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 $4$)