Definition:Binomial (Euclidean)/Third 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 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 $3$)


Example

Let $a = \sqrt {11}$ and $b = \sqrt {\dfrac {143} {49} }$.

Then:

\(\displaystyle \frac {\sqrt {a^2 - b^2} } a\) \(=\) \(\displaystyle \frac {\sqrt {11 - \frac {143} {49} } } {\sqrt {11} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sqrt {\frac {396} {49} } } {\sqrt {11} }\)
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {\frac {36} {49} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 6 7\) \(\displaystyle \in \Q\)

Therefore $\sqrt {11} + \sqrt {\dfrac {143} {49} }$ is a third 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.