Definition:Binomial (Euclidean)/Fourth 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 fourth binomial if and only if:

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

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

In the words of Euclid:

Again, if the square on the greater term be greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a fourth binomial;

(The Elements: Book $\text{X (II)}$: Definition $4$)


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


\(\ds \frac {\sqrt {a^2 - b^2} } a\) \(=\) \(\ds \frac {\sqrt {9 - 2} } 3\)
\(\ds \) \(=\) \(\ds \frac {\sqrt 7} 3\) \(\ds \notin \Q\)

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