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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

A number which cannot be described as an exact ratio between two proportions. Irrational numbers have an infinite and non-repeating string of digits after the decimal point.

Perhaps the most famous example of an irrational number is the square-root of 2. Two others are π (pi) and the 12-edo Semitone.

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[Charles Lucy]

From VAN NOSTRAND's Mathematics Dictionary:


Irrational algebraic surface. The graph of an algebraic function in which the variable (or variables) irreducibly under a radical sign..


A real number not expressible as an integer or quotient of integers; a nonrational number. The irrational numbers are those numbers defined by sets (A,B) of a Dedekind cut such that A has no greater member and B has no least member. Also, the irrational numbers are precisely those infinite decimals which are not repeating.

The irrational numbers are of two types, algebraic irrational numbers (irrational numbers are the roots of polynomial equations with rational coefficients) and transcendental numbers [e.g., e and π (pi)].

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