In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational.
When the ratio of lengths of two line segments is irrational, the line segments are also described as being incommensurable, meaning they share no measure in common.
Perhaps the best-known irrational numbers are: the ratio of a circle's circumference to its diameter π, Euler's number e, the golden ratio φ, and the square root of two √2.
Read more about Irrational Number: History, Transcendental and Algebraic Irrationals, Decimal Expansions, Irrational Powers, Open Questions, The Set of All Irrationals, See Also
Famous quotes containing the words irrational and/or number:
“How did reason enter the world? As is fitting, in an irrational way, accidentally. We will have to guess at it, like a riddle.”
—Friedrich Nietzsche (1844–1900)
“Computers are good at swift, accurate computation and at storing great masses of information. The brain, on the other hand, is not as efficient a number cruncher and its memory is often highly fallible; a basic inexactness is built into its design. The brain’s strong point is its flexibility. It is unsurpassed at making shrewd guesses and at grasping the total meaning of information presented to it.”
—Jeremy Campbell (b. 1931)