What is an irrational number?

An irrational number cannot be expressed as a fraction of two whole numbers. As a decimal, it goes on forever without repeating any pattern.

Contrast with rational numbers: 1/3 = 0.333... goes on forever but repeats (the 3 repeats) making it rational. An irrational number neither terminates nor repeats.

Famous examples:

• Pi (π) = 3.14159265358979... the ratio of a circle's circumference to its diameter
• √2 = 1.41421356... the length of the diagonal of a square with side 1. Proven irrational by ancient Greek mathematicians.
• e = 2.71828182... the base of natural logarithms
• The square root of any prime number

Simple proof that √2 is irrational: assume √2 = a/b in lowest terms. Then a² = 2b², so a is even (a = 2k). Then (2k)² = 2b², so b² = 2k², meaning b is also even. But if both a and b are even, a/b was not in lowest terms — a contradiction. Therefore √2 cannot be a fraction.