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class="existingWikiWord" href="/nlab/show/number+theory">number theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>, <a class="existingWikiWord" 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> </div> </div> <h1 id="irrational_numbers">Irrational numbers</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#in_the_real_numbers'>In the real numbers</a></li> <li><a href='#as_nonrepeating_radix_expansions'>As non-repeating radix expansions</a></li> <li><a href='#in_archimedean_integral_domains'>In Archimedean integral domains</a></li> <li><a href='#in_integral_domains_with_a_padic_norm'>In integral domains with a p-adic norm</a></li> </ul> <li><a href='#history'>History</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#continued_fractions'>Continued fractions</a></li> <ul> <li><a href='#theorems'>Theorems</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>An irrational number is of course a <a class="existingWikiWord" href="/nlab/show/number">number</a> that is not <a class="existingWikiWord" href="/nlab/show/rational+number">rational</a>. As such, the concept is perhaps uninteresting. However, the term ‘irrational number’ is often used for an irrational <em><a class="existingWikiWord" href="/nlab/show/real+number">real</a></em> number; in this case, it is interesting to consider such numbers for two reasons:</p> <ul> <li>Historically, it was an important discovery that irrational real numbers exist.</li> <li>The <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> of all irrational real numbers is interesting in <a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a>, <a class="existingWikiWord" href="/nlab/show/computability">computability</a> theory, and <a class="existingWikiWord" href="/nlab/show/descriptive+set+theory">descriptive set theory</a>.</li> </ul> <p>Of course, there are also various theorems about general classes of numbers that distinguish rational from irrational numbers.</p> <h2 id="definition">Definition</h2> <h3 id="in_the_real_numbers">In the real numbers</h3> <p>An <a class="existingWikiWord" href="/nlab/show/affine+function">affine function</a> on the real numbers consists of a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and real number <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> <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> such that for all real numbers <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">f(x) = a x + b</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/constant+function">constant function</a> at zero <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>x</mi><mn>.0</mn></mrow><annotation encoding="application/x-tex">\lambda x.0</annotation></semantics></math> is an affine function where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a = 0</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><mn>0</mn></mrow><annotation encoding="application/x-tex">b = 0</annotation></semantics></math>. A real number is <strong>irrational</strong> if for all affine functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/integer">integer</a> coefficients, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x) = 0</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>λ</mi><mi>x</mi><mn>.0</mn></mrow><annotation encoding="application/x-tex">f = \lambda x.0</annotation></semantics></math>. This is equivalent to saying that a real number is irrational if for all 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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≠</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">x \neq a</annotation></semantics></math>.</p> <p>Alternatively, a <a class="existingWikiWord" href="/nlab/show/real+number">real number</a> <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 <strong>irrational</strong> if given any <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a> <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> (thought of as a real number), the <a class="existingWikiWord" href="/nlab/show/absolute+value">absolute value</a> <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>a</mi><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{|x - a|}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a>.</p> <p>These two definitions are equivalent in <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a>. However, these two definitions no longer coincide in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>; the former definition of irrational number is called <strong>weakly irrational</strong> while the latter definition is called <strong>strongly irrational</strong> or <strong>strictly irrational</strong>. Strongly irrational numbers are most commonly used in constructive mathematics, since it uses the apartness relation or strict order relation of the real numbers, which, unlike equality, is what is detected of the real numbers in constructive mathematics.</p> <p>The <a class="existingWikiWord" href="/nlab/show/set">set</a> of irrational real numbers (a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of the set of real numbers) is variously denoted <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">\mathbb{I}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math>, or <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">\mathbb{B}</annotation></semantics></math> (in various fonts). The <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">\mathbb{I}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math> stand for ‘irrational’, while the <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">\mathbb{B}</annotation></semantics></math> stands for ‘Baire’ (see the next paragraph). Here we will use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math>.