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 *x*^{2} = 2 (Pythagorean theorem). No fraction
*m*/*n, *however, can satisfy (*m*/*n*)^{2}*
*=* *2; that is, *m*^{2}* *=*
*2*n*^{2} has no solution among the integers, because 2
divides *m*^{2} an even **number ** of times, and it
divides 2*n*^{2} 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.*