Quadratic irrational

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In mathematics, a quadratic irrational, also known as a 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 an irrational root of some quadratic equation whose coefficients are integers. They form a subset of the algebraic numbers. The quadratic irrationals, therefore, are all those numbers that can be expressed in this form:

${a+b\sqrt{c} \over d}$

for integers a, b, c, d; with b and d non-zero, and with c positive and not a perfect square. This implies that the quadratic irrationals have the same cardinality as ordered quadruples of integers, and are therefore countable. If b=1 in the above expression then the number is called a quadratic surd.^{[dubious – discuss]}

The quadratic irrationals with a given c form a field, called a quadratic field.

Quadratic irrationals have useful properties, 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

$\sqrt{3}=1.732\ldots=[1;1,2,1,2,1,2,\ldots]$