Harmony in Mathematics
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An Application
Table of Contents
Quite apart from its natural and historical interest, and its relevance to music, the Harmonic Mean pops up in some other interesting places. One application involves its use in approximating roots.
If we wished to find the square root of a, begin by letting a1 be a first approximation (say a12 < a). Then a/a1 = b1 will be an approximation such that b12 > a.
i.e. a12 < a < b12 Then if a2 is the arithmetic mean of a1 and b1, and b2 is the harmonic mean of a1 and b1, then
b22 < a < a22 That is, a2 and b2 will be better approximations by excess and defect.
This process is actually easier than it sounds. For example, to find approximations to the square root of 10:
Let the first approximation be 3.1 (3.12 < 10).
Once it is noted that the harmonic mean is 10 times the reciprocal of the arithmetic mean, calculations are even simpler. The sequence thus produced converges very rapidly, producing an accuracy of eight decimal places after only two applications. As a teaching tool, not only does it provide much-needed practice in the use of fraction operations, but allows students to calculate roots more accurately than the calculator, taking much of the mystery out of such numbers.
(Note that our Mean Calculator makes this process even easier! To estimate the square root of 2, for example, begin with a reasonable guess (say 1.4) and let this equal a. Then let b equal 2 divided by this estimate. The Geometric Mean actually gives the value for the square root of two, as accurately as your browser can calculate it. The other two values approximate it.)
Conclusion
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The Harmonic Mean provides an example of a topic in Mathematics which is rich in content and application. It is a unifying concept, linking the various branches of Mathematics, and so allowing students to better perceive of the discipline as a whole, rather than a series of unrelated parts.The historical and practical applications of the topic, when linked with the valuable manipulative practice which it affords in fraction work, algebra and geometric construction, make it potentially a motivating and worthwhile mathematical experience for students at various levels, from junior secondary through to the senior school. Such topics offer much which will enrich the teaching of mathematics for those willing to explore the possibilities.
References
Bell, E. T. (1937). Men of mathematics. Simon and Shuster, New York.
Boyer, C. B. (1968). A history of mathematics. John Wiley and Sons, New York.
Brice, J. (1990). Ropestretchers: Maths in their eyes. Longman Cheshire, Melbourne.
Eves, H. (1953). An introduction to the history of mathematics. Rinehart and Co. Ltd., New York.
Koestler, A. (1959). The sleepwalkers. Penguin Books. Harmondsworth, Middlesex.
National Council of Teachers of Mathematics (1969). Historical topics for the mathematics classroom (Thirty-first Yearbook). NCTM, Washington.
And some related links...
For comments & suggestions, please e-mail Steve Arnold.
