By R. J. Goult
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Extra resources for Applicable Mathematics: A Course for Scientists and Engineers
X . ;;;; 1. Note also that we have 2 2 2 1 sin- = +I when x = - 5rr ' 9rr' 13rr ' · · · x 2 2 2 . 1 sm- =-1 whenx=- 3rr'7rr'llrr'··· x It follows at once that given any number o > 0, however small, we can always find points x" x 2 , in 0 < x < 1 such that Jx1 -x21
3! n! The addition theorem for the exponential function then allows us to show that, for any positive integer n, exp n =en It is easy to deduce that the same result holds if n is replaced by any rational number, r = pjq. However, note that while exp xis defined even when xis irrational, this is not the case for ex. J 2 in the sense of a definition. The number e, like y2, is irrational; its decimal expansion is infmite and non-recurring. However, there is a further classification of the irrational numbers 34 APPLICABLE MATHEMATICS themselves which should be noted here.
Suppose the problem is to solve 8x 2 - 160lx + 200 = 0. The exact roots are 200 and l. The most obvious solution method is to use the formulae Xt and = 1601 + v[(l6oli- 64oo] 16 _ 1601 - v[(l601) 2 16 x2- - 64oo 1 The exact value of y[(l601) 2 - 6400]is 1599, but our simple computer with 10 binary digit accuracy will store both this number and 1601 as equivalent to 1600. Hence the answers it would produce are: x 1 = 200, x 2 = 0. The first of these is of course accurate to 3 significant figures; the second (since it involved the subtraction of two nearly equal numbers) is very inaccurate.