Some numbers cannot be written as fractions. The square root of 2 is one. Its decimal expansion never ends and never repeats. The Greeks discovered this around 500 BCE, and the discovery unsettled their belief that all numbers were rational.

Pi is irrational. So is e, the base of natural logarithms. The golden ratio, approximately 1.618, is also irrational. In fact, irrational numbers vastly outnumber rational ones, even though both sets are infinite.

Irrational numbers fill the gaps between rationals on the number line. Together they form the real numbers. In practice, we approximate them with decimals or fractions. But their exact nature—non-repeating, non-terminating, not a ratio—is what defines them. They are essential in geometry, calculus, and analysis.

• Cannot be written as a ratio of integers
• Decimal expansion never ends or repeats
• Examples: √2, π, e, golden ratio
• Together with rationals form the real numbers