group.</p> <h3 id="as_an_initial_object_in_a_category">As an initial object in a category</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/integers">integers</a> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub><mo>≔</mo><mo stretchy="false">{</mo><mi>a</mi><mo>∈</mo><mi>ℤ</mi><mo stretchy="false">|</mo><mi>a</mi><mo><</mo><mn>0</mn><mo>∨</mo><mn>0</mn><mo><</mo><mi>a</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{#0} \coloneqq \{a \in \mathbb{Z} \vert a \lt 0 \vee 0 \lt a \}</annotation></semantics></math></div> <p>be the set of integers apart from zero.</p> <p>For lack of a better name, let us define a <em>set with rational numbers</em> to be a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> with a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo>∈</mo><mi>ℤ</mi><mo>×</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\iota \in \mathbb{Z} \times \mathbb{Z}_{#0} \to A</annotation></semantics></math> with domain the Cartesian product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>×</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z} \times \mathbb{Z}_{#0}</annotation></semantics></math> and codomain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>∈</mo><mi>ℤ</mi><mo>.</mo><mo>∀</mo><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub><mo>.</mo><mo>∀</mo><mi>c</mi><mo>∈</mo><mi>ℤ</mi><mo>.</mo><mo>∀</mo><mi>d</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub><mo>.</mo><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>d</mi><mo>=</mo><mi>c</mi><mo>⋅</mo><mi>b</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mi>ι</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ι</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall a \in \mathbb{Z}. \forall b \in \mathbb{Z}_{#0}. \forall c \in \mathbb{Z}. \forall d \in \mathbb{Z}_{#0}. (a \cdot d = c \cdot b) \implies (\iota(a, b) = \iota(c, d))</annotation></semantics></math></div> <p>A <em>homomorphism of sets with rational numbers</em> between two sets with rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>B</mi> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">f \in B^A</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>∈</mo><mi>ℤ</mi><mo>.</mo><mo>∀</mo><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub><mo>.</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ι</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall a \in \mathbb{Z}. \forall b \in \mathbb{Z}_{#0}. f(\iota_A(a, b)) = \iota_B(a, b)</annotation></semantics></math></div> <p>The <em>category of sets with rational numbers</em> is the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SwRN</mi></mrow><annotation encoding="application/x-tex">SwRN</annotation></semantics></math> whose objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ob</mi><mo stretchy="false">(</mo><mi>SwRN</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ob(SwRN)</annotation></semantics></math> are sets with rational numbers and whose morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mor</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mor(A,B)</annotation></semantics></math> for sets with rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>SwDF</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in Ob(SwDF)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>SwRN</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \in Ob(SwRN)</annotation></semantics></math> are homomorphisms of sets with rational numbers. The set of <strong>rational numbers</strong>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>, is defined as the initial object in the category of sets with rational numbers.</p> <h3 id="in_dependent_type_theory"> In dependent type theory</h3> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, there are multiple different definitions of the type of rational numbers.</p> <h4 id="as_a_dependent_sum_type">As a dependent sum type</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mo>≠</mo><mn>0</mn></mrow></msub><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>a</mi><mo>:</mo><mi>ℤ</mi></mrow></munder><mi>max</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>a</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{\neq 0} \coloneqq \sum_{a:\mathbb{Z}} \max(a,-a) \gt 0</annotation></semantics></math></div> <p>be the non-zero integers, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gcd</mi><mo>:</mo><mi>ℤ</mi><mo>×</mo><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\gcd:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/greatest+common+divisor">greatest common divisor</a> on the integers. Then the rational numbers is defined as the type of pairs of integers and non-zero integers such that the greatest common divisor is equal to one:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>ℤ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>y</mi><mo>:</mo><msub><mi>ℤ</mi> <mrow><mo>≠</mo><mn>0</mn></mrow></msub></mrow></munder><mi>gcd</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>=</mo> <mi>ℤ</mi></msub><mn>1</mn></mrow><annotation encoding="application/x-tex">\mathbb{Q} \coloneqq \sum_{x:\mathbb{Z}} \sum_{y:\mathbb{Z}_{\neq 0}} \gcd(x, \pi_1(y)) =_\mathbb{Z} 1</annotation></semantics></math></div> <p>We define the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>ℤ</mi><mo>×</mo><msub><mi>ℤ</mi> <mrow><mo>≠</mo><mn>0</mn></mrow></msub><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">(-)/(-) : \mathbb{Z} \times \mathbb{Z}_{\neq 0} \rightarrow \mathbb{Q}</annotation></semantics></math> for all integers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a:\mathbb{Z}</annotation></semantics></math> and non-zero integers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><msub><mi>ℤ</mi> <mrow><mo>≠</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b:\mathbb{Z}_{\neq 0}</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo>≔</mo><mo stretchy="false">(</mo><mi>a</mi><mo>÷</mo><mi>gcd</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>÷</mo><mi>gcd</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a/b \coloneqq (a \div \gcd(a, \pi_1(b)), \pi_1(b) \div \gcd(a, \pi_1(b)), p(a, b))</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a, b)</annotation></semantics></math> is the witness that the greatest common divisor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>÷</mo><mi>gcd</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \div \gcd(a, \pi_1(b))</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>÷</mo><mi>gcd</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(b) \div \gcd(a, \pi_1(b))</annotation></semantics></math> is equal to 1, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>÷</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \div b</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/integer">integer</a> <a class="existingWikiWord" href="/nlab/show/division">division</a>, since integers are an <a class="existingWikiWord" href="/nlab/show/Euclidean+domain">Euclidean domain</a> and thus have a division and remainder operation.