In my lovely 0.999… = 1 post, I stated that you can’t divide by infinity. However, it’s not entirely intuitive why you can’t do math with infinity.

The real reason is that infinity is not a number. It is a concept of a quantity without a limit or an upper-bound.

But what would happen if we were to pretend infinity was a number and do math with it?

The answer: Math would break down.

Adding to Infinity

In every-day mathematics, all infinities are the same. They all represent the largest possible thing — the only number that goes on and on forever. It never ends.

This means that if we were to something simple and innocent like “infinity + 1″ we would get another number that is also the largest possible thing. Another number that goes on and on forever. Another number that never ends.

The only problem is that math just broke down. Both those two numbers go on forever. That means these two numbers are the same. Which means that adding one to infinity did absolutely nothing.

And if:
infinity + 1 = infinity
infinity + 1 – infinity = 0
1 = 0

Uh oh.

Multiplying Infinity

Multiplying infinity works on roughly the same principle as adding to infinity. You just get a larger infinity. Which is just a larger number that goes on for the same amount — forever. And with two numbers that both go on forever, neither can be larger. Thus multiplying infinity also breaks math.

3(infinity) = infinity
3 = infinity / infinity
3 = 1

Uh oh.

Subtracting Infinity

Subtracting numbers from infinity is not as intuitive as adding and multiplying, but it has the same breaking effect. Here’s the counter-intuitive statement: “If you take a number that goes on forever without a limit and subtract three, the result still goes on forever without a limit, and thus is still infinity.”

Think about it. What else could infinity – 3 be? If it’s a finite number, you should be able to write it. But you can’t.

Here’s another way to imagine it. Consider the following infinite series: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11…}
Now consider the same series with three elements removed: {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14…}

If you go to count the elements of each series, each series will go on forever. Each series will have infinite entries. And any two numbers that both go on forever have to be the same.

Thus:
infinity – 3 = infinity
infinity – 3 – infinity = 0
-3 = 0

Oh snap.

Dividing by Infinity

This is the common procedure used in the .999… does not equal 1 “argument”. They answer the question “What is 1 – 0.999…” with “1/infinity”. They say that “1/infinity” is the smallest possible number.

Another argument used by .999… = 1 deniers is that 1/3 does not equal “0.333333…” but — as they say, erroneously — “it is just slightly less than 1/3 in the same way that .999… is just slightly less than 1.” This would mean that:

1/3 = .333… + (1/infinity)

Which means that:

1 = 3/3 = 3(.333… + 1/infinity) = .999… + 3/infinity

But what is 3/infinity? Is it greater than 1/infinity? If so, then .999… and 1 differ by more than just 1/infinity, they differ by 3/infinity. This means the gap isn’t as small as you can possibly get. If the numbers are equal, then 1/infinity is behaving exactly like the number 0.

If not, then:
3/infinity = 1/infinity
3(infinity)/infinity = 1(infinity)/infinity
3 = 1

Or, as I like to say — math sucks when you try to use infinity as a number.

And therefore, infinity is not a number.

So ha!

Be Sociable, Share!

•