Why Simplify Square Roots?Date: 05/17/99 at 10:38:09 From: John Cendrowski Subject: Simplifying square roots--WHY? In a discussion with colleagues, the question arose regarding the rationale behind teaching simplification of radicals. We suggested addition, simplifying roots, and working with complex numbers. My colleague is still not convinced. He says students think of root 32 as a number between 5 and 6, not as 4 root 2. All three of us have been trained to simplify square roots, but we are now looking at the reasons why we should continue to teach the process. Thank you. Date: 05/17/99 at 13:28:12 From: Doctor Rick Subject: Re: Simplifying square roots--WHY? Hi, John. Thanks for the question. For some purposes - when all you want is a numerical answer to a single question - it may be sufficient to punch 32 into the calculator and see that it's 5.656854249. For some purposes, the most important thing about this number is that it lies between 5 and 6. But the real power of math - the power for understanding - is in its exactness. If you look at math history, you will find that the real innovation in Greek mathematics, compared with earlier Babylonian and Egyptian math, was that they made a distinction between approximations and exact results. The fact that two numbers are approximately equal, even to 9 decimal places, doesn't have any profound meaning. The number 5.656854249 doesn't mean anything to me, and in fact it ISN'T anything particular - just an APPROXIMATION to the square root of 32, to an arbitrary 9 decimal places. On the other hand, 4*sqrt(2) is an EXACT answer, with infinite precision; one can tell whether another number is EXACTLY equal to it. The expression can bring specific mathematical associations to my mind - for instance, it could be the diagonal of a square of side 4. Also, if the answer is not correct, I can tell a lot more about the nature of the student's error from the radical form than I could from a decimal. Thus, there is real mathematical benefit in retaining the exact value of an expression. For irrationals, this is only possible if a radical is retained in the expression - any finite decimal, which is necessarily a rational number, will be only an approximation. Simplification of a radical has the same value as reduction of a fraction to lowest terms. It permits us to recognize that two radical expressions are equivalent, even if their original forms appeared quite different. In particular, it allows a teacher to tell at a glance whether the answer is correct. I hope this gives you a rationale for passing on your training to the next generation. In the computer generation, much can be done with numerical methods; but real UNDERSTANDING of a problem calls for exact answers. Exact answers call for retension of radicals (along with pi and e), and recognition of a particular radical expression calls for simplification. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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