</p> <p>We may give <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+structure">topology</a> as a <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of the <a class="existingWikiWord" href="/nlab/show/real+line">real line</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">\mathbb{R}</annotation></semantics></math>. With this topology, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math> is sometimes called <strong><a class="existingWikiWord" href="/nlab/show/Baire+space+of+irrational+numbers">Baire space</a></strong>; however, one uses a different <a class="existingWikiWord" href="/nlab/show/uniform+structure">uniform structure</a>. (This should be distinguished from the sense of <a class="existingWikiWord" href="/nlab/show/Baire+space">Baire space</a> as a space to which the <a class="existingWikiWord" href="/nlab/show/Baire+category+theorem">Baire category theorem</a> applies; however, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math> is an example of such a space.)</p> <h3 id="as_nonrepeating_radix_expansions">As non-repeating radix expansions</h3> <p>There is another definition of irrational number, common in the prealgebra and high school algebra literature, which directly defines the irrational numbers in terms of base 10 infinite radix expansions. (see <a class="existingWikiWord" href="/nlab/show/prealgebra+real+number">prealgebra real number</a>). This can be done in every base greater than 1:</p> <p>Let the natural number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">b \gt 1</annotation></semantics></math> denote the base of the radix expansion, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[0, b)</annotation></semantics></math> denote the half-open <a class="existingWikiWord" href="/nlab/show/interval">interval</a> in the <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> of all natural numbers less than <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>. Base <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> infinite radix expansions are elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{Z} \times [0, b)^\mathbb{N}</annotation></semantics></math>, with the idea that each pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i, d)</annotation></semantics></math> consists of an integer <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> and a sequence of digits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(n)</annotation></semantics></math> in the base <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> infinite radix expansion. The <a class="existingWikiWord" href="/nlab/show/series">series</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow> <mn>∞</mn></munderover><mi>i</mi><mo>+</mo><mfrac><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mi>b</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\sum_{n = 0}^\infty i + \frac{d(n)}{b^{n + 1}}</annotation></semantics></math></div> <p>can be shown to be a <a class="existingWikiWord" href="/nlab/show/Cauchy+sequence">Cauchy sequence</a>.</p> <p>The set of repeating base <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> infinite radix expansion is the subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>b</mi><msup><mo stretchy="false">)</mo> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{Z} \times [0, b)^\mathbb{N}</annotation></semantics></math> such that for pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i, d)</annotation></semantics></math> in the subset, there exist natural number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> and positive natural number <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> such that the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>m</mi><mo>.</mo><mi>d</mi><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda m.d(m + k)</annotation></semantics></math> factors through the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/n\mathbb{Z}</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mi>m</mi><mo>.</mo><mi>d</mi><mo stretchy="false">(</mo><mi>m</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi><mi>ℤ</mi><mo>→</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda m.d(m + k):\mathbb{N} \to \mathbb{Z}/n\mathbb{Z} \to [0, b)</annotation></semantics></math></div> <p>Two base <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> infinite radix expansions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i, d)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(j, e)</annotation></semantics></math> are said to be <a class="existingWikiWord" href="/nlab/show/apartness+relation">apart from</a> each other if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>≠</mo><mi>j</mi></mrow><annotation encoding="application/x-tex">i \neq j</annotation></semantics></math> or there exists a natural number <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≠</mo><mi>e</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(n) \neq e(n)</annotation></semantics></math></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>i</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>#</mo><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi>i</mi><mo>≠</mo><mi>j</mi><mo stretchy="false">)</mo><mo>∨</mo><mo>∃</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo>.</mo><mi>d</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≠</mo><mi>e</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i, d) \# (j, e) \coloneqq (i \neq j) \vee \exists n \in \mathbb{N}.d(n) \neq e(n)</annotation></semantics></math></div> <p>A base <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> infinite radix expansion <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 <em>strictly non-repeating</em> if it is apart from every repeating base <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> infinite radix expansion. An <strong>irrational number</strong> is a strictly non-repeating base <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> infinite radix expansion.