Definition:Apotome/Sixth Apotome
Definition
Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome.
Then $a - b$ is a sixth apotome if and only if:
- $(1): \quad a \notin \Q$
- $(2): \quad b \notin \Q$
- $(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$
where $\Q$ denotes the set of rational numbers.
In the words of Euclid:
- and if neither, a sixth.
(The Elements: Book $\text{X (III)}$: Definition $6$)
Example
Let $a = \sqrt 7$ and $b = \sqrt 5$.
Then:
\(\ds \frac {\sqrt {a^2 - b^2} } a\) | \(=\) | \(\ds \frac {\sqrt {7 - 5} } {\sqrt 7}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\frac 2 7}\) | \(\ds \notin \Q\) |
Therefore $\sqrt 7 - \sqrt 5$ is a sixth apotome.
Also see
- Definition:First Apotome
- Definition:Second Apotome
- Definition:Third Apotome
- Definition:Fourth Apotome
- Definition:Fifth Apotome
Linguistic Note
The term apotome is archaic, and is rarely used nowadays.
It is pronounced a-POT-o-mee, just as "epitome" is pronounced e-PIT-o-mee.
It is transliterated directly from the Ancient Greek word ἀποτομή, which is the noun form of ἀποτέμνω, from ἀπο- (away) and τέμνω (to cut), meaning roughly to cut away.
Therefore, ἀποτομή means roughly (the portion) cut off.