Date: 12/19/96 at 18:48:09 From: David Gloeckner Subject: reciprocals Help! I'm home-schooling my 12-year-old and we're reviewing multiplication and division with fractions. He asks, "Of what use is the idea of a reciprocal?" Thanks for any help you can offer. Ken's befuddled mother, Carolyn Gloeckner
Date: 12/21/96 at 04:48:03 From: Doctor Pete Subject: Re: reciprocals Hi, The reciprocal is really little more than division in disguise. As you may know, the reciprocal of a number is 1 divided by that number. So the reciprocal of 5 is 1/5, for example. (Does 0 have a reciprocal? 1?) When students are introduced to division, they are not immediately told that division is really the same thing as multiplication; that would be somewhat confusing, since the methods are different (e.g., finding 375/5 is quite different from finding 15 x 5, although they both equal 75). But they are indeed the same operation; and the concept that explains this fact is the reciprocal. Division by a number is the same as multiplication by its reciprocal. To emphasize this, mathematicians sometimes refer to the reciprocal of a number as the *multiplicative inverse*. This is quite important, because this phrase describes one of the fundamental propeties of reciprocals, which is that the product of a number and its reciprocal is always 1. 1 is the *multiplicative identity*, the number which when multiplied to any other number, does not affect it. Hence the name "multiplicative inverse." The notion of reciprocals, then, explains the oft-quoted rule "invert the divisor and multiply." For example, 12/5 ------ = 12/5 x 10/3 = 4/1 x 2/1 = 8. 3/10 When first learning this rule, many students learn the rule but not its basis. The reason why it works is simple - it is simply the application of the idea of the multiplicative inverse. To see what's going on, there is a simple intermediate step I can insert in the above: 12/5 12/5 10/3 12/5 x 10/3 ------ = ------ x ------ = ------------- = 12/5 x 10/3 3/10 3/10 10/3 1 Clearly, any number divided by itself is 1. But what value do we choose for the numerator and denominator? We choose the unique value that when multiplied by the denominator, 3/10, gives 1. This is precisely the reciprocal, 10/3. What is interesting about all of this is that it is not unfamiliar, since addition and subtraction have similar properties. While 1 is the multiplicative identity, 0 is the additive identity (0+n = n for any number n). Then the *negative* of a number is to addition as the reciprocal is to multiplication. So the negative of 5 is, of course, -5, and their sum is always the additive identity 0. Naturally, "negative"'s synonym is "additive inverse." Hence the relation to multiplicative inverse. However, this analogy is not perfect, and the only place where we need to be careful is when the reciprocal is undefined. In particular, 0 has no reciprocal, whereas any number has a negative. (What is the negative of 0?) Suppose 1/0 was indeed some number. Call it n. Then 1 = 0 x n, but 0 x n = 0 for every n, so 1 = 0, a contradiction. Similarly, if we suppose 0/0 = n, then 0 = 0 x n = 0, but this is true for *any* n we choose, so n is not unique, and it makes no sense to assign it any one particular value. Hence division by 0 is undefined. There is no such exception for addition. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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