</p> <p>In <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, not every strictly irrational number is the limit of the series</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow> <mn>∞</mn></munderover><mi>i</mi><mo>+</mo><mfrac><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mi>b</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\sum_{n = 0}^\infty i + \frac{d(n)}{b^{n + 1}}</annotation></semantics></math></div> <p>given by a non-repeating base <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> infinite radix expansion. However, it is still the case that strictly irrational numbers which are also <a class="existingWikiWord" href="/nlab/show/Cauchy+real+numbers">Cauchy real numbers</a> are interdefinable with strictly non-repeating base <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> infinite radix expansions.</p> <h3 id="in_archimedean_integral_domains">In Archimedean integral domains</h3> <p>Let <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> be an <a class="existingWikiWord" href="/nlab/show/Archimedean+integral+domain">Archimedean integral domain</a> with 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><mo>⊆</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \subseteq R</annotation></semantics></math> being a integral subdomain of <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>.</p> <p>An element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> is irrational if for all <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>a</mi><mo stretchy="false">|</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\vert a \vert \gt 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>a</mi><mo>⋅</mo><mi>r</mi><mo>−</mo><mi>b</mi><mo stretchy="false">|</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\vert a \cdot r - b \vert \gt 0</annotation></semantics></math>.</p> <p>The set of <strong>irrational numbers</strong> in <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 defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝕁</mi> <mi>R</mi></msub><mo>≔</mo><mo stretchy="false">{</mo><mi>r</mi><mo>∈</mo><mi>R</mi><mo stretchy="false">|</mo><mo>∀</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>ℤ</mi><mo>.</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>a</mi><mo stretchy="false">|</mo><mo>></mo><mn>0</mn><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>a</mi><mo>⋅</mo><mi>r</mi><mo>−</mo><mi>b</mi><mo stretchy="false">|</mo><mo>></mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{J}_R \coloneqq \{r \in R \vert \forall a, b \in \mathbb{Z}. (\vert a \vert \gt 0) \wedge (\vert a \cdot r - b \vert \gt 0) \}</annotation></semantics></math></div> <h3 id="in_integral_domains_with_a_padic_norm">In integral domains with a p-adic norm</h3> <p>Let <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> be an <a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a> with a <a class="existingWikiWord" href="/nlab/show/p-adic+norm">p-adic norm</a> <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><msub><mo stretchy="false">|</mo> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\vert(-)\vert_p</annotation></semantics></math> for 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>, with 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><mo>⊆</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \subseteq R</annotation></semantics></math> being a integral subdomain of <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>.</p> <p>An element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math> is irrational if for all <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">b \in \mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>a</mi><msub><mo stretchy="false">|</mo> <mi>p</mi></msub><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\vert a \vert_p \gt 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>a</mi><mo>⋅</mo><mi>r</mi><mo>−</mo><mi>b</mi><msub><mo stretchy="false">|</mo> <mi>p</mi></msub><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\vert a \cdot r - b \vert_p \gt 0</annotation></semantics></math>.</p> <p>The set of <strong>irrational numbers</strong> in <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 defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝕁</mi> <mi>R</mi></msub><mo>≔</mo><mo stretchy="false">{</mo><mi>r</mi><mo>∈</mo><mi>R</mi><mo stretchy="false">|</mo><mo>∀</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>ℤ</mi><mo>.</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>a</mi><msub><mo stretchy="false">|</mo> <mi>p</mi></msub><mo>></mo><mn>0</mn><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>a</mi><mo>⋅</mo><mi>r</mi><mo>−</mo><mi>b</mi><msub><mo stretchy="false">|</mo> <mi>p</mi></msub><mo>></mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{J}_R \coloneqq \{r \in R \vert \forall a, b \in \mathbb{Z}. (\vert a \vert_p \gt 0) \wedge (\vert a \cdot r - b \vert_p \gt 0) \}</annotation></semantics></math></div> <h2 id="history">History</h2> <p>The followers of <a class="existingWikiWord" href="/nlab/show/Pythagoras">Pythagoras</a> believed that ‘All is number’, meaning what we now call (positive) <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>. In <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>, this meant that any two lengths (or other geometric magnitudes) <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 <span class="newWikiWord">commensurable<a href="/nlab/new/commensurable+quantities">?