Real Numbers:Irrational - Revision history

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Created page with "right|thumb|240px|The number [[Square root of 2|{{radic|2}} is irrational.]] In mathematics, the '''irrational numbers''' (from in-..."

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[[File:Square root of 2 triangle.svg|right|thumb|240px|The number [[Square root of 2|{{radic|2}}]] is irrational.]]

In [[mathematics]], the '''irrational numbers''' (from in- [[prefix]] assimilated to ir- (negative prefix, [[privative]]) + rational) are all the [[real number]]s which are not [[rational number]]s. That is, irrational numbers cannot be expressed as the ratio of two [[integer]]s. When the [[ratio]] of lengths of two line segments is an irrational number, the line segments are also described as being ''[[commensurability (mathematics)|incommensurable]]'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

Among irrational numbers are the ratio [[Pi|{{pi}}]] of a circle's circumference to its diameter, Euler's number [[E (mathematical constant)|''e'']], the golden ratio [[Golden ratio|''φ'']], and the [[square root of two]].<ref>[http://sprott.physics.wisc.edu/Pickover/trans.html The 15 Most Famous Transcendental Numbers]. by [[Clifford A. Pickover]]. URL retrieved 24 October 2007.</ref><ref>http://www.mathsisfun.com/irrational-numbers.html; URL retrieved 24 October 2007.</ref><ref>{{MathWorld|title=Irrational Number|urlname=IrrationalNumber}} URL retrieved 26 October 2007.</ref> In fact, all square roots of [[natural number]]s, other than of [[square number|perfect square]]s, are irrational.

Like all real numbers, irrational numbers can be expressed in [[positional notation]], notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor [[repeating decimal|end with a repeating sequence]]. For example, the decimal representation of {{pi}} starts with 3.14159, but no finite number of digits can represent {{pi}} exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics.

Irrational numbers can also be expressed as [[Continued fraction#Infinite continued fractions and convergents|non-terminating continued fractions]] and many other ways.

As a consequence of [[Cantor's diagonal argument|Cantor's proof]] that the real numbers are [[uncountable]] and the [[rational number|rationals]] countable, it follows that [[almost all]] real numbers are irrational.<ref>{{Cite book|last=Cantor|first=Georg|year=1955|orig-year=1915|title=Contributions to the Founding of the Theory of Transfinite Numbers|url=https://archive.org/details/contributionstot003626mbp|editor=Philip Jourdain|editor-link=Philip Jourdain|place=New York|publisher=Dover|isbn= 978-0-486-60045-1 }}</ref>

== Examples ==
{{unreferenced section|date=June 2013}}

=== Square roots ===
The [[square root of 2]] was the first number proved irrational, and that article contains a number of proofs. The [[golden ratio]] is another famous quadratic irrational number. The square roots of all natural numbers which are not [[perfect squares]] are irrational and a proof may be found in [[quadratic irrational]]s.

===General roots===
The proof above for the square root of two can be generalized using the [[fundamental theorem of arithmetic]]. This asserts that every integer has a [[unique factorization]] into primes. Using it we can show that if a rational number is not an integer then no integral power of it can be an integer, as in [[lowest terms]] there must be a [[Prime number|prime]] in the denominator that does not divide into the numerator whatever power each is raised to. Therefore, if an integer is not an exact {{mvar|k}}th power of another integer, then that first integer's [[nth root|{{mvar|k}}th root]] is irrational.

=== Logarithms ===
Perhaps the numbers most easy to prove irrational are certain [[logarithm]]s. Here is a [[proof by contradiction]] that log2 3 is irrational (log2 3 ≈ 1.58 > 0).

Assume log2 3 is rational. For some positive integers ''m'' and ''n'', we have

: <math>\log_2 3 = \frac{m}{n}.</math>

It follows that

: <math>2^{m/n}=3</math>

: <math>(2^{m/n})^n = 3^n</math>

: <math>2^m=3^n.</math>

However, the number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its [[prime factor]]s will be 2). Clearly, an integer cannot be both odd and even at the same time: we have a contradiction. The only assumption we made was that log2 3 is rational (and so expressible as a quotient of integers ''m''/''n'' with ''n'' ≠ 0). The contradiction means that this assumption must be false, i.e. log2 3 is irrational, and can never be expressed as a quotient of integers ''m''/''n'' with ''n'' ≠ 0.

Cases such as log10 2 can be treated similarly.

==Types==
=== Transcendental/algebraic ===
[[Almost all]] irrational numbers are [[Transcendental number|transcendental]] and all real transcendental numbers are irrational (there are also complex transcendental numbers): the article on transcendental numbers lists several examples. So ''e'' ''r'' and π ''r'' are irrational for all nonzero rational ''r'', and, e.g., ''e''π is irrational, too.

