Introduction to Rational Numbers for Middle School StudentsDate: 12/06/2006 at 12:59:28 From: Tracy Subject: rational numbers Why do we need to know about rational numbers? Aren't all fractions rational numbers? What's the difference? Date: 12/06/2006 at 13:30:27 From: Doctor Peterson Subject: Re: rational numbers Hi, Tracy. That's a good question! Too often books seem to make it sound as if there were some special magic about rational numbers, when they are really just fractions. Well, not quite. There is something new when you talk about rational numbers, though the books might not make it very clear. When you learned about fractions, you were focusing on a way of WRITING a number, with a numerator and a denominator; you learned how to add them, multiply them, and so on. You also learned other representations, such as decimals. But all that was just learning the mechanics of working with different ways to write numbers. Now you are getting older, and are ready to dig a little deeper. You are learning not about ways to write a number on paper; you are learning about the numbers themselves! A rational number is not just a fraction; it is any number that CAN be written as a fraction, regardless of how you happen to write it. And behind this statement lies something that is really profound: that not every number can be written that way! The ancient Greeks at one time assumed that any length could be measured as a fraction--that is, you could always find a small enough unit (say, a 16th of an inch, or a 1000th of an inch) so that the length is a whole number of those units (like 3/16 or 187/1000). Then they discovered that this was not true: that there were lengths that were not fractions. In particular, the first one they found was the square root of two, which is the length of the diagonal of a one-inch square. It can't be written as a fraction; it is not a rational number. So the important thing about rational numbers is that they form a specific set that is not the same thing as the set of all numbers. You're starting to embark on a voyage of discovery in math, finding new kinds of numbers and new things to do with them; you start that voyage by studying again the kinds of numbers you are already familiar with, in order to be well prepared for what comes next. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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