Definition:Binomial (Euclidean)/Fifth Binomial

From ProofWiki
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:

\(\displaystyle \frac {\sqrt {a^2 - b^2} } a\) \(=\) \(\displaystyle \frac {\sqrt {13 - 9} } {\sqrt {13} }\)
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {\frac 4 {13} }\) \(\displaystyle \notin \Q\)

Therefore $\sqrt {13} + 3$ is a fifth 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.