Definition:Binomial (Euclidean)/Second 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 second binomial if and only if:

$(1): \quad b \in \Q$
$(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$

where $\Q$ denotes the set of rational numbers.

In the words of Euclid:

but if the lesser term be commensurable in length with the rational straight line set out, let the whole be called a second binomial;

(The Elements: Book $\text{X (II)}$: Definition $2$)


Let $a = 2 \sqrt 3$ and $b = 3$.


\(\ds \frac {\sqrt {a^2 - b^2} } a\) \(=\) \(\ds \frac {\sqrt {12 - 9} } {2 \sqrt 3}\)
\(\ds \) \(=\) \(\ds \frac {\sqrt 3} {2 \sqrt 3}\)
\(\ds \) \(=\) \(\ds \frac 1 2\) \(\ds \in \Q\)

Therefore $2 \sqrt 3 + 3$ is a second 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.