View source for The square root of 2 is irrational

'''The square root of 2 is irrational''' is the claim in [[number theory]] that there is no rational number that when multiplied by itself equals the number 2.

One proof by [[reductio ad absurdum]] consists of first assuming that the square root of 2 can be written as a rational number. Thus, there is a pair of coprime integers <math>p</math> and <math>q</math> such that their ratio <math>\frac{p}q = \sqrt 2</math> ([[Every rational number has an irreducible representation]]). And so, <math>\frac{p^2}{q^2} = 2</math>, and <math>p^2=2q^2</math>. As <math>p^2</math> has a factor of <math>2</math>, it is even, and thus <math>p</math> is also even. Thus <math>p</math> can be written as <math>2k</math>, where <math>k</math> is an integer. It then follows that <math>(2k)^2=2q^2</math>, <math>4k^2=2q^2</math>, <math>2k^2=q^2</math>, and thus <math>q^2</math> also has a factor of <math>2</math>, and thus <math>q</math> is also even. If <math>p</math> and <math>q</math> are both even, then they share a common factor of <math>2</math>, and thus are not coprime, leading to a contradiction as we have already established that <math>p</math> and <math>q</math> are coprime. And so, no such <math>p,q</math> can exist, and the square root of 2 is irrational.

<div class="s-prose js-post-body" itemprop="text">
<p>It's not part of the definition, but the proof is easy.</p>
<p>Suppose one has a rational number <span class="math-container"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mfrac><mi>m</mi><mi>n</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-11" style="width: 1.072em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.009em; height: 0px; font-size: 106%;"><span style="position: absolute; clip: rect(1.198em, 1001.01em, 2.708em, -999.997em); top: -2.135em; left: 0em;"><span class="mrow" id="MathJax-Span-12"><span class="mfrac" id="MathJax-Span-13"><span style="display: inline-block; position: relative; width: 0.758em; height: 0px; margin-right: 0.129em; margin-left: 0.129em;"><span style="position: absolute; clip: rect(3.525em, 1000.63em, 4.217em, -999.997em); top: -4.462em; left: 50%; margin-left: -0.311em;"><span class="mi" id="MathJax-Span-14" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">m</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span><span style="position: absolute; clip: rect(3.525em, 1000.44em, 4.217em, -999.997em); top: -3.708em; left: 50%; margin-left: -0.186em;"><span class="mi" id="MathJax-Span-15" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span><span style="position: absolute; clip: rect(0.821em, 1000.76em, 1.261em, -999.997em); top: -1.255em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.758em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.072em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.142em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.463em; border-left: 0px solid; width: 0px; height: 1.337em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>m</mi><mi>n</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-6">\frac{m}{n}</script></span> that is reducible. We can assume <span class="math-container"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>n</mi><mo>&amp;#x2265;</mo><mn>1</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-16" style="width: 2.582em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.393em; height: 0px; font-size: 106%;"><span style="position: absolute; clip: rect(1.261em, 1002.33em, 2.456em, -999.997em); top: -2.135em; left: 0em;"><span class="mrow" id="MathJax-Span-17"><span class="mi" id="MathJax-Span-18" style="font-family: MathJax_Math; font-style: italic;">n</span><span class="mo" id="MathJax-Span-19" style="font-family: MathJax_Main; padding-left: 0.255em;">≥</span><span class="mn" id="MathJax-Span-20" style="font-family: MathJax_Main; padding-left: 0.255em;">1</span></span><span style="display: inline-block; width: 0px; height: 2.142em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.197em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-7">n \ge 1</script></span> (divide the numerator and denominator by <span class="math-container"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mo>&amp;#x2212;</mo><mn>1</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-21" style="width: 1.324em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.261em; height: 0px; font-size: 106%;"><span style="position: absolute; clip: rect(1.261em, 1001.2em, 2.393em, -999.997em); top: -2.135em; left: 0em;"><span class="mrow" id="MathJax-Span-22"><span class="mo" id="MathJax-Span-23" style="font-family: MathJax_Main;">−</span><span class="mn" id="MathJax-Span-24" style="font-family: MathJax_Main;">1</span></span><span style="display: inline-block; width: 0px; height: 2.142em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.13em; border-left: 0px solid; width: 0px; height: 0.937em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-8">-1</script></span> otherwise).</p>
