# Definition:Quadratic Irrational

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## 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:
- $-1 < \tilde{\alpha} < 0$

## Also known as

A **quadratic irrational** is also known as a **quadratic surd**.

## Also see

## Sources

- Weisstein, Eric W. "Quadratic Surd." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/QuadraticSurd.html