Rationalizing the DenominatorDate: 04/09/97 at 12:35:50 From: Katrina Subject: Fraction Denominators Why must all fractions have denominators that are rational numbers? Date: 07/22/97 at 00:31:57 From: Doctor Terrel Subject: Re: Fraction Denominators Katrina, That's a good question! If we have a fraction like 7.5 ----- 9.6 there's nothing really wrong with leaving it that way. It depends on where the fraction came from, what it means, and where we want to go with it. You see, fractions (in fact, most numbers), are just "things" or "mathematical ideas" until we give a meaning to them. Often they represent a ratio of two quantities, as opposed to the elementary idea of "parts of a whole." So in general, I'd say: don't worry about any requirement that the denominator be a rational or even a whole number until you know the purpose for using that fraction. However, I do recall from some of my earlier math courses that we used the phrase "rationalizing the denominator," and I think this may the reason you asked the question. You see, in trigonometry or algebra II we often had to simplify or even evaluate expressions such as: 13 300 -------- and --------------- sqrt 3 sqrt 5 - sqrt 2 Unless we do something tricky, we have ugly, LONG divisions on our hands, namely, 13 300 300 ------ = ? -------------- = ------- = ? 1.732 2.236 - 1.414 0.822 Of course, sqrt 3, sqrt 5, and sqrt 2 were always available in tables. But believe me, in the old days (B.C. = before calculators), long division was not enjoyed by many individuals. Instead, we were taught to "rationalize the denominator" like this (for the first example): 13 sqrt 3 13 (sqrt 3) 13 x 1.732 ------- x ------- = ------------ = ---------- sqrt 3 sqrt 3 (sqrt 3)^2 3 Because the rest does not involve a multi-digit divisor, I leave it to you to finish. The other problem is similar, but a little different. 300 sqrt 5 + sqrt 2 300(2.236 + 1.414) ---------------- x ---------------- = ----------------------- sqrt 5 - sqrt 2 sqrt 5 + sqrt 2 (sqrt 5)^2 - (sqrt 2)^2 300 x 3.650 300 x 3.650 = ------------- = -------------- = 100 x 3.650 = 365 5 - 2 3 I hope these two examples show that in the B.C. days we could calculate some pretty mean expressions, if we could make the denominator somehow simpler. Here the importance of the "rational denominator" was that the evaluation by long division was made quite a bit easier. Nowadays, if you need those values (as an engineer or architect might), a calculator does it directly, quickly, and more accurately. Here are your key sequences for a simple, non-scientific calculator: 1) 13 [/] 3 [sqrt] [=] 2) 5 [sqrt} [M+] 2 [sqrt] [M-] 300 [/] [MR] [=] By the way, could you reduce my original fraction to lowest terms, as a favor to me? Thanks. -Doctor Terrel, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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