Irrational Number
An irrational number is a real number that cannot be expressed as a fraction of two integers; its decimal form never terminates or repeats.
Definition
An irrational number is a number that cannot be written as a simple fraction, a real number that cannot be expressed as $p/q$ for any integers $p$ and $q$ with $q \neq 0$; equivalently, its decimal expansion is non-terminating and non-repeating. The irrationals $\mathbb{R}\setminus\mathbb{Q}$ are the complement of the rationals within the reals, and unlike $\mathbb{Q}$ they are uncountable (with the cardinality of the continuum), meaning "almost all" real numbers are irrational in a measure-theoretic sense.
Example
The square root of $2$ is about $1.41421356\ldots$, and those digits never repeat; pi ($3.14159\ldots$) is another famous irrational number. A classic proof that $\sqrt{2}$ is irrational assumes $\sqrt{2} = p/q$ in lowest terms: then $2q^2 = p^2$, so $p^2$ (and hence $p$) is even; writing $p = 2k$ gives $2q^2 = 4k^2$, so $q^2 = 2k^2$, making $q$ even too, which contradicts lowest terms. Transcendental numbers, those that are not roots of any polynomial with integer coefficients, are a proper subset of the irrationals: both $\pi$ and $e$ are transcendental, proven by Lindemann (1882) and Hermite (1873) respectively.
Key Insight
Irrational means "not a ratio": no matter how hard you try, you cannot find two whole numbers whose ratio equals the square root of $2$ exactly. Common irrationals include $\sqrt{2}$, $\sqrt{3}$, $\pi$, and $e$, and adding a rational to an irrational always yields an irrational number. Liouville (1844) first proved specific numbers irrational by constructing Liouville numbers whose rational approximations converge too quickly for algebraic numbers; the irrationals are dense in $\mathbb{R}$ and have measure $1$ within any interval, while the rationals have measure $0$, so together the two sets tile the real line perfectly.