# 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.

### 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;

### 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;

### 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.

### 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;

### 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;

### 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.

## 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.

## 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.