</p> <h4 id="as_a_higher_inductive_type">As a higher inductive type</h4> <p>In <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, the type of <strong>rational numbers</strong>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>, is defined (<a href="#UBP13">UBP13, §11.1</a>) as the <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a> generated by:</p> <ul> <li> <p>A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>ℤ</mi><mo>×</mo><msub><mi>ℤ</mi> <mrow><mo>≠</mo><mn>0</mn></mrow></msub><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">(-)/(-) : \mathbb{Z} \times \mathbb{Z}_{\neq 0} \rightarrow \mathbb{Q}</annotation></semantics></math>, where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mo>≠</mo><mn>0</mn></mrow></msub><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>a</mi><mo>:</mo><mi>ℤ</mi></mrow></munder><mi>max</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>a</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{\neq 0} \coloneqq \sum_{a:\mathbb{Z}} \max(a,-a) \gt 0</annotation></semantics></math></div></li> <li> <p>A dependent product of functions between identities representing that equivalent fractions are equal:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>equivfrac</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>ℤ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>b</mi><mo>:</mo><msub><mi>ℤ</mi> <mrow><mo>≠</mo><mn>0</mn></mrow></msub></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>c</mi><mo>:</mo><mi>ℤ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>d</mi><mo>:</mo><msub><mi>ℤ</mi> <mrow><mo>≠</mo><mn>0</mn></mrow></msub></mrow></munder><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo>⋅</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo>=</mo><mi>c</mi><mo stretchy="false">/</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex">equivfrac : \prod_{a:\mathbb{Z}} \prod_{b:\mathbb{Z}_{\neq 0}} \prod_{c:\mathbb{Z}} \prod_{d:\mathbb{Z}_{\neq 0}} (a \cdot \pi_1(d) = c \cdot \pi_1(b)) \to (a/b = c/d)\,,</annotation></semantics></math></div></li> <li> <p>A set-truncator</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>0</mn></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>a</mi><mo>:</mo><mi>ℚ</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>b</mi><mo>:</mo><mi>ℚ</mi></mrow></munder><mi>isProp</mi><mo stretchy="false">(</mo><mi>a</mi><mo>=</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_0: \prod_{a:\mathbb{Q}} \prod_{b:\mathbb{Q}} isProp(a=b)</annotation></semantics></math></div></li> </ul> <h2 id="properties">Properties</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> be the inclusion of the non-zero integers into the integers.</p> <h3 id="commutative_ring_structure_on_the_rational_numbers">Commutative ring structure on the rational numbers</h3> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>The rational number <strong>zero</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">0 \in \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≔</mo><mn>0</mn><mo stretchy="false">/</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \coloneqq 0/1</annotation></semantics></math></div></div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>The binary operation <strong>addition</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>ℚ</mi><mo>×</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">(-)+(-):\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo stretchy="false">/</mo><mi>d</mi><mo>≔</mo><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>i</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi><mo>⋅</mo><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>b</mi><mo>⋅</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a/b + c/d \coloneqq (a \cdot i(d) + c \cdot i(b))/(b \cdot d)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mspace width="thickmathspace"></mspace><mi>in</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \ in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">c \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d \in \mathbb{Z}_{#0}</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">q \in \mathbb{Q}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">r \in \mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>+</mo><mi>r</mi><mo>=</mo><mi>r</mi><mo>+</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">q + r = r + q</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>TODO</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>The unary operation <strong>negation</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">-(-):\mathbb{Q} \to \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">-(a/b) \coloneqq (-a)/b</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math></p> </div> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>The binary operation <strong>subtraction</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>ℚ</mi><mo>×</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">(-)-(-):\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo>−</mo><mi>c</mi><mo stretchy="false">/</mo><mi>d</mi><mo>≔</mo><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>i</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo>−</mo><mi>c</mi><mo>⋅</mo><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>b</mi><mo>⋅</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a/b - c/d \coloneqq (a \cdot i(d) - c \cdot i(b))/(b \cdot d)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">c \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d \in \mathbb{Z}_{#0}</annotation></semantics></math></p> </div> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p>The rational number <strong>one</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">1 \in \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≔</mo><mn>1</mn><mo stretchy="false">/</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \coloneqq 1/1</annotation></semantics></math></div></div> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>The binary operation <strong>multiplication</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>ℚ</mi><mo>×</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">(-)\cdot(-):\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">/</mo><mi>d</mi><mo>≔</mo><mo stretchy="false">(</mo><mi>a</mi><mo>⋅</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>b</mi><mo>⋅</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a/b \cdot c/d \coloneqq (a \cdot c)/(b \cdot d)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">c \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d \in \mathbb{Z}_{#0}</annotation></semantics></math></p> </div> <div class="num_defn"> <h6 id="definition_8">Definition</h6> <p>The right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>-action <strong>exponentiation</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mo>:</mo><mi>ℚ</mi><mo>×</mo><mi>ℕ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">(-)^{(-)}:\mathbb{Q} \times \mathbb{N} \to \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mo>≔</mo><mo stretchy="false">(</mo><msup><mi>a</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>b</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a/b)^n \coloneqq (a^n)/(b^n)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>.