68 Multiplication and Division

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you want to get, say, 3 How many times must you add 0 to itself to get anything but 0 No integer can do this trick In fact, no known number solves this problem Most mathematicians will tell you that division by 0 is not defined That s what my 7th-grade math teacher kept saying, and I pestered her about it I would ask, Why not or retort, Let s define it, then! She would repeat herself, Division by 0 is not defined I did not take her seriously, so I started trying to make division by 0 work I never came up with a well-defined way to do it But I came pretty close, and had a lot of fun trying

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Have you heard that dividing a positive integer by 0 gives you infinity If not, you probably will some day Be skeptical! The first thing you must do to figure out if it s really true is to define infinity That s not easy No meaningful, enduring definition of infinity produced by mathematicians has ever had anything to do with division by 0 Again, look at the problem inside-out If you want to multiply 0 by any positive integer n, you must add 0 to itself (n 1) times No matter how large you make n, you always get 0 when you add it to itself (n 1) times Why should adding 0 to itself forever make any difference It s tempting to suppose that it might, but that doesn t prove that it will In mathematics, we need proof before we can claim something is true!

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Manipulating equations Whenever we add or subtract a certain quantity to or from both sides of the equation, we still have a valid equation The same is true if we multiply both sides of an equation by a certain quantity, or divide either side by a certain quantity other than 0 The quantity you add, subtract, or multiply by can be a number, a variable, or a complicated expression, as long as it is the same for the left-hand side of the equation as for the righthand side If you divide both sides of an equation by anything, it s best to stick to nonzero numbers If you divide both sides of an equation by a variable or an expression containing a variable, you can get into trouble, as you ll see in Chap 11 Keep these rules in mind That way, you won t get confused later on when we do something like divide both sides of an equation by 999, or multiply both sides by (a + b) The fact that we can do these things makes solving equations and proving various facts far easier than they would be otherwise

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In terms of the number line and displacements, show what happens when you multiply the integer 1 over and over, endlessly, by 2

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Figure 5-3 illustrates this process Because the multiplier is negative, we jump to the opposite side of the number reflector each time we multiply Then the result becomes a new multiplicand Because the

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Moving Out and In

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To 32 8

Start at -1 -4

Figure 5-3 Here is what happens when we start at

1 and multiply by 2 over and over We jump back and forth across the number reflector, doubling our distance from it with each jump To avoid clutter, only the even-integer points are shown on the number line

absolute value of the multiplier is 2, we double our distance from the number reflector with each jump Expressed as equations, we have 1 ( 2) = 2 2 ( 2) = 4 4 ( 2) = 8 8 ( 2) = 16 16 ( 2) = 32 and so on, forever