Irrational Number
ArithmeticAn 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. Its decimal goes on forever without repeating any pattern.
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.
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.
Definition
An irrational number is 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 are the complement of the rationals within the reals.
Example
Proof that $\sqrt{2}$ is irrational: assume $\sqrt{2} = p/q$ in lowest terms. Then $2q^2 = p^2$, so $p^2$ is even, so $p$ is even. Write $p = 2k$; then $2q^2 = 4k^2$, so $q^2 = 2k^2$, so $q$ is even, contradicting lowest terms.
Key Insight
Common irrational numbers include $\sqrt{2}$, $\sqrt{3}$, $\pi$, and $e$. Adding a rational to an irrational always yields an irrational number.
Definition
The irrationals $\mathbb{R}\setminus\mathbb{Q}$ are uncountable (they have the cardinality of the continuum), while $\mathbb{Q}$ is countable. This means "almost all" real numbers are irrational in a measure-theoretic sense. Liouville (1844) first proved specific numbers irrational by constructing Liouville numbers whose rational approximations converge too quickly for algebraic numbers.
Example
Transcendental numbers (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
The irrationals are dense in $\mathbb{R}$ and yet have measure $1$ within any interval, while the rationals have measure $0$. The two sets together tile the real line perfectly.