Definition:Quadratic Irrational/Reduced

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Definition

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




Solutions of such quadratics can be written as:

$\alpha = \dfrac{\sqrt D + P} Q$

where $D, P, Q \in \Z$ 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.


Associated Quadratic Irrational

Let $\alpha$ be a reduced quadratic irrational expressed as:

$\alpha = \dfrac{P + \sqrt D} Q$

Then $\alpha$ is associated to $D$ if and only if $Q$ divides $D - P^2$.


Also see