</a></span> in the sense that there exists a <a class="existingWikiWord" href="/nlab/show/unit">unit</a> length <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>m</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">x = m u</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>n</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">y = n u</annotation></semantics></math> for some natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> and <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>. Identifying the ratios of geometric magnitudes with (positive) <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>, this becomes the claim that every real number is rational. The discovery that this is <em>false</em> is also attributed to the Pythagoreans (but the legends of punishment for this secret date from several hundred years later). Greek mathematicians developed further the theory of irrational numbers, up to the general theory of magnitudes (which we may now regard as a theory of <a class="existingWikiWord" href="/nlab/show/positive+real+numbers">positive real numbers</a>) attributed to <a class="existingWikiWord" href="/nlab/show/Eudoxus">Eudoxus</a> in Book X of <a class="existingWikiWord" href="/nlab/show/Euclid%27s+Elements">Euclid's Elements</a>.</p> <p>Mathematicians coming from the cultures of the Islamic Golden Age (particularly <span class="newWikiWord">Abu Kamil<a href="/nlab/new/Abu+Kamil">?</a></span>) were the first to treat irrational numbers algebraically as numbers (rather than geometrically as ratios of magnitudes); they applied the <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> of <span class="newWikiWord">Al-Khwarizmi<a href="/nlab/new/Al-Khwarizmi">?</a></span> to <a class="existingWikiWord" href="/nlab/show/square+roots">square roots</a>, cube <a class="existingWikiWord" href="/nlab/show/roots">roots</a>, etc. (Ultimately, <a class="existingWikiWord" href="/nlab/show/Omar+Khayyam">Omar Khayyam</a> developed a general method to find the real roots of any cubic <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a>.) However, they seem to have implicitly believed that all real numbers were expressible using such roots (<span class="newWikiWord">radical number<a href="/nlab/new/radical+number">?</a></span>s), which we now know is false even for some algebraic numbers, such as the real root of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mn>5</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^5 + 2 x + 1</annotation></semantics></math>. In any case, they only used such numbers.</p> <p>Later, European mathematicians of the early modern era (particularly <span class="newWikiWord">Cardano<a href="/nlab/new/Gerolamo+Cardano">?</a></span>, <span class="newWikiWord">Tartaglia<a href="/nlab/new/Tartaglia">?</a></span>, and <span class="newWikiWord">Ferrari<a href="/nlab/new/Lodovico+Ferrari">?</a></span>) had begun work with <a class="existingWikiWord" href="/nlab/show/imaginary+number">imaginary number</a>s, which are necessarily irrational. Following this, <span class="newWikiWord">Lambert<a href="/nlab/new/Johann+Lambert">?</a></span> and <a class="existingWikiWord" href="/nlab/show/Adrien-Marie+Legendre">Legendre</a> succeeded in proving the irrationality of <a class="existingWikiWord" href="/nlab/show/pi">pi</a>, <a class="existingWikiWord" href="/nlab/show/e">e</a>, and their powers, which ultimately led to the conjecture that they were transcendental (whereas radical numbers, and even the root of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mn>5</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^5 + 2x + 1</annotation></semantics></math>, are by definition <a class="existingWikiWord" href="/nlab/show/algebraic+number">algebraic</a>); this conjecture was later established by <a class="existingWikiWord" href="/nlab/show/Charles+Hermite">Hermite</a> and <span class="newWikiWord">Lindemann<a href="/nlab/new/Ferdinand+von+Lindemann">?</a></span>. Around this time, <a class="existingWikiWord" href="/nlab/show/Leonhard+Euler">Euler</a> and <a class="existingWikiWord" href="/nlab/show/Joseph-Louis+Lagrange">Lagrange</a> popularized <a class="existingWikiWord" href="/nlab/show/continued+fraction">continued fraction</a>s (see below) to study both rational and irrational numbers.</p> <p>During the <span class="newWikiWord">arithmetization of analysis<a href="/nlab/new/arithmetization+of+analysis">?</a></span> in the 19th century, people sometimes wrote of the problem of ‘defining irrational numbers’. The actual issue here was defining real numbers in general; one could define rational numbers algebraically, leaving only the irrational numbers as the problem. However, this may be a red herring; one could just as easily define <a class="existingWikiWord" href="/nlab/show/algebraic+number">algebraic number</a>s algebraically and say that the problem is defining transcendental numbers; indeed, it was only with the discovery that such numbers as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">e</mi></mrow><annotation encoding="application/x-tex">\mathrm{e}</annotation></semantics></math> are irrational that work on this problem came to life. On the other hand, it's not clear that anybody could completely work out the <a class="existingWikiWord" href="/nlab/show/order">order</a> properties of algebraic numbers without already coming upon <a class="existingWikiWord" href="/nlab/show/Richard+Dedekind">Dedekind</a>'s solution. In any case, specific irrational algebraic numbers such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mn>2</mn></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{2}</annotation></semantics></math> posed no difficulty to the <a class="existingWikiWord" href="/nlab/show/finitist+mathematics">finitist</a> methods used by such algebraists as <a class="existingWikiWord" href="/nlab/show/Leopold+Kronecker">Leopold Kronecker</a>.