# Definition:Apotome/Sixth Apotome

(Redirected from Definition: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.

## 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.

## 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.