Thursday, February 28, 2008

The Log-Antilog Procedure: Part II

Not too long later, I learned in additional mathematics, certain trigonometric identities of the form cos(A ± B) = –(cos A cos B ± sin A sin B), from which we can form another identity: cos A cos B = ½ [cos(A + B) + cos(A – B)].

This identity is the basis of an obsolete technique known as prosthaphaeretic multiplication, which is very similar but older than the log-antilog procedure taught by my teacher. To illustrate prosthaphaeretic multiplication on 10.8 x 87.85, we proceeds as follows: First write 10.8 x 87.85 as 0.108 x 0.8785 x 10000, then read the arccosine values for 0.108 and 0.8785 from table, they are 1.463 and 0.498 in radians. With the identity, multiplication of two cosines is converted to addition/subtraction of their respective arccosines, that is,

10.8 x 87.85 = cos 1.463 x cos 0.498 x 10000 = ½[cos(1.463 + 0.498) + cos(1.463 - 0.498)] x 10000

The result is 945, which is very close to the exact solution: 948.78.

At the time, I thought, the only sensible reason for retaining the log-antilog procedure in a modern mathematics textbook in the year when we were building the Petronas Twin Towers was that the syllabus designers want us to appreciate the hardship and tortuous path gone through by astronomers and arithmeticians a few centuries ago when they need to multiply numbers.

However, if they want us to fully appreciate how mathematics is done a few centuries ago; they should put the prosthaphaeresis in the textbook. The real reason, as I now believe, is because the syllabus designers are insensitive to technological environment which is ever changing. One or two decades ago, when electronic calculators became cheap and popular, they should have reformed the curriculum and remove the obsolete techniques like the log-antilog procedure.

Luckily, our syllabus designers have now changed their minds. The modern math syllabus was reformed a few years ago and the log-antilog technique for performing multiplication is now removed. However, the harm cannot be undone, as we students could have spent our time more productively a decade ago.

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