An irrational number is a real number that cannot be written as a fraction of two integers. Its decimal expansion never terminates and never repeats. Famous examples include β2, Ο, e, and the golden ratio Ο. The discovery that irrational numbers exist β attributed to the Pythagorean school around 500 BCE β reportedly shocked a mathematical worldview built on whole numbers and their ratios.
Take the diagonal of a square with side length 1. How long is it? By the Pythagorean theorem, it's β2. But if you try to write β2 as a fraction β any fraction at all, no matter how big the numbers β you will never succeed. β2 = 1.41421356... and the decimal digits go on forever without ever settling into a repeating pattern.
According to a story that has survived (with uncertain historical accuracy) since antiquity, the Pythagorean school in ancient Greece believed that "all is number" β meaning every quantity in the universe could be expressed as a ratio of whole numbers. When a member of the school (traditionally named Hippasus) proved that β2 could not be written as such a ratio, it directly contradicted their worldview. Legend has it he was thrown overboard at sea for revealing this secret. Historians treat the details of this story with skepticism, but the mathematical discovery itself β that β2 is irrational β is genuine and dates to around 500 BCE.
Here's something strange: there are actually more irrational numbers than rational numbers β infinitely more, in a precise mathematical sense (rationals are countable, irrationals are not). Yet on a calculator, you almost never encounter one exactly β calculators can only display terminating decimals, which are always rational approximations.
Suppose, for contradiction, that β2 = p/q with gcd(p,q) = 1. Then 2qΒ² = pΒ², so pΒ² is even, so p is even β write p = 2k. Then 2qΒ² = 4kΒ², so qΒ² = 2kΒ², making q even too. But then p and q share the factor 2, contradicting gcd(p,q) = 1. So no such fraction exists. This proof by contradiction (using infinite descent) appears in Euclid's Elements, Book X.
Irrational numbers split into two types:
Proved by Johann Lambert (1761) using continued fractions. Ο is also transcendental (Lindemann, 1882) β a stronger and much harder result, which also proved that squaring the circle by compass and straightedge is impossible.
Proved by Euler (1737) using the continued fraction expansion of e. e is also transcendental (Hermite, 1873).
Ο satisfies ΟΒ² = Ο + 1, and is the limit of the ratio of consecutive Fibonacci numbers. It is algebraic (not transcendental) since it's a root of xΒ² β x β 1 = 0.
Like the rationals, the irrationals are dense in the real line β between any two real numbers, there is an irrational number. In fact, "almost all" real numbers (in a measure-theoretic sense) are irrational, and moreover almost all are transcendental β algebraic numbers form a countable (hence "measure zero") subset of β.
Every irrational number has a unique infinite continued fraction expansion. The golden ratio has the simplest possible expansion: Ο = [1;1,1,1,1,...] β an infinite string of 1s, making it in a precise sense the "most irrational" number (hardest to approximate well by rationals). By contrast, Ο's continued fraction [3;7,15,1,292,...] has no discernible pattern, though the early terms give the famous approximation 355/113 (accurate to 6 decimal places), discovered by Zu Chongzhi (5th century China).
Joseph Liouville (1844) constructed the first numbers proven to be transcendental β before it was even known whether any transcendental numbers existed. He showed that algebraic numbers cannot be approximated "too well" by rationals, then explicitly constructed numbers (Liouville numbers) that violate this bound, proving they must be transcendental. This was decades before Hermite (e, 1873) and Lindemann (Ο, 1882) proved transcendence for naturally occurring constants.
The irrationality measure ΞΌ(x) quantifies how well a number can be approximated by rationals. Rational numbers have ΞΌ = 1 (trivially). Algebraic irrationals have ΞΌ = 2 (Roth's theorem, 1955, Fields Medal). Liouville numbers have ΞΌ = β. The irrationality measure of Ο is known to be at most about 7.6 (as of recent bounds), though its exact value remains unknown.
It is unknown whether Ο + e or Ο Β· e is irrational (though it's known at least one of them must be, since if both were rational, Ο and e would be roots of a specific quadratic, contradicting known transcendence β a nice example of an indirect proof yielding partial information). The irrationality of Euler-Mascheroni constant Ξ³ β 0.5772... is also unknown β it is not even known whether Ξ³ is irrational, despite Ξ³ appearing throughout analysis and number theory since Euler introduced it in 1734.