irrational number
Some numbers refuse to be written as a clean fraction no matter how hard you try. Their decimals run on forever without ever settling into a repeating pattern. These are the irrational numbers — real quantities that genuinely exist on the number line but slip through every attempt to capture them as a ratio.
Formally, an irrational number is a real number that cannot be written as a ratio of two integers. The most famous examples are sqrt(2) (the diagonal of a unit square), pi (the ratio of a circle’s circumference to its diameter), and e. Their decimal expansions are infinite and non-repeating, so you can never write them out completely or as a fraction a/b.
It is a striking fact, proved by the ancient Greeks, that sqrt(2) is irrational: assuming it equals a fraction in lowest terms leads to a contradiction. Even more surprising, there are far “more” irrational numbers than rational ones — between any two points on the line, irrationals are everywhere. Together the rationals and irrationals make up all of the real numbers, with no overlap between the two.
sqrt(2) = 1.41421356… never ends and never repeats, so it cannot equal any fraction a/b.
An endless, non-repeating decimal cannot be a ratio of integers.
Not every square root is irrational: sqrt(4) = 2 is a perfectly good integer. A root is irrational only when the number under it is not a perfect square (or perfect cube, etc.).