</p> </div> <p>This makes the rational numbers into a commutative ring.</p> <h3 id="order_structure_on_the_rational_numbers">Order structure on the rational numbers</h3> <div class="num_defn"> <h6 id="definition_9">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/predicate">predicate</a> <em>is positive</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>isPositive</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isPositive(a/b)</annotation></semantics></math>, is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>isPositive</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi>a</mi><mo>></mo><mn>0</mn><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn><mo stretchy="false">)</mo><mo>∨</mo><mo stretchy="false">(</mo><mi>a</mi><mo><</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isPositive(a/b) \coloneqq (a \gt 0) \wedge (i(b) \gt 0) \vee (a \lt 0) \wedge (i(b) \lt 0)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_10">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/predicate">predicate</a> <em>is negative</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>isNegative</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isNegative(a/b)</annotation></semantics></math>, is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>isNegative</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi>a</mi><mo>></mo><mn>0</mn><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo><</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∨</mo><mo stretchy="false">(</mo><mi>a</mi><mo><</mo><mn>0</mn><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isNegative(a/b) \coloneqq (a \gt 0) \wedge (i(b) \lt 0) \vee (a \lt 0) \wedge (i(b) \gt 0)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_11">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/predicate">predicate</a> <em>is zero</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>isZero</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isZero(a/b)</annotation></semantics></math>, is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>isZero</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>a</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">isZero(a/b) \coloneqq a = 0</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_12">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/predicate">predicate</a> <em>is non-positive</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>isNonPositive</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isNonPositive(a/b)</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>isNonPositive</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mo>¬</mo><mi>isPositive</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isNonPositive(a/b) \coloneqq \neg isPositive(a/b)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_13">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/predicate">predicate</a> <em>is non-negative</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>isNonNegative</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isNonNegative(a/b)</annotation></semantics></math>, is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>isNonNegative</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mo>¬</mo><mi>isNegative</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isNonNegative(a/b) \coloneqq \neg isNegative(a/b) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_14">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/predicate">predicate</a> <em>is non-zero</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>isNonZero</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isNonZero(a/b)</annotation></semantics></math>, is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>isNonZero</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>isPositive</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo><mo>∨</mo><mi>isNegative</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">isNonZero(a/b) \coloneqq isPositive(a/b) \vee isNegative(a/b)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_15">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/relation">relation</a> <em>is less than</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo><</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \lt q</annotation></semantics></math>, is defined as</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo><</mo><mi>q</mi><mo>≔</mo><mi>isPositive</mi><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \lt q \coloneqq isPositive(q - p)</annotation></semantics></math></p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">p:\mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">q:\mathbb{Q}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_16">Definition</h6> <p>The dependent type <em>is greater than</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>></mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \gt q</annotation></semantics></math>, is defined as</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo><</mo><mi>q</mi><mo>≔</mo><mi>isNegative</mi><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \lt q \coloneqq isNegative(q - p)</annotation></semantics></math></p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">p:\mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">q:\mathbb{Q}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_17">Definition</h6> <p>The dependent type <em>is apart from</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>#</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \# q</annotation></semantics></math>, is defined as</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo><</mo><mi>q</mi><mo>≔</mo><mi>isNonZero</mi><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \lt q \coloneqq isNonZero(q - p)</annotation></semantics></math></p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">p:\mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">q:\mathbb{Q}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_18">Definition</h6> <p>The