We are all familiar with the average, or arithmetic mean, of two numbers. Less frequently used is the notion of the geometric mean. In “Geometric Meaning in the Geometric Mean Means More Meaningful Mathematics” in the March 2001 issue of the Mathematics Teacher, Matt E. Fluster shows how the geometric mean, s = _ab, of two positive numbers, a and b, can be used in a first-year algebra course to tie together geometric, algebraic, and computational investigations. In this article, I add a bit of history and an example suitable for more advanced courses. The example uses the geometric mean to compute square roots with Newton's method and does not require calculus. The history surrounding these concepts provides an opportunity to point out the universal nature of mathematics as a human activity. The basic ideas in this article were alive in the minds of ancient peoples who lived in what are now India, Pakistan, Iraq, and Egypt.
Students who have grown up with computers and calculators may take these tools' capabilities for granted, but I find something magical about entering arbitrary values and computing transcendental functions such as the sine and cosine with the press of a button. Although the calculator operates mysteriously, students generally trust technology implicitly. However, beginning trigonometry students can compute the sine and cosine of any angle to any desired degree of precision using only simple geometry and a calculator with a square root key.
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