Definition:Binomial (Euclidean)/Second 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 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 $2$)
Example
Let $a = 2 \sqrt 3$ and $b = 3$.
Then:
\(\ds \frac {\sqrt {a^2 - b^2} } a\) | \(=\) | \(\ds \frac {\sqrt {12 - 9} } {2 \sqrt 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt 3} {2 \sqrt 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 2\) | \(\ds \in \Q\) |
Therefore $2 \sqrt 3 + 3$ is a second binomial.
Also see
- Definition:First 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.