My Favorite Number

Currently, my favorite number is -1/12 for one very particular reason.

But before you hear that reason, do me a favor and start adding together all the counting (mathematicians call them "natural") numbers.  You can use a calculator if you like.  There is a very particular reason for this.

But before you hear that reason, a story!

Once upon a time, there was a hotel with infinite rooms and no vacancies.  One day, a man came to the completely full hotel and asked for a room.  Now, the clerk working at the front desk was extremely clever.  He told the man that getting him a room would be no problem.  The clerk then went to the woman in room one and told her his plan.  The woman in the room agreed, packed up her things, and went to room two, sending its occupant to room three.  The person from room three then moved to room four, the person in room four went to room five, five went to six, and so on.  The clerk then showed the new guest to the newly vacant room one.

The infinite hotel paradox is fundamental in the explanation of why I have chosen my favorite number.

Now, consider the following equation:

U=1-1+1-1+1-1+ . . .

It's easy to see by looking at this that if you were to stop after an odd number of actions, U=1, and if you were to stop after an even number of actions, U=0.  What if, however, you never stopped?  What if you continued this infinitely?  What would your answer be then?

To answer these questions, I will add U to itself and use the principle from the infinite hotel paradox to shift the numbers by one position:

  U=1-1+1-1+1-1+ . . .
+U=    1-1+1-1+1- . . .

Set up this way, you can see that all of the positive 1's line up with a negative 1 and vice-versa except for the first one.  Thus, we know the following:

2U=1

Simple algebra then tells us that U=1/2.

So if you add and take away 1 an infinite number of times, you somehow end up with 1/2.  Math.

That result almost makes sense.  1/2 is the average of the two numbers the equation was bouncing between, so it seems appropriate that it would end up there.  I'm not done, though.  Consider this infinite sum:

X=1-2+3-4+5-6+ . . .

We're now going to use a similar premise to the one above.

  X=1-2+3-4+5-6+ . . .
+X=   1-2+3-4+5- . . .

If you do this arithmetic, you'll see that you end up with this:

2X=1-1+1-1+1-1+ . . .

Look familiar?  What I've just shown is that 2X=U.  We already know that U=1/2, so 2X=1/2.  Do your algebra and you'll find that X=1/4.

Alright, folks.  Here we go.  Hopefully, you've been adding up the natural numbers this whole time, because that's where we're going next.  When you get done adding all of them together, we're going to call that number N.  If you like equations better, N looks like this:

N=1+2+3+4+5+6+ . . .

Now, let's figure out what N is.  You just keep on adding so you can check my work, and I'll start subtracting X from N.

 N=1+2+3+4+5+6+ . . .
-X=-1+2-3+4-5+6- . . .

If you're still with me, you can see that all the odd numbers will disappear and all the even numbers will be doubled.  So you'll be left with this:

N-X=4+8+12+16+20+ . . .

Which some of you may recognize as this:

N-X=4(1+2+3+4+5+ . . . )

Now, notice that the right side of this equation is simply N being multiplied by 4.  We know that X=1/4, so I'm going to do some substitution and rearranging.

N-(1/4)=4N (I substituted in 1/4 and N.)
-(1/4)=3N (I subtracted N from both sides.)
N=-1/12 (I flipped the equation and divided by 3.)

So there you have it.  If you add all of the natural numbers together, the answer is -1/12.  Want to know something weirder?  Physicists have used this result in string theory.  That's right.  The people unravelling the mysteries of the universe are using the fact that 1+2+3+4+5+6+ . . . =-1/12!

No foolin'.

Results like this are the reason why I'm convinced that math is closer to philosophy than science.  In order to believe this, you have to accept that infinite sums are things we are allowed to work with.  Some mathematicians disagree.  That's when we enter this weird debate over whether or not numbers exist.  I tried to write an explanation for this debate, but these guys' videos do it so much better:

Do Numbers Exist?
Is Math a Feature of the Universe?

It ultimately comes down to this question:  Do we discover mathematics or do we create it?