Definition:Binomial (Euclidean)/Fifth Binomial
< Definition:Binomial (Euclidean)(Redirected from Definition:Fifth Binomial)
Jump to navigation
Jump to search
Definition
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 $5$)
Example
Let $a = \sqrt {13}$ and $b = 3$.
Then:
\(\ds \frac {\sqrt {a^2 - b^2} } a\) | \(=\) | \(\ds \frac {\sqrt {13 - 9} } {\sqrt {13} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\frac 4 {13} }\) | \(\ds \notin \Q\) |
Therefore $\sqrt {13} + 3$ is a fifth binomial.
Also see
- Definition:First Binomial
- Definition:Second Binomial
- Definition:Third Binomial
- Definition:Fourth 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.