The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This essential trains for: AMC-8, AMC-10, AMC-12, AIME, SAT-I, Math Kangaroo 9-10.

Irrational numbers are numbers that cannot be represented as fractions. That is, there is no division of integers that such a number can be the result of.

The existence of irrational numbers has been proven by the Phythagoreans by investigating the length of the diagonal of a square. If, after choosing a unit of length, we construct a square of side length 1, the length of its diagonal is, by the Pythagorean theorem:

equation

The proof that this number is not rational is a non-constructive proof by contradiction. Non-constructive means that, though it proves that the number is irrational in nature, it does not provide us with a method for finding the value of the number itself.

The strategy for the proof is to analyze the parity of the integers involved.

Proof that √2 is irrational:

Assume that √2 is a rational number. Then, it can be written uniquely as an irreducible fraction (i.e. the numerator and the denominator are relatively prime):

equation

Since a and b are relatively prime, then they cannot both be even.

It follows that the only parities that a and b can have are as follows:

  1. a even and b odd
  2. a odd and b even
  3. a odd and b odd

Square both sides of the equality:

equation

to obtain:

equation

and start case work.

Case 1:

Since a is even we can substitute it by:

equation

equation

equation

Since b is odd, so is its square. The left hand side is odd, the right hand side is clearly even. The equality above is impossible since there is no number that can be both even and odd.

Case 2:

Observe:

equation

to see that in this case, the left hand side must be even and the right hand side must be odd. Impossible.

Case 3:

Observe:

equation

to see that in this case, the left hand side must be even and the right hand side must be odd. Impossible.

Therefore, since there is no possible set of parities for the numerator and denominator it follows that our assumption that √2 can be written in fraction form is not true.

Problem: Prove that √3 is irrational.

Example: Certain expressions that look irrational are, however, rational.

equation

Though this number looks irrational, it is not. First, notice that:

equation

equation

equation

Always be on the lookout for perfect squares!