How Calculators Calculate

When you want to find the sine of a number, you probably type it into your calculator or computer and use the number that comes out. Have you ever thought about how the calculator/computer obtains the result? Since we are only good at computing integer powers of numbers, we have no direct way to evaluate transcendental functions, such as the exponential function or the sine function. We know that sin \(\pi =\ \) 0, but what is sin 3? Since sin 3 is an irrational number, it is represented by a nonrepeating, nonterminating decimal. So, computers and calculators work with approximations for transcendental functions that involve integer powers.

Most transcendental functions can be expressed as power series. As we shall see in Section 8.8, a power series is an infinite sum of monomials with integer exponents. Since we are good at raising numbers to integer powers, we can truncate these series and find good polynomial approximations to transcendental functions on restricted domains. This is how computers evaluate many transcendental functions.