# Definition:Binomial (Euclidean)

## Definition

Let $a$ and $b$ be two (strictly) positive real numbers such that:

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

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

Then $a + b$ is a **binomial**.

In the words of Euclid:

*If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be called***binomial***.*

(*The Elements*: Book $\text{X}$: Proposition $36$)

### First Binomial

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 $1$)

### Second Binomial

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 $2$)

### Third Binomial

Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial.

Then $a + b$ is a **third binomial** if and only if:

- $(1): \quad a \notin \Q$
- $(2): \quad b \notin \Q$
- $(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$

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

In the words of Euclid:

*and if neither of the terms be commensurable in length with the rational straight line set out, let the whole be called a***third binomial**.

(*The Elements*: Book $\text{X (II)}$: Definition $3$)

### Fourth Binomial

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 $4$)

### Fifth Binomial

Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial.

Then $a + b$ is a **fifth binomial** if and only if:

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

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

In the words of Euclid:

*if the lesser, a***fifth binomial**;

(*The Elements*: Book $\text{X (II)}$: Definition $5$)

### Sixth Binomial

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 $6$)

## Term

The **terms** of $a + b$ are the elements $a$ and $b$.

## Order

The **order** of $a + b$ is the name of its classification into one of the six categories: first, second, third, fourth, fifth or sixth.

## 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.