Definition:Binomial (Euclidean)/Sixth Binomial
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Definition
Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial.
Then $a + b$ is a sixth binomial if and only if:
- $(1): \quad: a \notin \Q$
- $(2): \quad: b \notin \Q$
- $(3): \quad: \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$
where $\Q$ denotes the set of rational numbers.
In the words of Euclid:
- and if neither, a sixth binomial.
(The Elements: Book $\text{X (II)}$: Definition $6$)
Example
Let $a = \sqrt 7$ and $b = \sqrt 5$.
Then:
\(\ds \frac {\sqrt {a^2 - b^2} } a\) | \(=\) | \(\ds \frac {\sqrt {7 - 5} } {\sqrt 7}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\frac 2 7}\) | \(\ds \notin \Q\) |
Therefore $\sqrt 7 + \sqrt 5$ is a sixth binomial.
Also see
- Definition:First Binomial
- Definition:Second Binomial
- Definition:Third Binomial
- Definition:Fourth Binomial
- Definition:Fifth Binomial
Linguistic Note
The term binomial arises from a word meaning two numbers.
This sense of the term is rarely used (if at all) outside of Euclid's The Elements nowadays.