Definition:Apotome/Warning

Definition

It is a mistake to suggest that $a$ and $b$ may be any real numbers such that $\dfrac a b \notin \Q$ and $\paren {\dfrac a b}^2 \in \Q$.

For example, let $a = \pi \sqrt 2$ and $b = \pi$.

Then we see that:

$(1): \quad \dfrac a b = \sqrt 2 \notin \Q$

$(2): \quad \paren {\dfrac a b}^2 = 2 \in \Q$

But $a - b = \pi \paren {\sqrt 2 - 1}$ is not an 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.