dependent type <em>is less than or equal to</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>≤</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \leq q</annotation></semantics></math>, is defined as</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo><</mo><mi>q</mi><mo>≔</mo><mi>isNonNegative</mi><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \lt q \coloneqq isNonNegative(q - p)</annotation></semantics></math></p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">p:\mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">q:\mathbb{Q}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_19">Definition</h6> <p>The dependent type <em>is greater than or equal to</em>, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>≤</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">p \leq q</annotation></semantics></math>, is defined as</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo><</mo><mi>q</mi><mo>≔</mo><mi>isNonPositive</mi><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \lt q \coloneqq isNonPositive(q - p)</annotation></semantics></math></p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">p:\mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">q:\mathbb{Q}</annotation></semantics></math>.</p> </div> <h3 id="pseudolattice_structure_on_the_rational_numbers">Pseudolattice structure on the rational numbers</h3> <div class="num_defn"> <h6 id="definition_20">Definition</h6> <p>The <strong>ramp function</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ramp</mi><mo>:</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">ramp:\mathbb{Q} \to \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ramp</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">/</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi>ramp</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>ramp</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>ramp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>ramp</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>b</mi><mo>⋅</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ramp(a/b) \coloneqq (ramp(a) \cdot ramp(i(b)) + ramp(-a) \cdot ramp(i(-b)))/(b \cdot b)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ℤ</mi> <mrow><mo>#</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}_{#0}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ramp</mi><mo>:</mo><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">ramp:\mathbb{Z} \to \mathbb{Z}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_21">Definition</h6> <p>The <strong>minimum</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>min</mi><mo>:</mo><mi>ℚ</mi><mo>×</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">min:\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>min</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>p</mi><mo>−</mo><mi>ramp</mi><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\min(p,q) \coloneqq p - ramp(p - q)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">p:\mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">q:\mathbb{Q}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_22">Definition</h6> <p>The <strong>maximum</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>max</mi><mo>:</mo><mi>ℚ</mi><mo>×</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">max:\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>max</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>p</mi><mo>+</mo><mi>ramp</mi><mo stretchy="false">(</mo><mi>q</mi><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\max(p, q) \coloneqq p + ramp(q - p)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">p:\mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">q:\mathbb{Q}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_23">Definition</h6> <p>The <strong>absolute value</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo>:</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\vert(-)\vert:\mathbb{Q} \to \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>p</mi><mo stretchy="false">|</mo><mo>≔</mo><mi>max</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\vert p \vert \coloneqq \max(p, -p)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">p:\mathbb{Q}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_24">Definition</h6> <p>The <strong>metric</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>ℚ</mi><mo>×</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\rho:\mathbb{Q} \times \mathbb{Q} \to \mathbb{Q}</annotation></semantics></math> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>max</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>−</mo><mi>min</mi><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(p, q) \coloneqq \max(p, q) - \min(p, q)</annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">p:\mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">q:\mathbb{Q}</annotation></semantics></math>.</p> </div> <h3 id="uniform_structure_on_the_rational_numbers">Uniform structure on the rational numbers</h3> <p>We define the ternary relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mi>ϵ</mi></msub><mi>y</mi><mo>≔</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo><</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">x \sim_\epsilon y \coloneqq \rho(x, y) \lt \epsilon</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">y \in \mathbb{Q}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math>, called “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> are within a distance of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>”. One could show that the rational numbers are a <a class="existingWikiWord" href="/nlab/show/uniform+space">uniform space</a> with respect to the <a class="existingWikiWord" href="/nlab/show/uniformity">uniformity</a> <a class="existingWikiWord" href="/nlab/show/ternary+relation">ternary relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mi>ϵ</mi></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">x \sim_\epsilon y</annotation></semantics></math>.</p> <ol> <li> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{Q}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mi>ϵ</mi></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">x \sim_\epsilon x</annotation></semantics></math></p> </li> <li> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">y \in \mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">z \in \mathbb{Q}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\delta \in \mathbb{Q}_+</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mi>δ</mi></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">x \sim_\delta y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><msub><mo>∼</mo> <mi>ϵ</mi></msub><mi>z</mi></mrow><annotation encoding="application/x-tex">y \sim_\epsilon z</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mrow><mi>δ</mi><mo>+</mo><mi>ϵ</mi></mrow></msub><mi>z</mi></mrow><annotation encoding="application/x-tex">x \sim_{\delta + \epsilon} z</annotation></semantics></math>.