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.sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks" style="width:18em; text-align:center; font-size:95%;"><tbody><tr><th class="sidebar-title" style="font-size:125%; font-weight:bold;">Fit approximation</th></tr><tr><td class="sidebar-image skin-invert-image"><span typeof="mw:File"><a href="/wiki/File:BigOnotationfunctionapprox_popular.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/BigOnotationfunctionapprox_popular.svg/80px-BigOnotationfunctionapprox_popular.svg.png" decoding="async" width="80" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/BigOnotationfunctionapprox_popular.svg/120px-BigOnotationfunctionapprox_popular.svg.png 1.5x, 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<ul><li><a href="/wiki/Approximation" title="Approximation">Approximation</a></li> <li><a href="/wiki/Generalization_error" title="Generalization error">Generalization error</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor polynomial</a></li> <li><a href="/wiki/Scientific_modelling" title="Scientific modelling">Scientific modelling</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Order-of-approx" title="Template:Order-of-approx"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Order-of-approx" title="Template talk:Order-of-approx"><abbr title="Discuss this 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class="mbox-text-span">This article <b>may be <a href="/wiki/Wikipedia:Vagueness" title="Wikipedia:Vagueness">confusing or unclear</a> to readers</b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarify the article</a>. There might be a discussion about this on <a href="/wiki/Talk:Order_of_approximation" title="Talk:Order of approximation">the talk page</a>.</span> <span class="date-container"><i>(<span class="date">March 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In science, engineering, and other quantitative disciplines, <b>order of approximation</b> refers to formal or informal expressions for how accurate an approximation is. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Usage_in_science_and_engineering">Usage in science and engineering</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_of_approximation&action=edit&section=1" title="Edit section: Usage in science and engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In formal expressions, the <a href="/wiki/English_numerals#Ordinal_numbers" title="English numerals">ordinal number</a> used before the word <a href="/wiki/Order_(mathematics)#Analysis" title="Order (mathematics)">order</a> refers to the highest <a href="/wiki/Power_function" class="mw-redirect" title="Power function">power</a> in the <a href="/wiki/Series_expansion" title="Series expansion">series expansion</a> used in the <a href="/wiki/Approximation#Etymology_and_usage" title="Approximation">approximation</a>. The expressions: a <i><b>zeroth-order</b> approximation</i>, a <i><b>first-order</b> approximation</i>, a <i><b>second-order</b> approximation</i>, and so forth are used as <a href="/wiki/Fixed_phrase" class="mw-redirect" title="Fixed phrase">fixed phrases</a>. The expression a <i>zero-order approximation</i> is also common. <a href="/wiki/Cardinal_numeral" title="Cardinal numeral">Cardinal numerals</a> are occasionally used in expressions like an <i>order-zero approximation</i>, an <i>order-one approximation</i>, etc. </p><p>The omission of the word <i>order</i> leads to <a href="/wiki/Phrase" title="Phrase">phrases</a> that have less formal meaning. Phrases like <b>first approximation</b> or <b>to a first approximation</b> may refer to <i>a roughly approximate value of a quantity</i>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The phrase <b>to a zeroth approximation</b> indicates <i>a wild guess</i>.<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The expression <i>order of approximation</i> is sometimes informally used to mean the number of <a href="/wiki/Significant_figure" class="mw-redirect" title="Significant figure">significant figures</a>, in increasing order of accuracy, or to the <a href="/wiki/Order_of_magnitude" title="Order of magnitude">order of magnitude</a>. However, this may be confusing, as these formal expressions do not directly refer to the order of derivatives. </p><p>The choice of series expansion depends on the <a href="/wiki/Scientific_method" title="Scientific method">scientific method</a> used to investigate a <a href="/wiki/Phenomenon#Scientific" title="Phenomenon">phenomenon</a>. The expression <b>order of approximation</b> is expected to indicate progressively more refined approximations of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> in a specified <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a>. The choice of order of approximation depends on the <a href="/wiki/Research" title="Research">research purpose</a>. One may wish to simplify a known <a href="/wiki/Closed-form_expression#Analytic_expression" title="Closed-form expression">analytic expression</a> to devise a new application or, on the contrary, try to <a href="/wiki/Curve_fitting" title="Curve fitting">fit a curve to data points</a>. Higher order of approximation is not always more useful than the lower one. For example, if a quantity is constant within the whole interval, approximating it with a second-order <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> will not increase the accuracy. </p><p>In the case of a <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth function</a>, the <i>n</i>th-order