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:
- $-1 < \tilde{\alpha} < 0$
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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 $D - P^2$.