Definition:Binomial (Euclidean)/Sixth Binomial

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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 $6$)


Let $a = \sqrt 7$ and $b = \sqrt 5$.


\(\ds \frac {\sqrt {a^2 - b^2} } a\) \(=\) \(\ds \frac {\sqrt {7 - 5} } {\sqrt 7}\)
\(\ds \) \(=\) \(\ds \sqrt {\frac 2 7}\) \(\ds \notin \Q\)

Therefore $\sqrt 7 + \sqrt 5$ is a sixth binomial.

Also see

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.