Irrational number

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being 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.The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram.The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:It will be a rational (magnitude) when we, for instance, say 10, 12, 3%, 6%, etc., because its value is pronounced and expressed quantitatively. What is not rational is irrational and it is impossible to pronounce and represent its value quantitatively. For example: the roots of numbers such as 10, 15, 20 which are not squares, the sides of numbers which are not cubes etc.their sums or differences, or results of their addition to a rational magnitude, or results of subtracting a magnitude of this kind from an irrational one, or of a rational magnitude from it.contained in a certain given magnitude once or many times, then this (given) magnitude corresponds to a rational number. . . . Each time when this (latter) magnitude comprises a half, or a third, or a quarter of the given magnitude (of the unit), or, compared with (the unit), comprises three, five, or three fifths, it is a rational magnitude. And, in general, each magnitude that corresponds to this magnitude (i.e. to the unit), as one number to another, is rational. If, however, a magnitude cannot be represented as a multiple, a part (l/n), or parts (m/n) of a given magnitude, it is irrational, i.e. it cannot be expressed other than by means of 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 irrationals.Almost all irrational numbers are 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., eπ is irrational, too.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, octal or hexadecimal expansions, and in general for expansions in every positional notation with natural bases.Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that ab is rational:It is not known if π + e {displaystyle pi +e} (or π − e {displaystyle pi -e} ) is irrational. In fact, there is no pair of non-zero integers m , n {displaystyle m,n} for which it is known whether m π + n e {displaystyle mpi +ne} is irrational. Moreover, it is not known whether or not the set { π , e } {displaystyle {pi ,e}} is algebraically independent over Q {displaystyle mathbb {Q} } .Since the reals form an uncountableset, of which the rationals are a countable subset, the complementary set ofirrationals is uncountable.