<p>Take an integer <span class="math-container"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>k</mi><mo>&amp;#x2265;</mo><mn>2</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-25" style="width: 2.456em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.33em; height: 0px; font-size: 106%;"><span style="position: absolute; clip: rect(1.261em, 1002.27em, 2.456em, -999.997em); top: -2.135em; left: 0em;"><span class="mrow" id="MathJax-Span-26"><span class="mi" id="MathJax-Span-27" style="font-family: MathJax_Math; font-style: italic;">k</span><span class="mo" id="MathJax-Span-28" style="font-family: MathJax_Main; padding-left: 0.255em;">≥</span><span class="mn" id="MathJax-Span-29" style="font-family: MathJax_Main; padding-left: 0.255em;">2</span></span><span style="display: inline-block; width: 0px; height: 2.142em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.197em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>≥</mo><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-9">k \ge 2</script></span> which divides both of <span class="math-container"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-10-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>m</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-30" style="width: 0.947em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.884em; height: 0px; font-size: 106%;"><span style="position: absolute; clip: rect(1.513em, 1000.88em, 2.33em, -999.997em); top: -2.135em; left: 0em;"><span class="mrow" id="MathJax-Span-31"><span class="mi" id="MathJax-Span-32" style="font-family: MathJax_Math; font-style: italic;">m</span></span><span style="display: inline-block; width: 0px; height: 2.142em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.603em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script type="math/tex" id="MathJax-Element-10">m</script></span> and <span class="math-container"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mi>n</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-33" style="width: 0.695em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.632em; height: 0px; font-size: 106%;"><span style="position: absolute; clip: rect(1.513em, 1000.63em, 2.33em, -999.997em); top: -2.135em; left: 0em;"><span class="mrow" id="MathJax-Span-34"><span class="mi" id="MathJax-Span-35" style="font-family: MathJax_Math; font-style: italic;">n</span></span><span style="display: inline-block; width: 0px; height: 2.142em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.063em; border-left: 0px solid; width: 0px; height: 0.603em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-11">n</script></span>, and so <span class="math-container"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>m</mi><mo>&amp;#x2032;</mo></msup><mo>=</mo><mi>m</mi><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>/</mo></mrow><mi>k</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-36" style="width: 4.594em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.343em; height: 0px; font-size: 106%;"><span style="position: absolute; clip: rect(1.198em, 1004.34em, 2.582em, -999.997em); top: -2.135em; left: 0em;"><span class="mrow" id="MathJax-Span-37"><span class="msup" id="MathJax-Span-38"><span style="display: inline-block; position: relative; width: 1.135em; height: 0px;"><span style="position: absolute; clip: rect(3.399em, 1000.88em, 4.217em, -999.997em); top: -4.022em; left: 0em;"><span class="mi" id="MathJax-Span-39" style="font-family: MathJax_Math; font-style: italic;">m</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span><span style="position: absolute; top: -4.399em; left: 0.884em;"><span class="mo" id="MathJax-Span-40" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span></span></span><span class="mo" id="MathJax-Span-41" style="font-family: MathJax_Main; padding-left: 0.255em;">=</span><span class="mi" id="MathJax-Span-42" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.255em;">m</span><span class="texatom" id="MathJax-Span-43"><span class="mrow" id="MathJax-Span-44"><span class="mo" id="MathJax-Span-45" style="font-family: MathJax_Main;">/</span></span></span><span class="mi" id="MathJax-Span-46" style="font-family: MathJax_Math; font-style: italic;">k</span></span><span style="display: inline-block; width: 0px; height: 2.142em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 1.203em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>m</mi><mo>′</mo></msup><mo>=</mo><mi>m</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-12">m' = m/k</script></span> and <span class="math-container"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><msup><mi>n</mi><mo>&amp;#x2032;</mo></msup><mo>=</mo><mi>n</mi><mrow class=&quot;MJX-TeXAtom-ORD&quot;><mo>/</mo></mrow><mi>k</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-47" style="width: 4.028em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.777em; height: 0px; font-size: 106%;"><span style="position: absolute; clip: rect(1.198em, 1003.78em, 2.582em, -999.997em); top: -2.135em; left: 0em;"><span class="mrow" id="MathJax-Span-48"><span class="msup" id="MathJax-Span-49"><span style="display: inline-block; position: relative; width: 0.884em; height: 0px;"><span style="position: absolute; clip: rect(3.399em, 1000.57em, 4.217em, -999.997em); top: -4.022em; left: 0em;"><span class="mi" id="MathJax-Span-50" style="font-family: MathJax_Math; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span><span style="position: absolute; top: -4.399em; left: 0.632em;"><span class="mo" id="MathJax-Span-51" style="font-size: 70.7%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span></span></span><span class="mo" id="MathJax-Span-52" style="font-family: MathJax_Main; padding-left: 0.255em;">=</span><span