<P> and there is a unique positive real number π with this property . A more abstract variation on the same idea, making use of sophisticated mathematical concepts of topology and algebra, is the following theorem: there is a unique (up to automorphism) continuous isomorphism from the group R / Z of real numbers under addition modulo integers (the circle group) onto the multiplicative group of complex numbers of absolute value one . The number π is then defined as half the magnitude of the derivative of this homomorphism . </P> <P> A circle encloses the largest area that can be attained within a given perimeter . Thus the number π is also characterized as the best constant in the isoperimetric inequality (times one - fourth). There are many other, closely related, ways in which π appears as an eigenvalue of some geometrical or physical process; see below . </P> <P> π is an irrational number, meaning that it cannot be written as the ratio of two integers (fractions such as 22 / 7 are commonly used to approximate π, but no common fraction (ratio of whole numbers) can be its exact value). Because π is irrational, it has an infinite number of digits in its decimal representation, and it does not settle into an infinitely repeating pattern of digits . There are several proofs that π is irrational; they generally require calculus and rely on the reductio ad absurdum technique . The degree to which π can be approximated by rational numbers (called the irrationality measure) is not precisely known; estimates have established that the irrationality measure is larger than the measure of e or ln (2) but smaller than the measure of Liouville numbers . </P> <P> The digits of π have no apparent pattern and have passed tests for statistical randomness, including tests for 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 π is normal has not been proven or disproven . </P>

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