Thursday, January 30, 2014

To Infinity...

...and beyond!

Do you spend much time thinking about infinity?

Probably not since you are wasting perfectly good infinity-thinking time reading this tripe. And ponder this: you don't have an infinite amount of time to ponder infinity. Due to your lack of infinity-thinking you probably just think of infinity as a lazy eight all laid over on it's side. Might make you think $$ \infty < 8 $$ or at least that infinity is somehow inferior. Perhaps inebriated.

If you think much about infinity then like most folks you may think of infinity as the largest number there is which is not a bad definition at times. Suppose you think of a large number, larger than you ever thunk before and you can give it a name. That number still isn't infinity because infinity is the largest there is and is therefore larger than yours. In fact, it is larger than yours plus one. Or wrap your noggin' around this: infinity is larger than your largest-ever number PLUS INFINITY! In fact infinity is bigger than infinity plus infinity. All because infinity is the largest number there is and like the national debt it just keeps getting bigger and just when you think it is getting big as fast as it possibly can it starts getting bigger even faster.

Now we just bounced around some quizzical ponderings about adding a couple of numbers and comparing to infinity to get you out of that self-absorbed all-about-Dunwoody mind-trap you've been stuck in and get a few of those remaining synapses fired up. Now it's time to get a bit more formal.

Now let's be clear before this goes any further. We're not talking arithmetic here--this will tread dangerously close to math. You know. That shit you forgot. And this is not modern math--the kind that seems to always involve toothpicks or pizza--this is that classic stuff with the Greek letters and what not.

And this is not the kind of question that you're kid is gonna see on the CRCT. No siree. Nothing at like:
If Johnny has 5 reefers and gives 3 to Suzie for a peek down her sweater, what can we say about Johnny?
  1. Johnny's gonna score with Suzie.
  2. Johnny is hanging with the wrong crowd.
  3. Johnny has 2 spliffs left.
  4. Johnny IS the wrong crowd. 
  5. All of the above.
Nope. This problem has fewer and much simpler answers. In fact, it only has one answer. And the question is "What is the sum of all integers greater than or equal to one?" Or, to get your geek on:
$$\sum\limits_{i=1}^\infty i = ? $$
While all the round-eyes are headed down to Kroger to buy more toothpicks before going all Rain Man on us let's get all the Asian kids to put down their violins and talk this thing thru.

First off we know the answer just has to be big. After all we start at 1 and add every other number on top of it so how can it not? We've already learned it can't be bigger than infinity--or can it?

This really makes your head hurt.

We're adding up an infinite number of numbers all of which are greater than one and the answer cannot be greater than infinity because infinity is the biggest number there is or ever will be but that is also the number of numbers we're adding up. Can it also be that the infinite sum of numbers greater than one cannot grow faster than infinity itself? Is this the mathematical equivalent of the physics conundrum of two objects approaching one another, each traveling at three quarters the speed of light and yet each sees the oncoming speed as less than the speed of light except that in this case we're talking about infinity and infinity is actually accelerating? Can we at least agree that infinity has no mass?

So if the sum of an infinite number of numbers greater than one cannot be infinity then surely it must be something else. If it is indeed less than infinity what other number could it be less than? If it is less than one other number can it be less than others as well? Could an infinite sum of numbers itself be less than an infinite number of numbers? But since we're adding integers then surely the sum of integers must also be an integer otherwise our entire understanding of the universe would collapse--dogs would meow, cats would bathe, pigs would fly and Democrats would pay taxes. The world as we know it would be over.

And so it is.

The answer to this infinite sum of integers is indeed less than a well known though relatively recently discovered number--zero. And just for fun it is a fraction as well.

Spoiler alert!

The answer is:
$$\sum\limits_{i=1}^\infty i = -1/12 $$
Don't believe it? Why, because half the shit you read here is made up? Fine. Check this out:

Of those who actually watched this video some probably have the same feeling you get when you see David Copperfield make the Eiffel Tower disappear--it looks believable but deep down you just know it isn't true. The rest are mumbling to themselves that one sixth really is bigger than one fourth and they knew it all along.

And the math world is all a-flutter as well but not for the reason you might think. Turns out the answer is correct but there are those who disagree with the method.

But given this answer perhaps the best way to look at this is if you had a dozen eggs and Johnny steals one this number represents the egg Johnny took.