Editing Pi (section)

=== Irrationality and normality ===
{{pi}} is an [[irrational number]], meaning that it cannot be written as the [[rational number|ratio of two integers]]. Fractions such as {{math|{{sfrac|22|7}}}} and {{math|{{sfrac|355|113}}}} are commonly used to approximate {{pi}}, but no [[common fraction]] (ratio of whole numbers) can be its exact value.{{sfn|Arndt|Haenel|2006|p=5}} Because {{pi}} is irrational, it has an infinite number of digits in its [[decimal representation]], and does not settle into an infinitely [[repeating decimal|repeating pattern]] of digits. There are several [[proof that π is irrational|proofs that {{pi}} is irrational]]; they are generally [[proofs by contradiction]] and require calculus. The degree to which {{pi}} can be approximated by [[rational number]]s (called the [[irrationality measure]]) is not precisely known; estimates have established that the irrationality measure is larger or at least equal to the measure of {{math|''e''}} but smaller than the measure of [[Liouville number]]s.<ref>{{cite journal |last1=Salikhov |first1=V. |year=2008 |title=On the Irrationality Measure of pi |journal=Russian Mathematical Surveys |volume=53 |issue=3 |pages=570–572 |doi=10.1070/RM2008v063n03ABEH004543 |bibcode=2008RuMaS..63..570S |s2cid=250798202 |issn=0036-0279}}</ref>

The digits of {{pi}} have no apparent pattern and have passed tests for [[statistical randomness]], including tests for [[normal number|normality]]; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture that {{pi}} is [[normal number|normal]] has not been proven or disproven.{{sfn|Arndt|Haenel|2006|pp=22–23}}

Since the advent of computers, a large number of digits of {{pi}} have been available on which to perform statistical analysis. [[Yasumasa Kanada]] has performed detailed statistical analyses on the decimal digits of {{pi}}, and found them consistent with normality; for example, the frequencies of the ten digits 0 to 9 were subjected to [[statistical significance test]]s, and no evidence of a pattern was found.{{sfn|Arndt|Haenel|2006|pp=22, 28–30}} Any random sequence of digits contains arbitrarily long subsequences that appear non-random, by the [[infinite monkey theorem]]. Thus, because the sequence of {{pi}}'s digits passes statistical tests for randomness, it contains some sequences of digits that may appear non-random, such as a [[Six nines in pi|sequence of six consecutive 9s]] that begins at the 762nd decimal place of the decimal representation of {{pi}}.{{sfn|Arndt|Haenel|2006|p=3}} This is also called the "Feynman point" in [[mathematical folklore]], after [[Richard Feynman]], although no connection to Feynman is known.