The easy way to "prove" that zero is one and -1 is infinity

So you've been set a task: prove something which is clearly not true. That sounds much tougher than it is. All you need is basic math skills, an infinite series of numbers, and a playful spirit. Take a look.

We're going to look at how easy it is to "prove" impossible things if you can write a few whole numbers, juggle them around, and add them together. To be fair, there need to be a lot of numbers for you to be able to do this, but you don't have to write them all. You can just write a few terms in a series, an infinite series, and people will get the idea.

For example:

1/2 + 1/4 + 1/8 + 1/16 + . . .

That series will go on to forever, and you understand how it will go. You also understand what the sum of all those infinite terms will be. As the number of terms increase, the series will add up to a number that gets closer and closer to one. Because it adds up to a definite number, it's called a series that displays "absolute convergence." It converges on a single, unchanging number.

Not all infinite series will do the same. And if you work with series that don't absolutely converge, you can use simple tricks to prove all sorts of things. For example, let's look at a series that proves that one is equal to zero.

1 - 1 + 1 - 1 + 1 - 1 + . . .

Adding and subtracting one is the simplest possible thing to do in math, so how is this series going to prove an impossible thing? Well, first let's say the series adds up to a number that we'll call N. Then, let's group the numbers just the right way.

1 - 1 + 1 - 1 + 1 - 1 + . . . = N

(1 - 1) + (1 - 1) + (1 - 1) + . . . = N

All I've added are parentheses, and those don't make a difference if all we're doing is adding a series of terms, but this makes the explanation of the series obvious. All of those grouped (1 - 1) terms are equal to zero, and even if you add an infinite number of zeroes together, you'll get zero.

1 - 1 + 1 - 1 + 1 - 1 + . . . = (1 - 1) + (1 - 1) + (1 - 1) + . . . = (0) + (0) + (0) + . . . = 0 = N

That's simple enough. But let's look at another way to group the pairing.

1 + (-1) + 1 + (-1) + 1 + (-1) . . . = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) . . . = 1 + 0 + 0 + 0 . . . = N

Since N is equal to N, and both zero and one are N, then one is equal to zero. Here's an even simpler one.

1 + 1 + 1 + 1 . . . = M

It's pretty obvious that M is infinity, right? Well, what if we change it around a little? What if we look at terms that are equivalent to one? (-1 + 2) is equal to one, right? And (-2 + 3) is also equal to one. Here's another pattern you could use to write this series:

(-1 + 2) + (-2 + 3) + (-3 + 4) + (-4 + 5) + . . . = M

This can be re-written like this:

-1 + (2 - 2) + (3 - 3) + (4 - 4) + . . . = M = -1

In this way, we've "proved" M is equal to infinity and M is equal to negative one. So negative one is equal to infinity. Meaning you only have to go slightly into debt to have an infinite amount of money.

[Via Magnificent Mistakes in Mathematics]