</p> </li> <li> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">y \in \mathbb{Q}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mi>ϵ</mi></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">x \sim_\epsilon y</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><msub><mo>∼</mo> <mi>ϵ</mi></msub><mi>x</mi></mrow><annotation encoding="application/x-tex">y \sim_\epsilon x</annotation></semantics></math>.</p> </li> <li> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">y \in \mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mn>1</mn></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">x \sim_1 y</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo><</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\rho(x, y) \lt 1</annotation></semantics></math>.</p> </li> <li> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">y \in \mathbb{Q}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\delta \in \mathbb{Q}_+</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mrow><mi>min</mi><mo stretchy="false">(</mo><mi>δ</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">x \sim_{\min(\delta, \epsilon)} y</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mi>δ</mi></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">x \sim_{\delta} y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mi>ϵ</mi></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">x \sim_{\epsilon} y</annotation></semantics></math>.</p> </li> <li> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{Q}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">y \in \mathbb{Q}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\delta \in \mathbb{Q}_+</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\delta \leq \epsilon</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mi>δ</mi></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">x \sim_{\delta} y</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mi>ϵ</mi></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">x \sim_{\epsilon} y</annotation></semantics></math>.</p> </li> </ol> <h3 id="algebraic_closure">Algebraic closure</h3> <p>The <a class="existingWikiWord" href="/nlab/show/algebraic+closure">algebraic closure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ℚ</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{\mathbb{Q}}</annotation></semantics></math> of the rational numbers is called the <a class="existingWikiWord" href="/nlab/show/field">field</a> of <em><a class="existingWikiWord" href="/nlab/show/algebraic+numbers">algebraic numbers</a></em>. The <a class="existingWikiWord" href="/nlab/show/absolute+Galois+group">absolute Galois group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gal</mi><mo stretchy="false">(</mo><mover><mi>ℚ</mi><mo>¯</mo></mover><mo stretchy="false">|</mo><mi>ℚ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Gal(\overline{\mathbb{Q}}\vert \mathbb{Q})</annotation></semantics></math> has some curious properties, see <a href="absolute+Galois+group#OfTheRationalNumbers">there</a>.</p> <h3 id="topologies">Topologies</h3> <p>There are several interesting <a class="existingWikiWord" href="/nlab/show/topological+space">topologies</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> that make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> under addition, allowing us to define interesting fields by taking the <a class="existingWikiWord" href="/nlab/show/complete+space">completion</a> with respect to this topology:</p> <ol> <li> <p>The <em><a class="existingWikiWord" href="/nlab/show/discrete+topology">discrete topology</a></em> is the most obvious, which is already complete.</p> </li> <li> <p>The <em>absolute-value topology</em> is defined by the <a class="existingWikiWord" href="/nlab/show/metric+space">metric</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">d(x,y) \coloneqq {|x - y|}</annotation></semantics></math>; the completion is the field of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>. The rational numbers are thus a <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>.</p> <p>(This topology is <a class="existingWikiWord" href="/nlab/show/totally+disconnected+space">totally disconnected</a> (<a href="totally+disconnected+space#RationalNumbersAreTotallyDisconnected">this exmpl.</a>))</p> </li> <li> <p>Fixing a <a class="existingWikiWord" href="/nlab/show/prime+number">prime number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, the <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-adic topology</em> is defined by the <a class="existingWikiWord" href="/nlab/show/ultrametric">ultrametric</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≔</mo><mn>1</mn><mo stretchy="false">/</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">d(x,y) \coloneqq 1/n</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is the highest exponent on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/prime+factorization">prime factorization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|x - y|}</annotation></semantics></math>; the completion is the field of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adic+number">adic numbers</a>.</p> </li> </ol> <p>According to <a class="existingWikiWord" href="/nlab/show/Ostrowski%27s+theorem">Ostrowski's theorem</a> this are the only possibilities.</p> <p>Interestingly, (2) cannot be interpreted as a <a class="existingWikiWord" href="/nlab/show/locale">localic</a> group, although the completion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> can. (Probably the same holds for (3); I need to check.)</p> <h3 id="analysis">Analysis</h3> <p>One could analytically define the concepts of <a class="existingWikiWord" href="/nlab/show/limit+of+a+function">limit of a function</a> and <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> with respect to the absolute-value topology, and prove that the limit of a function satisfy the <a class="existingWikiWord" href="/nlab/show/algebraic+limit+theorem">algebraic limit theorem</a>. Since the <a class="existingWikiWord" href="/nlab/show/reciprocal+function">reciprocal function</a> on the rational numbers is well defined for non-zero rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℚ</mi> <mrow><mo>≠</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Q}_{\neq 0}</annotation></semantics></math>, given a continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f:I \to \mathbb{Q}</annotation></semantics></math> for open interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊆</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">I \subseteq \mathbb{Q}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/difference+quotient">difference quotient</a> function exists and thus the <a class="existingWikiWord" href="/nlab/show/derivative">derivative</a> is well-defined for continuous functions. One could thus define <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on the rational numbers, and because the rational numbers are a Hausdorff space, <a class="existingWikiWord" href="/nlab/show/analytic+functions">analytic functions</a> on the rational numbers, despite the fact that the rational numbers are a totally disconnected space.