</p> <p>To this day, there are various specific real numbers (such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>+</mo><mi mathvariant="normal">e</mi></mrow><annotation encoding="application/x-tex">\pi + \mathrm{e}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Euler+number">Euler–Mascheroni constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math>, etc.) whose rationality or irrationality is unknown. In <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, this makes it unproved that these numbers are rational or irrational (although the <a class="existingWikiWord" href="/nlab/show/double+negation">double negation</a> of this statement can be proved for any real number). The question of whether <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><msqrt><mn>2</mn></msqrt> <msqrt><mn>2</mn></msqrt></msup></mrow><annotation encoding="application/x-tex">{\sqrt{2}}^{\sqrt{2}}</annotation></semantics></math> is rational or irrational is part of a famous illustration of the nature of constructive vs nonconstructive proof. (Namely, there is a cheap and easy nonconstructive proof that there exist irrational 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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mi>b</mi></msup></mrow><annotation encoding="application/x-tex">a^b</annotation></semantics></math> is rational: let <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> be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mn>2</mn></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{2}</annotation></semantics></math> and 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 either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mn>2</mn></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{2}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><msqrt><mn>2</mn></msqrt> <msqrt><mn>2</mn></msqrt></msup></mrow><annotation encoding="application/x-tex">{\sqrt{2}}^{\sqrt{2}}</annotation></semantics></math>, depending on whether the latter is rational or irrational. A constructive proof that decides which of these is the case is much harder: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><msqrt><mn>2</mn></msqrt> <msqrt><mn>2</mn></msqrt></msup></mrow><annotation encoding="application/x-tex">{\sqrt{2}}^{\sqrt{2}}</annotation></semantics></math> turns out to be irrational, by a constructive version of the <span class="newWikiWord">Gelfond–Schneider theorem<a href="/nlab/new/Gelfond-Schneider+theorem">?</a></span>.<sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup>)</p> <h2 id="properties">Properties</h2> <p>The Baire space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/product+space">product space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℕ</mi> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{N}^{\mathbb{N}}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℵ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\aleph_0</annotation></semantics></math> copies of the <a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>. The homeomorphism is given by <a class="existingWikiWord" href="/nlab/show/continued+fraction">continued fraction</a>s (see below).</p> <p>Every <a class="existingWikiWord" href="/nlab/show/inhabited+space">inhabited</a> <a class="existingWikiWord" href="/nlab/show/Polish+space">Polish space</a> is a <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math> is itself a Polish space.</p> <p>As a subset of the real line, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/full+set">full set</a> (meaning that its complement, the set of rational numbers, is <a class="existingWikiWord" href="/nlab/show/null+set">null</a>).</p> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a> may be identified with a <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math>, consisting of those irrational numbers whose continued fraction expansion consists only of <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math> (but this does not agree with the usual inclusions into <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>).</p> <p>The <a class="existingWikiWord" href="/nlab/show/fan+theorem">fan theorem</a> states precisely that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math> (when thought of as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>) is <a class="existingWikiWord" href="/nlab/show/sober+space">sober</a> or that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math> (when thought of as a <a class="existingWikiWord" href="/nlab/show/locale">locale</a>) is <a class="existingWikiWord" href="/nlab/show/topological+locale">topological/spatial/has enough points</a>. This is true in <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a> and in <a class="existingWikiWord" href="/nlab/show/intuitionistic+mathematics">intuitionistic mathematics</a> but fails in other forms of <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>.</p> <h2 id="continued_fractions">Continued fractions</h2> <p>(Main article: <a class="existingWikiWord" href="/nlab/show/continued+fraction">continued fraction</a>.)</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>;</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>3</mn></msub><mo>,</mo><mi>…</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a_0;a_1,a_2,a_3,\ldots]</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/infinite+sequence">infinite sequence</a> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, all positive except (possibly) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">a_0</annotation></semantics></math>. We interpret this as the number</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>0</mn></msub><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>a</mi> <mn>2</mn></msub><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>a</mi> <mn>3</mn></msub><mo>+</mo><mi>⋯</mi></mrow></mfrac></mrow></mfrac></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex"> a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \cdots}}} .