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.Are you confused?
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!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 compli-
cated expression, as long as it is the same for the left-hand side of the equation as for the right-
hand 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 some-
thing 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.Here’s a challenge!
In terms of the number line and displacements, show what happens when you multiply the integer − 1
over and over, endlessly, by −2.Solution
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 the68 Multiplication and Division