Definition:Apotome/Fifth Apotome

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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 fifth apotome 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 annex be so commensurable, a fifth;

(The Elements: Book $\text{X (III)}$: Definition $5$)


Let $a = \sqrt {13}$ and $b = 3$.


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

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

Also see

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.