Irrational number

In mathematics, an irrational number is any real number that is not a rational number, i.e., one that cannot be written as a ratio of two integers, i.e., it is not of the form

Irrationality of the square root of 2

One proof of the irrationality of the square root of 2 is the following reductio ad absurdum. The proposition is proved by assuming the negation and showing that that leads to a contradiction, which means that the proposition must be true.

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• Assume that √2 is a rational number. This would mean that there exist integers a and b such that a / b = √2.
• Then √2 can be written as an irreducible fraction (the fraction is shortened as much as possible) a / b such that a and b are coprime integers and (a / b)² = 2.
• It follows that a² / b² = 2 and a² = 2 b².
• Therefore a² is even because it is equal to 2 b² which is obviously even.
• It follows that a must be even. (Odd numbers have odd squares and even numbers have even squares.)
• Because a is even, there exists a k that fulfills: a = 2k.
• We insert the last equation of (3) in (6): 2b² = (2k)² is equivalent to 2b² = 4k² is equivalent to b² = 2k².
• Because 2k² is even it follows that b² is also even which means that b is even because only even numbers have even squares.
• By (5) and (8) a and b are both even, which contradicts that a / b is irreducible as stated in (2).

Since we have found a contradiction the assumption (1) that √2 is a rational number must be false. The opposite is proven. √2 is irrational.

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This proof can be generalized to show that any root of any natural number is either a natural number or irrational.

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A different proof

Another reductio ad absurdum showing that √2 is irrational is less well-known and has sufficient charm that it is worth including here. It proceeds by observing that if √2 = m/n then √2 = (2n − m)/(m − n), so that a fraction in lowest terms is reduced to yet lower terms. That is a contradiction if n and m are positive integers, so the assumption that √2 is rational must be false. It is possible to construct from an isosceles right triangle whose leg and hypotenuse have respective lengths n and m, by a classic straightedge-and-compass construction, a smaller isosceles right triangle whose leg and hypotenuse have respective lengths m − n and 2n − m. That construction proves the irrationality of √2 by the kind of method that was employed by ancient Greek geometers.

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A simpler proof of a similar result

When a line segment is divided into two disjoint subsegments in such a way that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part, then that ratio is the golden ratio, equal to

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:arphi={1+sqrt{5} over 2}.

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Assume this is a rational number n/m in lowest terms. Take n to be the length of the whole and m the length of the longer part. Then the length of the shorter part is n − m. Then we have

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:{n over m}={mathrm{whole} over mathrm{longer} mathrm{part}}

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