Definition:Quadratic Irrational

Definition
A quadratic irrational is an irrational number of the form:
 * $r + s \sqrt n$

where $r, s$ are rational and $n$ is a positive integer which is not a square.

Reduced Form
An irrational root \(\alpha\) of a quadratic equation with integer coefficients is called reduced if \(\alpha > 1\) and its conjugate \(\tilde{\alpha}\) satisfies \(-1 < \tilde{\alpha} < 0\).

Solutions of such quadratics can be written as $$\alpha = \frac{\sqrt{D} + P}{Q}$$ where \(D, P, Q \in \mathbf{Z}\) and \(D, Q > 0\). It is also possible (though not required) to ensure that \(Q\) divides \(D - P^2\). This is actually a necessary assumption for some proofs and warrants its own definition. As an example for some enlightenment, notice $$\alpha = \frac{2 + \sqrt{7}}{4}$$ is reduced but \(4\) does not divide \(7 - 2^2\). However, if we write this as \(\frac{8 + \sqrt{112}}{16}\), we have our desired condition.

Association
We say a reduced quadratic irrational \(\alpha\) is associated to \(D\) if we can write $$\alpha = \frac{P + \sqrt{D}}{Q}$$ and \(Q\) divides \(D - P^2\).

Also see

 * Quadratic Equation
 * Quadratic Irrational is Root of Quadratic Equation