Irrational Numbers:

Fractions were used as early as 1700 BC by the ancient Egyptians, but it was not until Pythagoras (530 BC) that the need for other numbers, such as {sqroot 2}, was discovered. The need for such irrational numbers is corroborated in modern mathematical analysis, in which they play a fundamental role in the integral calculus, trigonometry, and other fields. Pythagoras showed that the ratio x of the diagonal of an isosceles right triangle to the length of a side must satisfy the equation x2 = 2 (Pythagorean theorem). No fraction m/n, however, can satisfy (m/n)2 = 2; that is, m2 = 2n2 has no solution among the integers, because 2 divides m2 an even number of times, and it divides 2n2 an odd number of times.

Eudoxus of Cnidus pointed out in about 367 BC that, although {sqroot 2} could not be represented exactly by any one fraction, it could be represented as a limit of a sequence of fractions. Thus, {sqroot 2} can be represented in the form of an infinite decimal: {sqroot 2} = 1.4142 . . . ; this amounts to specifying {sqroot 2} as the limit of the sequence of decimal fractions.

Excerpt from the Encyclopedia Britannica without permission.