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Difference between revisions of "Irrational number"

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A number that is not a rational number (i.e. an integer or a fraction). Geometrically, an irrational number expresses the length of a segment that is incommensurate with the segment of unit length. Already the ancient mathematicians knew of the existence of incommensurate segments. They knew, e.g., that the diagonal and side of a square are incommensurate, which is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i0525601.png" /> being irrational.
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A number that is not a rational number (i.e. an integer or a fraction). Geometrically, an irrational number expresses the length of a segment that is incommensurate with the segment of unit length. Already the ancient mathematicians knew of the existence of incommensurate segments. They knew, e.g., that the diagonal and side of a square are incommensurate, which is equivalent to $\sqrt2$ being irrational.
  
Every real number can be written as an infinite decimal fraction, and the irrational numbers (and only they) can be written as non-periodic decimal fractions, e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i0525602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i0525603.png" />. Irrational numbers determine cuts (cf. [[Dedekind cut|Dedekind cut]]) in the set of rational numbers for which there is no largest number in the lower class and no smallest number in the upper class. The set of irrational numbers is everywhere dense on the real axis: Between any two numbers there is an irrational number. The set of irrational numbers is uncountable, is a set of the second category and has type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i0525604.png" /> (cf. [[Category of a set|Category of a set]]; [[Set of type F sigma(G delta)|Set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i0525605.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i0525606.png" />)]]).
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Every real number can be written as an infinite decimal fraction, and the irrational numbers (and only they) can be written as non-periodic decimal fractions, e.g. $\sqrt2=1.41\ldots$, $\pi=3.14\ldots$. Irrational numbers determine cuts (cf. [[Dedekind cut|Dedekind cut]]) in the set of rational numbers for which there is no largest number in the lower class and no smallest number in the upper class. The set of irrational numbers is everywhere dense on the real axis: Between any two numbers there is an irrational number. The set of irrational numbers is uncountable, is a set of the second category and has type $G_\delta$ (cf. [[Category of a set|Category of a set]]; [[Set of type F sigma(G delta)|Set of type $F_\sigma$ ($G_\delta$)]]).
  
Irrational algebraic numbers (in contrast to transcendental numbers) do not allow for approximation of arbitrary order by rational fractions. More precisely, for any irrational [[Algebraic number|algebraic number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i0525607.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i0525608.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i0525609.png" /> such that for any integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256012.png" />) one has
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Irrational algebraic numbers (in contrast to transcendental numbers) do not allow for approximation of arbitrary order by rational fractions. More precisely, for any irrational [[Algebraic number|algebraic number]] $\xi$ of degree $n$ there exists a $c>0$ such that for any integers $p$ and $q$ ($q>0$) one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256013.png" /></td> </tr></table>
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$$\left|\xi-\frac pq\right|>\frac{c}{q^n}.$$
  
 
Quadratic irrationalities, and only they, can be expressed by periodic continued fractions.
 
Quadratic irrationalities, and only they, can be expressed by periodic continued fractions.
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====Comments====
 
====Comments====
In fact, one can prove that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256014.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256016.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256017.png" /> is algebraic (Roth's theorem). The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256020.png" /> are known to be irrational (even transcendental, cf. [[Transcendental number|Transcendental number]]). However, it is not known whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052560/i05256022.png" /> are irrational or not.
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In fact, one can prove that for any $\epsilon>0$ there is a $c(\epsilon)>0$ such that $|\xi-p/q|>c(\epsilon)q^{-2-\epsilon}$ if $\xi\not\in\mathbf Q$ is algebraic (Roth's theorem). The numbers $e$, $\pi$, $e^\pi$ are known to be irrational (even transcendental, cf. [[Transcendental number|Transcendental number]]). However, it is not known whether $e+\pi$, $e\pi$ are irrational or not.

Latest revision as of 10:10, 13 April 2014

A number that is not a rational number (i.e. an integer or a fraction). Geometrically, an irrational number expresses the length of a segment that is incommensurate with the segment of unit length. Already the ancient mathematicians knew of the existence of incommensurate segments. They knew, e.g., that the diagonal and side of a square are incommensurate, which is equivalent to $\sqrt2$ being irrational.

Every real number can be written as an infinite decimal fraction, and the irrational numbers (and only they) can be written as non-periodic decimal fractions, e.g. $\sqrt2=1.41\ldots$, $\pi=3.14\ldots$. Irrational numbers determine cuts (cf. Dedekind cut) in the set of rational numbers for which there is no largest number in the lower class and no smallest number in the upper class. The set of irrational numbers is everywhere dense on the real axis: Between any two numbers there is an irrational number. The set of irrational numbers is uncountable, is a set of the second category and has type $G_\delta$ (cf. Category of a set; Set of type $F_\sigma$ ($G_\delta$)).

Irrational algebraic numbers (in contrast to transcendental numbers) do not allow for approximation of arbitrary order by rational fractions. More precisely, for any irrational algebraic number $\xi$ of degree $n$ there exists a $c>0$ such that for any integers $p$ and $q$ ($q>0$) one has

$$\left|\xi-\frac pq\right|>\frac{c}{q^n}.$$

Quadratic irrationalities, and only they, can be expressed by periodic continued fractions.


Comments

In fact, one can prove that for any $\epsilon>0$ there is a $c(\epsilon)>0$ such that $|\xi-p/q|>c(\epsilon)q^{-2-\epsilon}$ if $\xi\not\in\mathbf Q$ is algebraic (Roth's theorem). The numbers $e$, $\pi$, $e^\pi$ are known to be irrational (even transcendental, cf. Transcendental number). However, it is not known whether $e+\pi$, $e\pi$ are irrational or not.

How to Cite This Entry:
Irrational number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Irrational_number&oldid=31666
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article