class="mi" id="MathJax-Span-53" style="font-family: MathJax_Math; font-style: italic; padding-left: 0.255em;">n</span><span class="texatom" id="MathJax-Span-54"><span class="mrow" id="MathJax-Span-55"><span class="mo" id="MathJax-Span-56" style="font-family: MathJax_Main;">/</span></span></span><span class="mi" id="MathJax-Span-57" style="font-family: MathJax_Math; font-style: italic;">k</span></span><span style="display: inline-block; width: 0px; height: 2.142em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.33em; border-left: 0px solid; width: 0px; height: 1.203em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mo>′</mo></msup><mo>=</mo><mi>n</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-13">n'=n/k</script></span> are smaller integers satisfying <span class="math-container"><span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" style="position: relative;" data-mathml="<math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;><mfrac><msup><mi>m</mi><mo>&amp;#x2032;</mo></msup><msup><mi>n</mi><mo>&amp;#x2032;</mo></msup></mfrac><mo>=</mo><mfrac><mi>m</mi><mi>n</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-58" style="width: 3.777em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.525em; height: 0px; font-size: 106%;"><span style="position: absolute; clip: rect(1.009em, 1003.52em, 2.77em, -999.997em); top: -2.135em; left: 0em;"><span class="mrow" id="MathJax-Span-59"><span class="mfrac" id="MathJax-Span-60"><span style="display: inline-block; position: relative; width: 0.947em; height: 0px; margin-right: 0.129em; margin-left: 0.129em;"><span style="position: absolute; clip: rect(3.274em, 1000.82em, 4.217em, -999.997em); top: -4.462em; left: 50%; margin-left: -0.374em;"><span class="msup" id="MathJax-Span-61"><span style="display: inline-block; position: relative; width: 0.821em; height: 0px;"><span style="position: absolute; clip: rect(3.525em, 1000.63em, 4.217em, -999.997em); top: -4.022em; left: 0em;"><span class="mi" id="MathJax-Span-62" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">m</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span><span style="position: absolute; top: -4.274em; left: 0.632em;"><span class="mo" id="MathJax-Span-63" style="font-size: 50.3%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span><span style="position: absolute; clip: rect(3.336em, 1000.63em, 4.217em, -999.997em); top: -3.582em; left: 50%; margin-left: -0.311em;"><span class="msup" id="MathJax-Span-64"><span style="display: inline-block; position: relative; width: 0.632em; height: 0px;"><span style="position: absolute; clip: rect(3.525em, 1000.44em, 4.217em, -999.997em); top: -4.022em; left: 0em;"><span class="mi" id="MathJax-Span-65" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span><span style="position: absolute; top: -4.211em; left: 0.443em;"><span class="mo" id="MathJax-Span-66" style="font-size: 50.3%; font-family: MathJax_Main;">′</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span><span style="position: absolute; clip: rect(0.821em, 1000.95em, 1.261em, -999.997em); top: -1.255em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.947em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.072em;"></span></span></span></span><span class="mo" id="MathJax-Span-67" style="font-family: MathJax_Main; padding-left: 0.255em;">=</span><span class="mfrac" id="MathJax-Span-68" style="padding-left: 0.255em;"><span style="display: inline-block; position: relative; width: 0.758em; height: 0px; margin-right: 0.129em; margin-left: 0.129em;"><span style="position: absolute; clip: rect(3.525em, 1000.63em, 4.217em, -999.997em); top: -4.462em; left: 50%; margin-left: -0.311em;"><span class="mi" id="MathJax-Span-69" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">m</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span><span style="position: absolute; clip: rect(3.525em, 1000.44em, 4.217em, -999.997em); top: -3.708em; left: 50%; margin-left: -0.186em;"><span class="mi" id="MathJax-Span-70" style="font-size: 70.7%; font-family: MathJax_Math; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 4.028em;"></span></span><span style="position: absolute; clip: rect(0.821em, 1000.76em, 1.261em, -999.997em); top: -1.255em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.758em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.072em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.142em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.53em; border-left: 0px solid; width: 0px; height: 1.603em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mo>′</mo></msup><msup><mi>n</mi><mo>′</mo></msup></mfrac><mo>=</mo><mfrac><mi>m</mi><mi>n</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-14">\frac{m'}{n'} = \frac{m}{n}</script></span>.</p>
<p>Repeat this process as long as the fraction remains reducible.</p>
<p>One cannot repeat forever, because the sequence of denominators is a sequence of positive integers that gets smaller and smaller, and no such sequence can continue infinitely (essentially one is using mathematical induction in this step).</p>
<p>When the process stops, one has an irreducible fraction representing the same rational number</p>
    </div>

{{Claim
|Claim=The square root of 2 is irrational
|Level=Proven
|Nature=Theoretical
|Counterclaim=
|DependentOn1=Every rational number has an irreducible representation
|DependencyOf1=
}}
[[Category:Mathematics]]