irrational number

An irrational number is a real number that is not rational, that is, cannot be expressed as a fraction (or ratio ) of the form p / q , where p and q are integers.

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proofs that the square root of 2 is irrational

Pythagorean proof, as given by Euclid in his Elements

proof by contradiction:

1. Assume that 2 is rational, that is, there exists integers p and q such that 2 = p / q ; take the irreducible form of this fraction, so that p and q have no factors in common
2. square both sides, to give 2 = p ² / q ²
3. rearrange, to give 2 q ² = p ²
4. hence p ² is even
5. hence p is even (trivial proof left as an excercise for the reader); write p = 2 m
6. substitute for p in (3), to give 2 q ² = (2 m ) ² = 4 m ²
7. divide through by 2, to give q ² = 2 m ²
8. hence q ² is even
9. hence q is even

(1) assumes that p and q have no factors in common; (5) and (9) show they they both have 2 as a factor. This is a contradiction. Hence the assumption (1) is false, and 2 is not rational.

proof based on unique factorisation

[I first saw this rather elegant little proof in Chaitin's Meta Math! , p98. It uses the result of unique factorisation into powers of primes (the fundamental theorem of arthimetic ).]

1. Assume that 2 is rational, that is, there exists integers p and q such that 2 = p / q ; take the irreducible form of this fraction, so that p and q have no factors in common
2. square both sides, and rearrange, to give 2 q ² = p ²
3. factorise p and q into their unique prime factoriastions, p = 2 ^a 3 ^b 5 ^c ..., q = 2 ^x 3 ^y 5 ^z ...
4. substitute into (2), and simplify, to give 2 ^{2 x +1} 3 ^{2 y} 5 ^{2 z} ... = 2 ^{2 a} 3 ^{2 b} 5 ^{2 c} ...

Since the LHS and the RHS refer to the same number, and since such factorisation is unique, we have 2 x +1 = 2 a (or odd = even ). This is a contradiction. Hence the assumption (1) is false, and 2 is not rational.