A number is irrational if it cannot be expressed as a fraction p/q where p and q are integers. Its decimal expansion never ends and never repeats. sqrt(2), pi, e, and phi are all irrational. They are not exceptions or curiosities: the vast majority of real numbers are irrational.
Blue: rational numbers (exact fractions). Red: irrational numbers (non-repeating decimals). Between any two rationals lies an irrational, and vice versa.
Comparison table of rational numbers with repeating or terminating decimals versus irrational numbers with non-repeating non-terminating decimals
| RATIONAL: terminates or repeats | IRRATIONAL: never repeats |
|---|---|
| 1/4 = 0.25000... | sqrt(2) = 1.4142135... |
| terminates | no pattern, ever |
| 1/3 = 0.3333... | pi = 3.1415926... |
| repeating block: {3} | no pattern, ever |
| 22/7 = 3.142857... | e = 2.7182818... |
| repeating block: {142857} | no pattern, ever |
| 5/11 = 0.454545... | phi = 1.6180339... |
| repeating block: {45} | no pattern, ever |
The rational numbers, despite being infinitely numerous, can be listed (they are countable). The irrationals cannot be listed. If you picked a real number at random, the probability of it being rational is exactly zero.
A number is irrational if it cannot be written as a fraction p/q with integers p and q. Its decimal expansion never ends and never repeats. The Pythagoreans proved sqrt(2) irrational around 500 BC, a shocking discovery at the time. Pi was proved irrational by Lambert in 1761, and e by Euler in 1737. Most real numbers are irrational: the rationals are countably infinite but the irrationals are uncountable, so picking a real number at random gives an irrational with probability 1. Algebraic irrationals satisfy polynomial equations; transcendentals do not.