General Proofs

http://math.tutorvista.com/number-system/irrational-numbers.html

Theorem: √2 is irrational

Notation:
Below, I will use gcd (m, n) to denote the greatest common divisor of m and n. In addition, d | a will be used to denote that d divides a

Background knowledge:
(#1) Recall that a number is rational if it can be written as a fraction, where the numerator and denominator are integers.

(#2) Definition¹:
Suppose that c and d are integers. Then d divides c, if and only if there is an integer k such that c = kd.

(#3) Theorem²:
If p is a prime and p | (a₁a₂^… a_k), then for some j, 1 =j =k, p | a_j. As a special case, if p | a^k, then p|a.
For example: 2 | 4² so 2 | 4.
   
Using this information, we will now prove that √2 is irrational by using proof by contradiction.

Proof:

Assume that √2 is a rational number. Then by definition of rationality, we can write √2 = m/n for some positive integers m and n. Furthermore, we can assume that the fraction m/n is in reduced form, which means that the gcd (m, n) = 1. Then n√2 = m. Squaring both sides, we see that 2n² = m². From (#2) above, we see that 2|m². Hence, by (#3) above, we have 2|m. Squaring both sides, this means that 4|m². Since 2n² = m², it must also be that 4|2n². Then dividing both sides by 2 we get 2|n². By (#3), this implies that 2|n. Thus, we have established that 2|m and 2|n. As a result, gcd (m, n) ≠ 1. However, this contradicts with gcd (m, n) = 1. Hence our original assumption that √2 was a rational number must be false. Thus, √2 is irrational.

1. This definition was obtained from “Number Theory Through Inquiry” by David Marshall, Edward Odell, and Michael Starbird. http://vault.hanover.edu/~wahl/documents/NumberTheoryText_121106.pdf
2. This Theorem was taken directly from “An Introduction to Number Theory” by Harold Stark.