*Acta Math. Acad. Paed. Nyíregyháziensis*

**15** (1999), 1-7

`www.bgytf.hu/~amapn`

# Pythagorean side and diagonal numbers

**László Filep**

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### Abstract:

The Pythagorean discovery that the ratio of the diagonal (diameter) and
side of a square cannot be expressed as the ratio of two positive integers
means in modern language that is not a rational number. The above incommensurability
had a great effect on Greek philosophy and mathematics. Many papers deal
with it and its impact, but relatively few with the approximation of the
ratio in question. Even not Euclid's Elements that was purely theoretical
and neglected the calculation methods. The aim of our paper to analyse the
sources and the literature on this problem and give a new (and hopefully
a satisfactory) explanation of Pythagorean's approximation method, i.e.
of their so-called side and diagonal (or diameter) numbers.

*Mathematics Subject Classification.* 01A20.

*Key words and phrases. *Incommensurability,
approximation, Pythagoreans.