</annotation></semantics></math></div> <p>By truncating this expression after <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math>, we produce a rational number; altogether, this is an infinite sequence of rational numbers.</p> <div class="un_theorem"> <h6 id="theorems">Theorems</h6> <p>This is a <a class="existingWikiWord" href="/nlab/show/Cauchy+sequence">Cauchy sequence</a> whose limit is irrational. Furthermore, every irrational number has a unique representation in this way. Yet more, the <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> thus shown between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕁</mi></mrow><annotation encoding="application/x-tex">\mathbb{J}</annotation></semantics></math> and the infinitary <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>×</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>×</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>×</mo><msup><mi>ℕ</mi> <mo>+</mo></msup><mo>×</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \times \mathbb{N}^+ \times \mathbb{N}^+ \times \mathbb{N}^+ \times \cdots</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> when the two sets are given their usual <a class="existingWikiWord" href="/nlab/show/topological+structure">topologies</a>.</p> </div> <p>The usual proofs of these theorems are entirely <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a>. Accordingly, in the <a class="existingWikiWord" href="/nlab/show/foundations+of+mathematics">foundations of mathematics</a>, one may define Baire space either as the space of irrational numbers or as the infinite product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℕ</mi> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{N}^{\mathbb{N}}</annotation></semantics></math>. However, to treat Baire space as a <a class="existingWikiWord" href="/nlab/show/uniform+space">uniform space</a> or as a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, one uses the structure from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℕ</mi> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{N}^{\mathbb{N}}</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pi">pi</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/e">e</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/golden+ratio">golden ratio</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li>Wikipedia (English): <ul> <li><a href="http://en.wikipedia.org/wiki/Baire_space_%28set_theory%29">Baire space (set theory)</a></li> <li><a href="http://en.wikipedia.org/wiki/Irrational_number#History">Irrational numbers → History</a></li> </ul> </li> </ul> <p>For the definition of the irrational numbers in terms of non-repeating base 10 infinite radix expansions</p> <ul> <li> <p>Nichols, Eugene D, et al. Holt Algebra with Trigonometry. Holt, Rinehart and Winston : Harcourt Brace Jovanovich, 1992.</p> </li> <li> <p>Marecek, Lynn, et al. Prealgebra 2e. OpenStax, Rice University, 2020.</p> </li> </ul> <p>That Cauchy irrational numbers have radix expansions in constructive mathematics</p> <ul> <li id="Swan24">Andrew Swan (2024) on Category Theory Zulip, <a href="https://categorytheory.zulipchat.com/#narrow/stream/229199-learning.3A-questions/topic/Radix.20expansions.20in.20constructive.20mathematics/near/456366041">Radix expansions in constructive mathematics</a></li> </ul> <div class="footnotes"><hr /><ol><li id="fn:1"> <p>Of course, one could also take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><msqrt><mn>2</mn></msqrt></mrow><annotation encoding="application/x-tex">a = \sqrt{2}</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><mn>2</mn><mfrac><mrow><mi>log</mi><mn>3</mn></mrow><mrow><mi>log</mi><mn>2</mn></mrow></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b = 2\frac{\log 3}{\log 2})</annotation></semantics></math>, which are both irrational by easy constructive proofs, if one is after definite irrational 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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mi>b</mi></msup></mrow><annotation encoding="application/x-tex">a^b</annotation></semantics></math> is rational (here the result is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mi>b</mi></msup><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">a^b = 3</annotation></semantics></math>). The irrationality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>log</mi><mn>3</mn></mrow><mrow><mi>log</mi><mn>2</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\log 3}{\log 2}</annotation></semantics></math> follows, as easily as that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mn>2</mn></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{2}</annotation></semantics></math>, from the <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+arithmetic">fundamental theorem of arithmetic</a>: there do not exist nonzero integers <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mi>p</mi></msup><mo>=</mo><msup><mn>3</mn> <mi>q</mi></msup></mrow><annotation encoding="application/x-tex">2^p = 3^q</annotation></semantics></math>. <a href="#fnref:1" rev="footnote">↩</a></p> </li></ol></div></body></html> </div> <div class="revisedby"> <p> Last revised on August 4, 2024 at 20:19:33. 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