Definition:Binomial (Euclidean)/First 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 first binomial if and only if:

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

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


In the words of Euclid:

Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable 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 first binomial straight line;

(The Elements: Book $\text{X (II)}$: Definition $1$)


Example

Let $a = 9$ and $b = \sqrt {17}$.

Then:

\(\displaystyle \frac {\sqrt {a^2 - b^2} } a\) \(=\) \(\displaystyle \frac {\sqrt {81 - 17} } 9\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sqrt {64} } 9\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 8 9\) \(\displaystyle \in \Q\)

Therefore $9 + \sqrt {17}$ is a first 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.