Proof that π is irrational

A mathematical argument showing that π cannot be expressed as the ratio of two integers. The result distinguishes π from rational numbers, whose decimal expansions terminate or eventually repeat.

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Proof by contradiction: A proof method that assumes a claim is false and derives a contradiction. The irrationality argument begins by assuming π is a ratio of integers.
Johann Heinrich Lambert: An eighteenth-century Swiss mathematician who proved that π is irrational. Lambert gave the first published proof, using a continued fraction for the tangent function.
Integer: A whole number, including zero and the negative whole numbers. The assumed numerator and denominator are integers.
Transcendental number: A number that is not a root of any nonzero polynomial with integer coefficients. Transcendence is stronger than irrationality, and π has both properties.
Squaring the circle: The classical problem of constructing a square equal in area to a given circle using compass and straightedge. π’s transcendence, a stronger fact, makes this construction impossible.
Taylor series: A representation of a function as an infinite power series centered at a point. Some historical irrationality arguments use series for trigonometric functions.
Adrien-Marie Legendre: A French mathematician whose number theory and geometry shaped nineteenth-century mathematics. Legendre presented an irrationality proof for π and conjectured its transcendence.
Divisibility: The relation in which one integer is an exact multiple of another. Divisibility constraints turn the constructed integral into an integer.
Lindemann–Weierstrass theorem: A theorem establishing the transcendence of certain values of the exponential function at algebraic inputs. Lindemann’s result later proved π transcendental, far beyond the claim considered here.
Circle: The set of points in a plane at a fixed distance from a center. The circumference-to-diameter ratio defining π comes from circle geometry.