22nd August 2011 - How Infinity Messes Up Stuff

I just love this video. It's caused me much amusement for me in tormenting my friends so I thought maybe I should write about it.

On Minute Physics, Henry Reich 'proves' that

 1+ 2 + 4 + 8 + 16+... = -1.

You can see how in the video so I'll just let you watch it before proceeding.

Now, according to the mathematics you're taught at school, every step is correct yet something seems terribly wrong here. How can you be adding positive numbers and end up with a negative number?

Let's look at what he does. 

1. He multiplies by 1 which doesn't change anything.
2. He rewrites 1 as (2 - 1) which is also fine since 2 - 1 = 1.
3. He applies the distributive law. The distributive law says that (a + b)c = ac + bc and a(b + c) = ab + bc). For example, (3+5)x2 = 3x2 + 5x2. If you check this, you will see that (3 + 5)2 = 8 x 2 =16 and also 3x2 + 5x2 = 6 + 10 = 16. To apply this to the example, he does 2(1+2+4+8+...) - 1(1+2+4+8+...) and then applies it again to multiply each number inside the bracket by the number outside to get 2 + 4 + 8 + 16 + ... - 1 - 2 - 4 - 8 - 16 - ...
4. He cancels all the terms and is left with -1.

So where is the flaw? It's in step 4 because when we deal with infinity strange things happen, and the distributive law doesn't work. The method in itself is a useful mathematical tool, as we'll see below, but unfortunately it doesn't work when we are dealing with infinite series.

Let's look at how it should work. What we will do is use another mathematical trick, and by trick I don't mean something bad or deceitful, but something clever. We will look at a finite series which has N+1 terms, and then we will let N get as big as we want, infinite in fact, and see what happens. This will show us how infinity is confusing us in the video above. We're going to follow the Minute Physics method 

Now, as we're doubling each time, so we have

1st Term: 1 = 2⁰

2nd Term: 2 = 2¹

3rd Term: 4 = 2x2=2²

4th Term: 8 = 2x2x2=2³

5th Term: 16 =2x2x2x2=2⁴

Notice that the power of 2 for each term is one less than its position in the series. This means for the N th term, where N is some number (we don't care what number it is at the moment, just some large positive whole number.)

N th Term: 2x2x2x..x2=2^N-1.

Right, back to the method given in the video, but this time for a finite series.

1 + 2 + 4 + 8 + ...+2^N-1 = 1(1 + 2 + 4 + 8 + ...+2^N-1)

= (2-1)(1 + 2 + 4 + 8 + ...+2^N-1)

= 2(1 + 2 + 4 + 8 + ...+2^N-1)-1(1 + 2 + 4 + 8 + ...+2^N-1)

= 2 + 4 + 8 + ...+ 2^N-1+2^N - 1 - 2 - 4 - 8 - ...- 2^N-1

= 2^N - 1.

Well, that means that 1 + 2 + 4 + 8 + ...+2^N-1= 2^N - 1, which is quite a nice formula but how does it help us with the infinite series in the video?

What we'll do now we've got this nicely written down is make N get bigger and bigger so it tends to infinity. That means the left hand side becomes 1 + 2 + 4+ 8 + ... where the dots mean carry on forever. What happens to the right hand side? Since N tends to infinite, then 2^N tends to infinity even more quickly, so the right hand side tends to infinity. This means that the sum 1+2+ 4+ 8 + ... tends to infinity too, and most definitely not -1.