Square Root of Two Proof

My father and I recently discussed one of many proofs that sqrt(2), the square root of two, is an irrational number. By irrational number, of course, I mean a number that cannot be expressed as the quotient of two integers. Here is the proof, as I understand it:

Let us assume that sqrt(2) is a rational number. That means that it can be expressed as p/q, where p and q are integers.

So we get:

  sqrt(2) = p/q

Using simple algebra, the above equation can be rearranged as follows:

  2 = (p/q)²
  2q² = p²

Now, according to the Fundamental Theory of Arithmetic (or Unique Factorization Theorem), "every natural number greater than one either is itself a prime number or can be written as a product of prime numbers" (wikipedia).

q and p are no exception to this rule. Accordingly, they can each be expressed in terms of one or more prime numbers. q² will have exactly twice as many prime factors as q, and p² will have exactly twice as many prime factors as p. That is, q² and p² will each have an even number of prime factors. The expression 2q² must, then, have an odd number of prime factors (an even number + one extra factor of 2).

This is where the contradiction comes in. The Fundamental Theory of Arithmetic also states that each number's set of prime factors is unique (i.e. there is only one way to express a number in terms of its prime factors). So, if p² is equal to 2q², then the two expressions must have the same number of prime factors.

Yet, we have just shown (two paragraphs above) that p² has an even number of prime factors, while 2q² has an odd number of prime factors. That means that p² and 2q² can't possibly have the same number of prime factors. In turn, this implies that p² does not equal 2q², so our original equation can't have been correct. sqrt(2) cannot be expressed as the quotient of two integers, so it cannot be a rational number; it must be irrational.

QED :o)

There are all sorts of other ways to prove that the sqrt(2) is an irrational number. One, for example, involves geometry (specifically, the 45-45-90 triangle with a hypotenuse equal to the square root of two).