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.
Both rationals and irrationals are dense on the number line: between any two numbers you can always find both types. But the irrationals are uncountably more numerous.
The proof by contradiction: if sqrt(2) = p/q in lowest terms, both p and q end up even, but a fraction in lowest terms cannot have both parts even. The contradiction proves no such fraction exists.
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.