In mathematics, a quadratic irrational, also known as a quadratic surd or quadratic irrationality, is an irrational number that is the solution to some quadratic equation with rational coefficients. Since fractions can be cleared from a quadratic equation by multiplying both sides by their common denominator, this is the same as saying it is a root of a quadratic equation whose coefficients are integers. Quadratic irrationals therefore have the form Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... In mathematics, an irrational number is any real number that is not a rational number, i. ... Graph of a quadratic function: y = x2 - x - 2 = (x+1)(x-2) The x-coordinatess of the points where the graph crosses the x-axis, x = -1 and x = 2, are the roots of the quadratic equation: x2 - x - 2 = 0. ... In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the least common multiple of the denominators of a set of vulgar fractions. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

for integers a, b, c. This implies that the quadratic rationals have the same cardinality as ordered triples of integers, and are therefore countable. The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics the term countable set is used to describe the size of a set, e. ...

Quadratic irrationals are special numbers, especially in relation to continued fractions, where we have the result that all quadratic irrationals, and only quadratic irrationals, have periodic continued fraction forms. For example In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

A special case includes Pell's equation. Pells equation is any Diophantine equation of the form where n is a nonsquare integer. ...

External links

• Mathworld article
• Continued fraction calculator for quadratic irrationals
• Proof that e is not a quadratic irrational