</p> <p>For example,</p> <ul> <li> <p>A rational metric space is a type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><msub><mi>ℚ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_X:X \times X \to \mathbb{Q}_{\geq 0}</annotation></semantics></math> such that</p> <ul> <li>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">y \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_X(x, y) = d_X(y, x)</annotation></semantics></math></li> <li>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">y \in X</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">z \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>≤</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_X(x, z) \leq d_X(x, y) + d_X(y, z)</annotation></semantics></math></li> <li>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>y</mi><mo>⇔</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x = y \iff d_X(x, y) = 0</annotation></semantics></math></li> </ul> </li> </ul> <p>One could show that any <a class="existingWikiWord" href="/nlab/show/open+interval">open interval</a>, <a class="existingWikiWord" href="/nlab/show/half-open+interval">half-open interval</a>, and <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a> in the rational numbers is a rational metric space, with its metric inherited from the Euclidean metric on the rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>ℚ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo stretchy="false">|</mo><mi>a</mi><mo>−</mo><mi>b</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">d_\mathbb{Q}(x, y) = {|a - b|}</annotation></semantics></math>. Thus one could do analysis with <a class="existingWikiWord" href="/nlab/show/partial+functions">partial functions</a> in the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>, such as the <a class="existingWikiWord" href="/nlab/show/reciprocal+function">reciprocal function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><annotation encoding="application/x-tex">x \mapsto \frac{1}{x}</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/rational+functions">rational functions</a> on the rational numbers.</p> <p>Then, given a rational metric space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, d_X)</annotation></semantics></math>:</p> <ul> <li> <p>if it exists, the (Weierstrass) <a class="existingWikiWord" href="/nlab/show/limit+of+a+function">limit of a function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f:X \to \mathbb{Q}</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the rational numbers approaching an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">a \in X</annotation></semantics></math> is a rational number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">L \in \mathbb{Q}</annotation></semantics></math> with a function on the positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>:</mo><msub><mi>ℚ</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">M:\mathbb{Q}_+ \to \mathbb{Q}_+</annotation></semantics></math> such that for all positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math> and for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">b \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo><</mo><mi>M</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0 \lt d_X(a, b) \lt M(\epsilon)</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>L</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">{|f(b) - L|} \lt \epsilon</annotation></semantics></math></p> </li> <li> <p>similarly, if it exists, the (Bourbaki) <a class="existingWikiWord" href="/nlab/show/limit+of+a+function">limit of a function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f:X \to \mathbb{Q}</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the rational numbers approaching an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">a \in X</annotation></semantics></math> is a rational number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">L \in \mathbb{Q}</annotation></semantics></math> with a function on the positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>:</mo><msub><mi>ℚ</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">M:\mathbb{Q}_+ \to \mathbb{Q}_+</annotation></semantics></math> such that for all positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math> and for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">b \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo><</mo><mi>M</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_X(a, b) \lt M(\epsilon)</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>L</mi><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">{|f(b) - L|} \lt \epsilon</annotation></semantics></math></p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/pointwise+continuous+function">pointwise continuous function</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the rational numbers is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f:X \to \mathbb{Q}</annotation></semantics></math> with a function from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the set of endofunctions on the positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>:</mo><mi>X</mi><mo>→</mo><msub><mi>ℚ</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">M:X \to \mathbb{Q}_+ \to \mathbb{Q}_+</annotation></semantics></math> such that for all positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math> and for all rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">a \in X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">b \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo><</mo><mi>M</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\delta_X(a, b) \lt M(a, \epsilon)</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">{|f(a) - f(b)|} \lt \epsilon</annotation></semantics></math>.