Definition:Binomial (Euclidean)/Third Binomial/Example
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Example
Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial.
By definition, $a + b$ is a third binomial if and only if:
- $(1): \quad a \notin \Q$
- $(2): \quad b \notin \Q$
- $(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$
where $\Q$ denotes the set of rational numbers.
Let $a = \sqrt {11}$ and $b = \sqrt {\dfrac {143} {49} }$.
Then:
| \(\ds \frac {\sqrt {a^2 - b^2} } a\) | \(=\) | \(\ds \frac {\sqrt {11 - \frac {143} {49} } } {\sqrt {11} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sqrt {\frac {396} {49} } } {\sqrt {11} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {\frac {36} {49} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 6 7\) | \(\ds \in \Q\) |
Therefore $\sqrt {11} + \sqrt {\dfrac {143} {49} }$ is a third binomial.