approximation is a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> of <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> <i>n</i>, which is obtained by truncating the Taylor series to this degree. The formal usage of <i>order of approximation</i> corresponds to the omission of some terms of the <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> used in the <a href="/wiki/Series_expansion" title="Series expansion">expansion</a>. This affects <a href="/wiki/Accuracy_and_precision" title="Accuracy and precision">accuracy</a>. The error usually varies within the interval. Thus the terms (<i>zeroth</i>, <i>first</i>, <i>second,</i> etc.) used above meaning do not directly give information about <a href="/wiki/Percent_error" class="mw-redirect" title="Percent error">percent error</a> or <a href="/wiki/Significant_figures" title="Significant figures">significant figures</a>. For example, in the <a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor series</a> expansion of the <a href="/wiki/Exponential_function#Formal_definition" title="Exponential function">exponential function</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}=\underbrace {1} _{0^{\text{th}}}+\underbrace {x} _{1^{\text{st}}}+\underbrace {\frac {x^{2}}{2!}} _{2^{\text{nd}}}+\underbrace {\frac {x^{3}}{3!}} _{3^{\text{rd}}}+\underbrace {\frac {x^{4}}{4!}} _{4^{\text{th}}}+\ldots \;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mn>1</mn> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>th</mtext> </mrow> </msup> </mrow> </munder> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mi>x</mi> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>st</mtext> </mrow> </msup> </mrow> </munder> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>nd</mtext> </mrow> </msup> </mrow> </munder> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>rd</mtext> </mrow> </msup> </mrow> </munder> <mo>+</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow> <mn>4</mn> <mo>!</mo> </mrow> </mfrac> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>th</mtext> </mrow> </msup> </mrow> </munder> <mo>+</mo> <mo>…</mo> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}=\underbrace {1} _{0^{\text{th}}}+\underbrace {x} _{1^{\text{st}}}+\underbrace {\frac {x^{2}}{2!}} _{2^{\text{nd}}}+\underbrace {\frac {x^{3}}{3!}} _{3^{\text{rd}}}+\underbrace {\frac {x^{4}}{4!}} _{4^{\text{th}}}+\ldots \;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c96ff2a2e75692eab761a4445f3b8660edcb514e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:43.214ex; height:9.676ex;" alt="{\displaystyle e^{x}=\underbrace {1} _{0^{\text{th}}}+\underbrace {x} _{1^{\text{st}}}+\underbrace {\frac {x^{2}}{2!}} _{2^{\text{nd}}}+\underbrace {\frac {x^{3}}{3!}} _{3^{\text{rd}}}+\underbrace {\frac {x^{4}}{4!}} _{4^{\text{th}}}+\ldots \;,}"></span> the zeroth-order term is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a434da1f18b0ffcefba3174899069259da671c87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1;}"></span> the first-order term is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> second-order is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}/2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}/2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71ac48bc22f4b0ef0bc32c374ba23863af137797" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.356ex; height:3.176ex;" alt="{\displaystyle x^{2}/2,}"></span> and so forth. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|<1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|<1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e598c844b4d52aafd507beeab10b9b7e7d1119" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.531ex; height:2.843ex;" alt="{\displaystyle |x|<1,}"></span> each higher order term is smaller than the previous. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x|<<1,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><<</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|<<1,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dde616f14beb8852fddfee1151e18fbbad3a974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.726ex; height:2.843ex;" alt="{\displaystyle |x|<<1,\,}"></span> then the first order approximation, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{x}\approx 1+x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>≈</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{x}\approx 1+x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f49dca5972e8264f7a37e33f06b7d2ff1eae5fab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.334ex; height:2.676ex;" alt="{\displaystyle e^{x}\approx 1+x,}"></span> is often sufficient. But at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0954524da2f040331897141e1bfa00761a40126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x=1,}"></span> the first-order term, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> is not smaller than the zeroth-order term, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4e445819b13a052647aa3eb2be990b0a4b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 1.}"></span> And at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83025c6189f0cf8917e9d0193f34ca8a4533afa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.237ex; height:2.509ex;" alt="{\displaystyle x=2,}"></span> even the second-order term, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{3}/3!=4/3,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>!</mo> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}/3!