What is a Reciprocal?Date: 10/10/2002 at 11:44:59 From: George Best Subject: Nature of the Reciprocal Dear Dr Math, I was thinking about reciprocals because they seem to pop up in a lot of areas of everyday mathematics, physics, and electronics. I was wondering about their nature. I am trying to formulate a rule in my head as to what they do. Dividing something into 1. What exactly am I achieving by this in plain English? Here are my thoughts so far: 1 The Whole - = --------------- = How many parts you would need to x The Part make a whole. Is it some kind of ratio or the inverse of its fraction? Can you clarify what the reciprocal is? George Date: 10/10/2002 at 13:06:40 From: Doctor Peterson Subject: Re: Nature of the Reciprocal Hi, George. The reciprocal is also called the multiplicative inverse, which means it is what you multiply by to undo a multiplication (that is, to divide). So 1/2 is the number you multiply by in order to divide by 2; that makes it the multiplicative inverse of 2. And of course, you can find the reciprocal of a number either by just dividing 1 by the number, or by writing the number as a fraction (2/1) and flipping the fraction over (1/2). Similarly, the negative of a number is the additive inverse: the number you add in order to subtract your number. The additive inverse of 4 is -4, since if you add -4 to something, it is the same as subtracting 4. Now these two concepts have a lot in common. The negative can be thought of as a reflection in a mirror placed at the origin: | | <-----------2---1---0---1---2----------> | | mirror The negative is on the opposite side of the mirror and equally far back, just as your image in a mirror is opposite to you. The multiplicative inverse can also be seen as a sort of reflection, but in a curved mirror centered at 0, passing through 1 and -1: | | | ....+.... ... | ... . | . . | . . | . <----------+---------+----o----+----------o--> -1 | 1/2 1 2 . | . . | . ... | ... ....+.... | | | This reflection makes very distant objects look as if they are near the center of the mirror: the reciprocal of 100 is 1/100, and the reciprocal of 1,000,000 is 1/1,000,000. You can imagine that the image of "infinity" is right at the center, at 0. Numbers at the mirror's surface, at 1 and -1, are left unmoved: the reciprocal of 1 is 1. Remember also that the reciprocal of the reciprocal takes you back where you started; the reciprocal of 2 is 1/2, and the reciprocal of 1/2 is 2. So the reciprocal turns the number line inside out: everything farther than 1 unit from the origin moves inside, and everything inside goes out. This is called "inversion" in geometry, and is a very useful concept. (I should mention that a real circular (or cylindrical) mirror would not do exactly this; the mirror idea is just a useful way to picture what is happening.) 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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