## 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 a reduced quadratic irrational if and only if

$(1): \quad \alpha > 1$
$(2): \quad$ its conjugate $\tilde{\alpha}$ satisfies:
$-1 < \tilde{\alpha} < 0$