=4/3,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eee84632a55c83c5f47752d4025e0a6d63110b1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.808ex; height:3.176ex;" alt="{\displaystyle 2^{3}/3!=4/3,\,}"></span> is greater than the zeroth-order term. </p> <div class="mw-heading mw-heading3"><h3 id="Zeroth-order">Zeroth-order</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_of_approximation&action=edit&section=2" title="Edit section: Zeroth-order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Zeroth-order approximation</i> is the term <a href="/wiki/Scientist" title="Scientist">scientists</a> use for a first rough answer. Many <a href="/wiki/Approximation#Science" title="Approximation">simplifying assumptions</a> are made, and when a number is needed, an order-of-magnitude answer (or zero <a href="/wiki/Significant_figure" class="mw-redirect" title="Significant figure">significant figures</a>) is often given. For example, "the town has <b>a few thousand</b> residents", when it has 3,914 people in actuality. This is also sometimes referred to as an <a href="/wiki/Order_of_magnitude" title="Order of magnitude">order-of-magnitude</a> approximation. The zero of "zeroth-order" represents the fact that even the only number given, "a few", is itself loosely defined. </p><p>A zeroth-order approximation of a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> (that is, <a href="/wiki/Mathematics" title="Mathematics">mathematically</a> determining a <a href="/wiki/Formula" title="Formula">formula</a> to fit multiple <a href="/wiki/Data_point" class="mw-redirect" title="Data point">data points</a>) will be <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a>, or a flat <a href="/wiki/Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">line</a> with no <a href="/wiki/Slope" title="Slope">slope</a>: a polynomial of degree 0. For example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=[0,1,2],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=[0,1,2],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772dac285229995430ec70eecdd3f4aa914ae601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.924ex; height:2.843ex;" alt="{\displaystyle x=[0,1,2],}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=[3,3,5],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=[3,3,5],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c07cfac788dfa94a5d50c8161eadf9826263bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.75ex; height:2.843ex;" alt="{\displaystyle y=[3,3,5],}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim f(x)=3.67}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∼</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3.67</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\sim f(x)=3.67}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc74fea0ac890ed8dc2fac0df43ff486202ac829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.904ex; height:2.843ex;" alt="{\displaystyle y\sim f(x)=3.67}"></span></dd></dl> <p>could be – if data point accuracy were reported – an approximate fit to the data, obtained by simply averaging the <i>x</i> values and the <i>y</i> values. However, data points represent <a href="/wiki/Unit_of_observation#Data_point" title="Unit of observation">results of measurements</a> and they do differ from <a href="/wiki/Point_(geometry)#Points_in_Euclidean_geometry" title="Point (geometry)">points in Euclidean geometry</a>. Thus quoting an average value containing three significant digits in the output with just one significant digit in the input data could be recognized as an example of <a href="/wiki/False_precision" title="False precision">false precision</a>. With the implied accuracy of the data points of ±0.5, the zeroth order approximation could at best yield the result for <i>y</i> of ~3.7 ± 2.0 in the interval of <i>x</i> from −0.5 to 2.5, considering the <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>. </p><p>If the data points are reported as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=[0.00,1.00,2.00],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0.00</mn> <mo>,</mo> <mn>1.00</mn> <mo>,</mo> <mn>2.00</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=[0.00,1.00,2.00],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32145730a0801b92b6f76eac2515890f7be6567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.839ex; height:2.843ex;" alt="{\displaystyle x=[0.00,1.00,2.00],}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=[3.00,3.00,5.00],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3.00</mn> <mo>,</mo> <mn>3.00</mn> <mo>,</mo> <mn>5.00</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=[3.00,3.00,5.00],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6029bd23e5b2bc702ab7cc77f00554fa2727346d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.665ex; height:2.843ex;" alt="{\displaystyle y=[3.00,3.00,5.00],}"></span></dd></dl> <p>the zeroth-order approximation results in </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim f(x)=3.67.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∼</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3.67.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\sim f(x)=3.67.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dc3ef38332dbff4fdbf81140c2534d23c095f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.551ex; height:2.843ex;" alt="{\displaystyle y\sim f(x)=3.67.}"></span></dd></dl> <p>The accuracy of the result justifies an attempt to derive a multiplicative function for that average, for example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim x+2.67.