</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the rational numbers is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f:X \to \mathbb{Q}</annotation></semantics></math> with a function on the positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>:</mo><msub><mi>ℚ</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">M:\mathbb{Q}_+ \to \mathbb{Q}_+</annotation></semantics></math> such that for all positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math> and for all elements numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">a \in X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">b \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo><</mo><mi>M</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\delta_X(a, b) \lt M(\epsilon)</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">{|f(a) - f(b)|} \lt \epsilon</annotation></semantics></math>.</p> </li> </ul> <p>Now, given an <a class="existingWikiWord" href="/nlab/show/injection">injection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>X</mi><mo>↪</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">i:X \hookrightarrow \mathbb{Q}</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of the rational numbers,</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/pointwise+differentiable+function">pointwise differentiable function</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the rational numbers is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f:X \to \mathbb{Q}</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/derivative">derivative</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f':X \to \mathbb{Q}</annotation></semantics></math> on the rational numbers and a function from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the set of endofunctions on the positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>:</mo><mi>X</mi><mo>→</mo><msub><mi>ℚ</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">M:X \to \mathbb{Q}_+ \to \mathbb{Q}_+</annotation></semantics></math> such that for all positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi mathvariant="normal">Q</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathrm{Q}_+</annotation></semantics></math> and for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">a \in X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">b \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo><</mo><mi>M</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{|f(a) - f(b)|} \lt M(a, \epsilon)</annotation></semantics></math> implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|f(b) - f(a) - f'(a)(i(b) - i(a)))|} \lt \epsilon {|f(a) - f(b)|}</annotation></semantics></math></div></li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/uniformly+differentiable+function">uniformly differentiable function</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the rational numbers is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f:X \to \mathbb{Q}</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/derivative">derivative</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f':X \to \mathbb{Q}</annotation></semantics></math> on the rational numbers and a function on the positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>:</mo><msub><mi>ℚ</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">M:\mathbb{Q}_+ \to \mathbb{Q}_+</annotation></semantics></math> such that for all positive rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi mathvariant="normal">Q</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathrm{Q}_+</annotation></semantics></math> and for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">a \in X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">b \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo><</mo><mi>M</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{|f(a) - f(b)|} \lt M(\epsilon)</annotation></semantics></math> implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|f(b) - f(a) - f'(a)(i(b) - i(a)))|} \lt \epsilon {|f(a) - f(b)|}</annotation></semantics></math></div></li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> in rational numbers is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f:\mathbb{Q} \to \mathbb{Q}</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a> of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mi>f</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mo stretchy="false">(</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D^{(-)}f:\mathbb{N} \to (\mathbb{Q} \to \mathbb{Q})</annotation></semantics></math> and a sequence of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msup><mi>f</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mo stretchy="false">(</mo><msub><mi>ℚ</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>ℚ</mi> <mo>+</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M^{(-)}f:\mathbb{N} \to (\mathbb{Q}_+ \to \mathbb{Q}_+)</annotation></semantics></math> in the positive rational numbers, such that</p> <ul> <li> <p>for every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>D</mi> <mn>0</mn></msup><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D^{0}f)(x) = f(x)</annotation></semantics></math></p> </li> <li> <p>for every natural number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, for every positive rational number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math>, for every rational number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">h \in \mathbb{Q}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mo stretchy="false">|</mo><mi>h</mi><mo stretchy="false">|</mo><mo><</mo><msup><mi>M</mi> <mi>n</mi></msup><mi>f</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0 \lt | h | \lt M^{n}f(\epsilon)</annotation></semantics></math>, and for every rational number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{Q}</annotation></semantics></math>,</p> </li> </ul> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></munderover><mfrac><mrow><msup><mi>h</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>i</mi></msup><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mi>i</mi><mo>!</mo></mrow></mfrac><mo>|</mo></mrow><mo><</mo><mi>ϵ</mi><mo stretchy="false">|</mo><msup><mi>h</mi> <mi>n</mi></msup><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">\left|f(x + h) - \sum_{i=0}^n \frac{h^i (D^{i}f)(x)}{i!}\right| \lt \epsilon |h^n|</annotation></semantics></math></div> <p>Alternatively, we could use <a class="existingWikiWord" href="/nlab/show/infinitesimals">infinitesimals</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Q}[\epsilon]/(\epsilon^2 = 0)</annotation></semantics></math> be the dual rational numbers, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Q}[[\epsilon]]</annotation></semantics></math> be the commutative ring of <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> on the rational numbers. The former is equivalent to the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mo>×</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q} \times \mathbb{Q}</annotation></semantics></math> and the latter is equivalent to the <a class="existingWikiWord" href="/nlab/show/function+set">function set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℚ</mi> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{Q}^\mathbb{N}</annotation></semantics></math>.