# The Irrationals

## Synopsis

The ancient Greeks discovered them, but it wasn't until the nineteenth century that irrational numbers were properly understood and rigorously defined, and even today not all their mysteries have been revealed. In The Irrationals, the first popular and comprehensive book on the subject, Julian Havil tells the story of irrational numbers and the mathematicians who have tackled their challenges, from antiquity to the twenty-first century. Along the way, he explains why irrational numbers are surprisingly difficult to define--and why so many questions still surround them. Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.

## Excerpt

It is terrifying to think how much research is needed to
determine the truth of even the most unimportant fact.

Stendhal

Sources and Apologia

The birth of irrational numbers took place in the cradle of European mathematics: the Greece of several centuries B.C.E. For this conviction and for much more we must place reliance on a few fragments of contemporary papyrus, some complete but much later manuscripts, and the scholarship of many specialists who, even between themselves, sometimes disagree in fundamental ways. Of great importance is the following passage:

Thales, who had travelled to Egypt, was the first to introduce
this science [geometry] into Greece. He made many discoveries
himself and taught the principles for many others to his suc
cessors, attacking some problems in a general way and others
more empirically. Next after him Mamercus, brother of the poet
Stesichorus, is remembered as having applied himself to the
study of geometry; and Hippias of Elis records that he acquired
a reputation in it. Following upon these men, Pythagoras trans
formed mathematical philosophy into a scheme of liberal edu
cation, surveying its principles from the highest downwards
and investigating its theorems in an immaterial [abstract] and
intellectual manner. He it was who discovered the doctrine of
proportionals and the structure of the cosmic figures.

Thay-leez, but originally T-hay-leez.

From the translation by Glenn R. Morrow: Proclus, A Commentary on the First Book of Euclid’s Elements (Princeton University Press, 1992).

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