Rationalizing the DenominatorDate: 05/03/99 at 20:31:23 From: Karen Subject: Fractions and square roots Why can't you have a square root in the denominator of a fraction? Date: 05/04/99 at 12:35:30 From: Doctor Peterson Subject: Re: Fractions and square roots Hi, Karen. Technically, it's not a fraction unless the numerator and denominator are integers, but I think you're probably asking why we have to simplify an algebraic fraction by rationalizing the denominator. Why aren't you allowed to leave a square root in the basement if it isn't doing any harm? It's not really that you CAN'T, but that you don't want to, or rather that it's traditional not to. In part, the concept of rationalizing the denominator is left over from the days when all arithmetic was done by hand. Which would you rather calculate: 1 1 ------- = -------------- sqrt(2) 1.414213562373 or sqrt(2) 1.414213562373 ------- = -------------- 2 2 I think you'll find the latter is much easier to do! Even with computers, I suspect the latter is likely to be more accurate. Secondly, though, it's a good idea to have a standard form in which to write expressions like this, even if you aren't going to evaluate them, because it lets you compare them more easily. Think of it as a "mug shot" at the police station. They always line you up the same way and take pictures of the front and side, so different people can be compared easily. Here, if I asked you to tell me whether 1/sqrt(2) and sqrt(2)/2 are the same, they hardly look like the same "person" - but when you make them both "turn the same way" by rationalizing the denominator, the fact that they are the same becomes obvious. The same two reasons explain why we reduce fractions to lowest terms: ease of calculation and standardization of form. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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