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© Copyright 1997, Jim Loy
You may have seen this little proof that 2=1:
a = x [true for some a's and x's]
a+a = a+x [add a to both sides]
2a = a+x [a+a = 2a]
2a-2x = a+x-2x [subtract 2x from both sides]
2(a-x) = a+x-2x [2a-2x = 2(a-x)]
2(a-x) = a-x [x-2x = -x]
2 = 1 [divide both sides by a-x]
You may doubt that 2=1. So, where is the mistake? Think about it.
You may not like the first step (a=x). But, we do this kind of thing all the time in Algebra. It's true for plenty of a's and x's. Assume that a is the number of ears on my head, and x is the number of ears on your head. In that case a=x (if a is not equal to x, forgive me for mentioning it).
Anyway, all of the steps are perfectly legal except for the last one, dividing both sides by a-x. What is a-x? Well, a=x (step 1), so a-x=0. In the last step, we divided by zero. That's not allowed. And this puzzle is a good example of why it is not allowed.
What's 1/0? Infinity, right? We said above that we can't divide by zero. But, can't we divide by zero, if we're careful? Let's look at 1/0, more closely. In Calculus, we deal with problems like this by using limits. In other words, we don't look at 1/0, we look at 1/x (the graph of y = 1/x is shown on the left) when x gets close to zero. Well, when x gets close to zero, 1/x gets very large without bounds, it is infinity. Not so fast, x also gets close to zero on the negative side. Then 1/x becomes a very large negative number, without bounds, it is negative infinity. So, the answer to the question, "What is 1/0?" is "plus-or-minus infinity." Kind of a wild answer, isn't it? It is not exactly simple.
So, an apparently simple situation like 1/0 blows up in our faces. That's another good reason for not allowing division by zero.
Infinity is not a number, anyway (not in arithmetic or algebra, see Transfinite Numbers). We can add infinities, and multiply them (sort of). But, we don't get bigger infinities, we get the same infinity. That's interesting. But, if we multiply infinity to both sides of an equation, we are in big trouble. It is the same as dividing by zero. In our little puzzle, when we divided both sides by a-x, that was the same as multiplying both sides by infinity. It is meaningless. It is not allowed in mathematics.
Arithmetic with infinity is not allowed, because infinity is not a number. And, just like our little puzzle, we get answers that make no sense. Calculus is essentially the field in which we deal with infinity and division by zero. And, we never deal directly with infinity or division by zero. We always see what happens when a number gets large without bound or gets closer and closer to zero.
Calculus is fascinating. I call it "algebra with limits." I recommend it to anyone who likes numbers.
Addendum:
I received email asking if there was an actual proof that you can't divide by zero. Well, the above 2=1 proof should do quite well, as a counter example. We divide by zero and we get the wrong answer. More simply, assume you can divide by zero. Then we start with 2(0)=1(0), which is true. Divide by zero, and you get 2=1 which is false. We have our contradiction. It doesn't work.