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∼</mo> <mi>x</mi> <mo>+</mo> <mn>2.67.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\sim x+2.67.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4662260e6422c41151bb0f265d08d1a74c785eda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.205ex; height:2.509ex;" alt="{\displaystyle y\sim x+2.67.}"></span></dd></dl> <p>One should be careful though, because the multiplicative function will be defined for the whole interval. If only three data points are available, one has no knowledge about the rest of the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a>, which may be a large part of it. This means that <i>y</i> could have another component which equals 0 at the ends and in the middle of the interval. A number of functions having this property are known, for example <i>y</i> = sin π<i>x</i>. <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> is useful and helps predict an <a href="/wiki/Closed-form_expression" title="Closed-form expression">analytic solution</a>, but the approximation alone does not provide conclusive evidence. </p> <div class="mw-heading mw-heading3"><h3 id="First-order">First-order</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_of_approximation&action=edit&section=3" title="Edit section: First-order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>First-order approximation</i> is the term scientists use for a slightly better answer.<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has <span class="nowrap"><span data-sort-value="7003400000000000000♠"></span>4<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>3</sup></span>, or <i>four thousand</i>, residents"). In the case of a first-order approximation, at least one number given is exact. In the zeroth-order example above, the quantity "a few" was given, but in the first-order example, the number "4" is given. </p><p>A first-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a linear approximation, straight line with a slope: a polynomial of degree 1. For example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=[0.00,1.00,2.00],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0.00</mn> <mo>,</mo> <mn>1.00</mn> <mo>,</mo> <mn>2.00</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=[0.00,1.00,2.00],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32145730a0801b92b6f76eac2515890f7be6567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.839ex; height:2.843ex;" alt="{\displaystyle x=[0.00,1.00,2.00],}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=[3.00,3.00,5.00],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3.00</mn> <mo>,</mo> <mn>3.00</mn> <mo>,</mo> <mn>5.00</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=[3.00,3.00,5.00],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6029bd23e5b2bc702ab7cc77f00554fa2727346d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.665ex; height:2.843ex;" alt="{\displaystyle y=[3.00,3.00,5.00],}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim f(x)=x+2.67}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∼</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>2.67</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\sim f(x)=x+2.67}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287d6f009a56580055a9380a3f00db774cd9ae84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.074ex; height:2.843ex;" alt="{\displaystyle y\sim f(x)=x+2.67}"></span></dd></dl> <p>is an approximate fit to the data. In this example there is a zeroth-order approximation that is the same as the first-order, but the method of getting there is different; i.e. a wild stab in the dark at a relationship happened to be as good as an "educated guess". </p> <div class="mw-heading mw-heading3"><h3 id="Second-order">Second-order</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_of_approximation&action=edit&section=4" title="Edit section: Second-order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Second-order approximation</i> is the term scientists use for a decent-quality answer. Few simplifying assumptions are made, and when a number is needed, an answer with two or more significant figures ("the town has <span class="nowrap"><span data-sort-value="7003390000000000000♠"></span>3.9<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>3</sup></span>, or <i>thirty-nine hundred</i>, residents") is generally given. As in the examples above, the term "2nd order" refers to the number of exact numerals given for the imprecise quantity. In this case, "3" and "9" are given as the two successive levels of precision, instead of simply the "4" from the first order, or "a few" from the zeroth order found in the examples above. </p><p>A second-order approximation of a function (that is, mathematically determining a formula to fit multiple data points) will be a <a href="/wiki/Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic polynomial</a>, geometrically, a <a href="/wiki/Parabola" title="Parabola">parabola</a>: a polynomial of degree 2. For example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=[0.00,1.00,2.00],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>0.00</mn> <mo>,</mo> <mn>1.00</mn> <mo>,</mo> <mn>2.00</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=[0.00,1.00,2.00],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32145730a0801b92b6f76eac2515890f7be6567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.839ex; height:2.843ex;" alt="{\displaystyle