Irrational number

An irrational number is a real number that cannot be written as a ratio of two integers. Its decimal expansion is infinite and nonrepeating.

Rational number: A number expressible as the ratio of two integers, with a nonzero denominator. Irrational numbers are precisely the real numbers excluded from this class.
Proof by contradiction: A proof method that derives a contradiction from the negation of the desired claim. The classic proof for the square root of two assumes a rational fraction and reaches impossible parity.
Square root of 2: The positive number whose square is 2. Its irrationality has a short proof based on parity.
Hippasus: A figure traditionally associated with the early Greek discovery of incommensurable magnitudes. Later accounts connect him with the discovery that √2 cannot be a ratio of integers.
Dedekind cut: A partition of the rational numbers into lower and upper sets that can define a real number. Cuts construct irrational numbers as precise gaps among rational numbers.
Real number: A number represented by a point on the continuous number line. Irrational numbers form part of the real numbers alongside rational numbers.
Euclid's proof of the irrationality of the square root of 2: A proof that assuming √2 is a fraction in lowest terms forces both numerator and denominator to be even. It is the canonical demonstration that a specific real number is irrational.
Pi: The ratio of a circle's circumference to its diameter in Euclidean geometry. Its irrationality was proved in the eighteenth century.
Pythagoreanism: An ancient Greek philosophical and mathematical tradition associated with Pythagoras and his followers. Its emphasis on whole-number ratios frames the historical shock of incommensurable lengths.
Density of rational numbers: The property that between any two distinct real numbers lies a rational number. Irrational numbers coexist with rational numbers at every scale on the number line.

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