# Definition:Binomial (Euclidean)/First Binomial

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## Contents

## 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

- Definition:Second Binomial
- Definition:Third Binomial
- Definition:Fourth Binomial
- Definition:Fifth Binomial
- Definition:Sixth 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.