x=[0.00,1.00,2.00],}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=[3.00,3.00,5.00],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>3.00</mn> <mo>,</mo> <mn>3.00</mn> <mo>,</mo> <mn>5.00</mn> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=[3.00,3.00,5.00],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6029bd23e5b2bc702ab7cc77f00554fa2727346d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.665ex; height:2.843ex;" alt="{\displaystyle y=[3.00,3.00,5.00],}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\sim f(x)=x^{2}-x+3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∼</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\sim f(x)=x^{2}-x+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4cb4b2f9c073a051cd7c39f15c239a525374523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.327ex; height:3.176ex;" alt="{\displaystyle y\sim f(x)=x^{2}-x+3}"></span></dd></dl> <p>is an approximate fit to the data. In this case, with only three data points, a parabola is an exact fit based on the data provided. However, the data points for most of the interval are not available, which advises caution (see "zeroth order"). </p> <div class="mw-heading mw-heading3"><h3 id="Higher-order">Higher-order</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_of_approximation&action=edit&section=5" title="Edit section: Higher-order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While higher-order approximations exist and are crucial to a better understanding and description of reality, they are not typically referred to by number. </p><p>Continuing the above, a third-order approximation would be required to perfectly fit four data points, and so on. See <a href="/wiki/Polynomial_interpolation" title="Polynomial interpolation">polynomial interpolation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Colloquial_usage">Colloquial usage</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_of_approximation&action=edit&section=6" title="Edit section: Colloquial usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These terms are also used <a href="/wiki/Colloquialism" title="Colloquialism">colloquially</a> by scientists and engineers to describe phenomena that can be neglected as not significant (e.g. "Of course the rotation of the Earth affects our experiment, but it's such a high-order effect that we wouldn't be able to measure it." or "At these velocities, relativity is a fourth-order effect that we only worry about at the annual calibration.") In this usage, the ordinality of the approximation is not exact, but is used to emphasize its insignificance; the higher the number used, the less important the effect. The terminology, in this context, represents a high level of precision required to account for an effect which is inferred to be very small when compared to the overall subject matter. The higher the order, the more precision is required to measure the effect, and therefore the smallness of the effect in comparison to the overall measurement. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_of_approximation&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Linearization" title="Linearization">Linearization</a></li> <li><a href="/wiki/Perturbation_theory" title="Perturbation theory">Perturbation theory</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a></li> <li><a href="/wiki/Chapman%E2%80%93Enskog_theory#Mathematical_Formulation" title="Chapman–Enskog theory"> Chapman–Enskog method</a></li> <li><a href="/wiki/Big_O_notation" title="Big O notation">Big O notation</a></li> <li><a href="/wiki/Order_of_accuracy" title="Order of accuracy">Order of accuracy</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Order_of_approximation&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><i>first approximation</i> in Webster's Third New International Dictionary, Könemann, <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-8290-5292-8" title="Special:BookSources/3-8290-5292-8">3-8290-5292-8</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.webster-dictionary.org/definition/to%20a%20first%20approximation"><i>to a first approximation</i></a> in Online Dictionary and Translations Webster-dictionary.org.</span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.webster-dictionary.org/definition/to%20a%20zeroth%20approximation"><i>to a zeroth approximation</i></a> in Online Dictionary and Translations Webster-dictionary.org.</span> </li> </ol></div></div> <div class="navbox-styles"><link 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title="Tangent half-angle substitution">Tangent half-angle substitution</a></li></ul></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions in integration</a> <ul><li><a href="/wiki/Quadratic_integral" title="Quadratic integral">Quadratic integral</a></li></ul></li> <li><a href="/wiki/Trapezoidal_rule" title="Trapezoidal rule">Trapezoidal rule</a></li> <li>Volumes <ul><li><a href="/wiki/Disc_integration" title="Disc integration">Washer method</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Shell method</a></li></ul></li> <li><a href="/wiki/Integral_equation" title="Integral equation">Integral equation</a></li> <li><a href="/wiki/Integro-differential_equation" title="Integro-differential equation">Integro-differential equation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Derivatives <ul><li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li></ul></li> <li>Basic theorems <ul><li><a href="/wiki/Fundamental_Theorem_of_Line_Integrals" class="mw-redirect" title="Fundamental Theorem of Line Integrals">Line