Irrational numbers can also be found within the [[countable set|countable]] set of real [[algebraic number]]s (essentially defined as the real [[zero of a function|roots]] of [[polynomial]]s with integer coefficients), i.e., as real solutions of polynomial equations

:<math>p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 = 0\;, </math>

where the coefficients <math>a_i</math> are integers and <math>a_n \ne 0</math>. [[rational root theorem|Any rational root]] of this polynomial equation must be of the form ''r'' /''s'', where ''r'' is a [[divisor]] of ''a''0 and ''s'' is a divisor of ''a''''n''. If a real root <math>x_0</math> of a polynomial <math>p</math> is not among these finitely many possibilities, it must be an irrational algebraic number. An exemplary proof for the existence of such algebraic irrationals is by showing that ''x''0  = (21/2 + 1)1/3 is an irrational root of a polynomial with integer coefficients: it satisfies (''x''3 − 1)2 = 2 and hence ''x''6 − 2''x''3 − 1 = 0, and this latter polynomial has no rational roots (the only candidates to check are ±1, and ''x''0, being greater than 1, is neither of these), so ''x''0 is an irrational algebraic number.

Because the algebraic numbers form a [[field (mathematics)|subfield]] of the real numbers, many irrational real numbers can be constructed by combining transcendental and algebraic numbers. For example, 3{{pi}} + 2, {{pi}} + {{radic|2}} and ''e''{{radic|3}} are irrational (and even transcendental).

==Decimal expansions==
The decimal expansion of an irrational number never repeats or terminates (the latter being equivalent to repeating zeroes), unlike any rational number. The same is true for [[Binary numeral system|binary]], [[octal]] or [[hexadecimal]] expansions, and in general for expansions in every [[Positional notation|positional]] [[numeral system|notation]] with [[natural number|natural]] bases.

To show this, suppose we divide integers ''n'' by ''m'' (where ''m'' is nonzero). When [[long division]] is applied to the division of ''n'' by ''m'', only ''m'' remainders are possible. If 0 appears as a remainder, the decimal expansion terminates. If 0 never occurs, then the algorithm can run at most ''m'' − 1 steps without using any remainder more than once. After that, a remainder must recur, and then the decimal expansion repeats.

Conversely, suppose we are faced with a [[repeating decimal]], we can prove that it is a fraction of two integers. For example, consider:

:<math>A=0.7\,162\,162\,162\,\ldots</math>

Here the repetend is 162 and the length of the repetend is 3. First, we multiply by an appropriate power of 10 to move the decimal point to the right so that it is just in front of a repetend. In this example we would multiply by 10 to obtain:

:<math>10A = 7.162\,162\,162\,\ldots</math>

Now we multiply this equation by 10''r'' where ''r'' is the length of the repetend. This has the effect of moving the decimal point to be in front of the "next" repetend. In our example, multiply by 103:

:<math>10,000A=7\,162.162\,162\,\ldots</math>

The result of the two multiplications gives two different expressions with exactly the same "decimal portion", that is, the tail end of 10,000''A'' matches the tail end of 10''A'' exactly. Here, both 10,000''A'' and 10''A'' have {{gaps|.162|162|162|...}} after the decimal point.

Therefore, when we subtract the 10''A'' equation from the 10,000''A'' equation, the tail end of 10''A'' cancels out the tail end of 10,000''A'' leaving us with:

:<math>9990A=7155.</math>

Then

:<math>A= \frac{7155}{9990} = \frac{53}{74}</math>

is a ratio of integers and therefore a rational number.

== Irrational powers ==
Dov Jarden gave a simple non-[[constructive proof]] that there exist two irrational numbers ''a'' and ''b'', such that ''a''''b'' is rational:<ref>{{cite book |url=http://condor.depaul.edu/mash/atotheamg.pdf|title=Philosophies of mathematics| first1=Alexander |last1=George |first2=Daniel J. |last2=Velleman |isbn=0-631-19544-0 |publisher=Blackwell |year=2002 |pages=3–4}}</ref>

Consider {{radic|2}}{{radic|2}}; if this is rational, then take ''a'' = ''b'' = {{radic|2}}. Otherwise, take ''a'' to be the irrational number {{radic|2}}{{radic|2}} and ''b'' = {{radic|2}}. Then ''a''''b'' = ({{radic|2}}{{radic|2}}){{radic|2}} = {{radic|2}}{{radic|2}}·{{radic|2}} = {{radic|2}}2 = 2, which is rational.

Although the above argument does not decide between the two cases, the [[Gelfond–Schneider theorem]] shows that {{radic|2}}{{radic|2}} is [[Transcendental number|transcendental]], hence irrational. This theorem states that if ''a'' and ''b'' are both [[algebraic number]]s, and ''a'' is not equal to 0 or 1, and ''b'' is not a rational number, then any value of ''a''''b'' is a transcendental number (there can be more than one value if [[Exponentiation#Powers of complex numbers|complex number exponentiation]] is used).