</p> <p>Both of these are commutative <a class="existingWikiWord" href="/nlab/show/rational+algebras">rational algebras</a> with ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h:\mathbb{Q} \to \mathbb{Q}[\epsilon]/(\epsilon^2 = 0)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><mi>ℚ</mi><mo>→</mo><mi>ℚ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k:\mathbb{Q} \to \mathbb{Q}[[\epsilon]]</annotation></semantics></math>.</p> <ul> <li>A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f:X \to \mathbb{Q}</annotation></semantics></math> is <strong>differentiable</strong> if it has a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">lift</mi> <mi>f</mi></msub><mo>:</mo><mi>ℚ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>→</mo><mi>ℚ</mi><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{lift}_f:\mathbb{Q}[\epsilon]/(\epsilon^2 = 0) \to \mathbb{Q}[\epsilon]/(\epsilon^2 = 0)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">lift</mi> <mi>f</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{lift}_f(h(i(a))) = h(i(a))</annotation></semantics></math> and a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f':X \to \mathbb{Q}</annotation></semantics></math> such that for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">a \in X</annotation></semantics></math>,</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">lift</mi> <mi>f</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><mo stretchy="false">(</mo><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\mathrm{lift}_f(h(i(a)) + \epsilon) = h(i(a)) + h(f'(a)) \epsilon</annotation></semantics></math></div> <ul> <li>A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">f:X \to \mathbb{Q}</annotation></semantics></math> is <strong>smooth</strong> if it has a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">lift</mi> <mi>f</mi></msub><mo>:</mo><mi>ℚ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>→</mo><mi>ℚ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ϵ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathrm{lift}_f:\mathbb{Q}[[\epsilon]] \to \mathbb{Q}[[\epsilon]]</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">lift</mi> <mi>f</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{lift}_f(h(i(a))) = h(i(a))</annotation></semantics></math> and a sequence of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mi>d</mi> <mo lspace="verythinmathspace" rspace="0em">−</mo></msup><mi>f</mi></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mo lspace="verythinmathspace" rspace="0em">−</mo></msup></mrow></mfrac><mo>:</mo><mi>ℕ</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times X \to \mathbb{Q}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mi>d</mi> <mn>0</mn></msup><mi>f</mi></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mn>0</mn></msup></mrow></mfrac><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\frac{d^0 f}{d x^0}\left(a\right) = a</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">a \in X</annotation></semantics></math>, such that for all natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">lift</mi> <mi>f</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mn>∞</mn></munderover><mfrac><mn>1</mn><mrow><mi>i</mi><mo>!</mo></mrow></mfrac><mi>h</mi><mrow><mo>(</mo><mfrac><mrow><msup><mi>d</mi> <mi>i</mi></msup><mi>f</mi></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mi>i</mi></msup></mrow></mfrac><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>)</mo></mrow><msup><mi>ϵ</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\mathrm{lift}_f(h(j(a)) + \epsilon) = \sum_{i = 0}^{\infty} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i</annotation></semantics></math></div></li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integer">integer</a></p> </li> <li> <p><strong>rational number</strong></p> <ul> <li> <p>a finite <a class="existingWikiWord" href="/nlab/show/field+extension">field extension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> is called a <em><a class="existingWikiWord" href="/nlab/show/number+field">number field</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dyadic+rational+number">dyadic rational number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/decimal+rational+number">decimal rational number</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Q%2FZ">Q/Z</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+number">algebraic number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gaussian+number">Gaussian number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/irrational+number">irrational number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quadratic+irrational+number">quadratic irrational number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+number">real number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/p-adic+number">p-adic number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+numbers+object">rational numbers object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+interval+coalgebra">rational interval coalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+root+theorem">rational root theorem</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Leo+Corry">Leo Corry</a>, <em>A Brief History of Numbers</em>, Oxford University Press (2015) [<a href="https://global.oup.com/academic/product/a-brief-history-of-numbers-9780198702597">ISBN:9780198702597</a>]</li> </ul> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Rational_number">Rational number</a></em></li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/univalent+foundations+of+mathematics">univalent foundations of mathematics</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> with the <a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a>):</p> <ul> <li id="UBP13"><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, §11.1 of: <em><a class="existingWikiWord" href="/nlab/show/HoTT+book">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em> (2013)</li> </ul> <p>and specifically in <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/UniMath+project">UniMath project</a>, <em><a href="https://unimath.github.io/agda-unimath/elementary-number-theory.rational-numbers.html">agda-unimath/elementary-number-theory.rational-numbers</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 5, 2023 at 02:34:23. 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