integrals</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Gauss'</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable calculus</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence theorem</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix and determinant</a></li> <li><a href="/wiki/Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li>Advanced topics <ul><li><a href="/wiki/Differential_form" title="Differential form">Differential forms</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Generalized_Stokes%27_theorem" class="mw-redirect" title="Generalized Stokes' theorem">Generalized Stokes' theorem</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sequences and series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetico-geometric_sequence" title="Arithmetico-geometric sequence">Arithmetico-geometric sequence</a></li> <li>Types of series <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier</a></li> <li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">Infinite</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a> <ul><li><a href="/wiki/Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul></li> <li>Tests of convergence <ul><li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel's</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet's</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Term</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Special functions<br />and numbers</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli numbers</a></li> <li><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e (mathematical constant)</a></li> <li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li> <li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li> <li><a href="/wiki/Stirling%27s_approximation" title="Stirling's approximation">Stirling's approximation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/History_of_calculus" title="History of calculus">History of calculus</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adequality" title="Adequality">Adequality</a></li> <li><a href="/wiki/Brook_Taylor" title="Brook Taylor">Brook Taylor</a></li> <li><a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a></li> <li><a href="/wiki/Generality_of_algebra" title="Generality of algebra">Generality of algebra</a></li> <li><a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a></li> <li><a href="/wiki/Infinitesimal" title="Infinitesimal">Infinitesimal</a></li> <li><a href="/wiki/Infinitesimal_calculus" class="mw-redirect" title="Infinitesimal calculus">Infinitesimal calculus</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a></li> <li><a href="/wiki/Fluxion" title="Fluxion">Fluxion</a></li> <li><a href="/wiki/Law_of_Continuity" class="mw-redirect" title="Law of Continuity">Law of Continuity</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</a></i></li> <li><i><a href="/wiki/The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Integrals32" scope="row" class="navbox-group" style="width:1%;text-align:left"><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Integrals</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_integrals_of_rational_functions" title="List of integrals of rational functions">rational functions</a></li> <li><a href="/wiki/List_of_integrals_of_irrational_functions" title="List of integrals of irrational functions">irrational functions</a></li> <li><a href="/wiki/List_of_integrals_of_exponential_functions" title="List of integrals of exponential functions">exponential functions</a></li> <li><a href="/wiki/List_of_integrals_of_logarithmic_functions" title="List of integrals of logarithmic functions">logarithmic functions</a></li> <li><a href="/wiki/List_of_integrals_of_hyperbolic_functions" title="List of integrals of hyperbolic functions">hyperbolic functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_hyperbolic_functions" title="List of integrals of inverse hyperbolic functions">inverse</a></li></ul></li> <li><a href="/wiki/List_of_integrals_of_trigonometric_functions" title="List of integrals of trigonometric functions">trigonometric functions</a> <ul><li><a href="/wiki/List_of_integrals_of_inverse_trigonometric_functions" title="List of integrals of inverse trigonometric functions">inverse</a></li> <li><a href="/wiki/Integral_of_the_secant_function" title="Integral of the secant function">Secant</a></li> <li><a href="/wiki/Integral_of_secant_cubed" title="Integral of secant cubed">Secant cubed</a></li></ul></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_limits" title="List of limits">List of limits</a></li> <li><a href="/wiki/Differentiation_rules" title="Differentiation rules">List of derivatives</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Complex calculus <ul><li><a href="/wiki/Contour_integral" class="mw-redirect" title="Contour integral">Contour integral</a></li></ul></li> <li>Differential geometry <ul><li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">of curves</a></li> <li><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">of surfaces</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li></ul></li> <li><a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a></li> <li><a href="/wiki/Gabriel%27s_horn" title="Gabriel's horn">Gabriel's horn</a></li> <li><a href="/wiki/Integration_Bee" title="Integration Bee">Integration Bee</a></li> <li><a href="/wiki/Proof_that_22/7_exceeds_%CF%80" title="Proof that 22/7 exceeds π">Proof that 22/7 exceeds π</a></li> <li><a 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