An example that provides a simple constructive proof is<ref>Lord, Nick, "Maths bite: irrational powers of irrational numbers can be rational", ''Mathematical Gazette'' 92, November 2008, p. 534.</ref>

:<math>\left(\sqrt{2}\right)^{\log_{\sqrt{2}}3}=3.</math>

The base of the left side is irrational and the right side is rational, so one must prove that the exponent on the left side, <math>\log_{\sqrt{2}}3</math>, is irrational. This is so because, by the formula relating logarithms with different bases,

:<math>\log_{\sqrt{2}}3=\frac{\log_2 3}{\log_2 \sqrt{2}}=\frac{\log_2 3}{1/2} = 2\log_2 3</math>

which we can assume, for the sake of establishing a [[proof by contradiction|contradiction]], equals a ratio ''m/n'' of positive integers. Then <math>\log_2 3 = m/2n</math> hence <math>2^{\log_2 3}=2^{m/2n}</math> hence <math>3=2^{m/2n}</math> hence <math>3^{2n}=2^m</math>, which is a contradictory pair of prime factorizations and hence violates the [[fundamental theorem of arithmetic]] (unique prime factorization).

A stronger result is the following:<ref name=Marshall>Marshall, Ash J., and Tan, Yiren, "A rational number of the form ''a''''a'' with ''a'' irrational", ''[[Mathematical Gazette]]'' 96, March 2012, pp. 106-109.</ref> Every rational number in the interval <math>((1/e)^{1/e}, \infty)</math> can be written either as ''a''''a'' for some irrational number ''a'' or as ''n''''n'' for some natural number ''n''. Similarly,<ref name=Marshall/> every positive rational number can be written either as <math>a^{a^a}</math> for some irrational number ''a'' or as <math>n^{n^n}</math> for some natural number ''n''.

== Set of all irrationals ==
Since the reals form an [[uncountable]]
set, of which the rationals are a [[Countable set|countable]] subset, the complementary set of
irrationals is uncountable.

Under the usual ([[Euclidean distance|Euclidean]]) distance function d(''x'', ''y'') = |''x'' − ''y''|, the real numbers are a [[metric space]] and hence also a [[topological space]]. Restricting the Euclidean distance function gives the irrationals the structure of a metric space. Since the subspace of irrationals is not closed,
the induced metric is not [[complete (topology)|complete]]. However, being a [[G-delta set]]—i.e., a countable intersection of open subsets—in a complete metric space, the space of irrationals is [[completely metrizable]]: that is, there is a metric on the irrationals inducing the same topology as the restriction of the Euclidean metric, but with respect to which the irrationals are complete. One can see this without knowing the aforementioned fact about G-delta sets: the [[continued fraction]] expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seen to be completely metrizable.

Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals equipped with the subspace topology have a basis of [[clopen set]]s so the space is [[Zero-dimensional space|zero-dimensional]].

@@ Line 127: / Line 127: @@
 Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals equipped with the subspace topology have a basis of clopen sets so the space is zero-dimensional.
-==Resources==
+==Licensing==
-* [https://en.wikipedia.org/wiki/Irrational_number Irrational number], Wikipedia
+Content obtained and/or adapted from:
-* [https://en.wikipedia.org/wiki/Square_root_of_2#Proofs_of_irrationality Square root of 2], Wikipedia
+* [https://en.wikipedia.org/wiki/Irrational_number Irrational number, Wikipedia] under a CC BY-SA license

← Older revision		Revision as of 21:30, 19 October 2021
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	==Resources==		==Resources==
	* [https://en.wikipedia.org/wiki/Irrational_number Irrational number], Wikipedia		* [https://en.wikipedia.org/wiki/Irrational_number Irrational number], Wikipedia
		+	* [https://en.wikipedia.org/wiki/Square_root_of_2#Proofs_of_irrationality Square root of 2], Wikipedia

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 === Square roots ===
 The [[square root of 2]] was the first number proved irrational, and that article contains a number of proofs. The [[golden ratio]] is another famous quadratic irrational number. The square roots of all natural numbers which are not [[perfect squares]] are irrational and a proof may be found in [[quadratic irrational]]s.
 ===General roots===

@@ Line 1: / Line 1: @@
-[[File:Square root of 2 triangle.svg|right|thumb|240px|The number [[Square root of 2|{{radic|2}}]] is irrational.]]
+[[File:Square root of 2 triangle.svg|right|thumb|240px|The number <math> \sqrt{2} </math> is irrational.]]
 In [[mathematics]], the '''irrational numbers''' (from in- [[prefix]] assimilated to ir- (negative prefix, [[privative]]) + rational) are all the [[real number]]s which are not  [[rational number]]s. That is, irrational numbers cannot be expressed as the ratio of two [[integer]]s. When the [[ratio]] of lengths of two line segments is an irrational number, the line segments are also described as being ''[[commensurability (mathematics)|incommensurable]]'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.
@@ Line 109: / Line 109: @@
 Furthermore, the set of all irrationals is a disconnected metrizable space. In fact, the irrationals equipped with the subspace topology have a basis of [[clopen set]]s so the space is [[Zero-dimensional space|zero-dimensional]].