The Theorem of Pythagoras

The Pythagorean theorem is the most appropriate starting point for a book on mathematics and its history. It is not only the oldest mathematical theorem, but also the source of three great streams of mathematical thought numbers, geometry, and infinity. The number stream begins with

Pythagorean triples

; triples of integers

$$(a, b, c)$$

such that

$$a^2 + b^2 = c^2$$

. The geometry stream begins with the interpretation of

$$a^2, b^2,\ \rm{and}\ c^2$$

as squares on the sides of a right-angled triangle with sides

a, b

, and hypotenuse

c

. The infinity stream begins with the discovery that

$$\sqrt{2}$$

, the hypotenuse of the right-angled triangle whose other sides are of length 1, is an

irrational

number. These three streams are followed separately through Greek mathematics in Chapters 2, 3, and 4. The geometry stream resurfaces in Chapter 7, where it takes an algebraic turn. The basis of algebraic geometry is the possibility of describing points by numbers—their coordinates—and describing each curve by an equation satisfied by the coordinates of its points. This fusion of numbers with geometry is briefly explored at the end of this chapter, where we use the formula

$$a^2 + b^2 = c^2$$

to define the concept of distance in terms of coordinates.