numbers, maths, infinity

Infinity and Incomprehensible Numbers

Counting To Infinity
No one can truly fathom the concept of infinity. Well actually you can fathom it, all day long if you like, but that doesn’t mean you’ll ever fully grasp it, or that it’s even possible to.

No one, not even the greatest scientific and mathematical minds, truly understands infinity.  I prefer to think of it as a concept rather than a number, though given I’m no mathematician, my preference is of little consequence.  It is interesting though to ponder the largest numbers and the subject of infinity.  So let’s have at it.

A Mere Trillion?  Now That’s Thinking Small

The largest numbers we’re familiar with in terms of everyday life fall into the range of millions, billions and trillions.  In terms of money, there’s about 400 billionaires in the US.  Not too shabby.  But this isn’t about money or income inequality, so let’s explore a bit farther along the spectrum of numbers, to places that make a trillion, let alone a billion, look like pocket change.

A Googol

Many people will be familiar with the number googol. It’s a one, followed by one hundred zeroes.  Now we’re just starting out, but to give you an idea of just how incredibly large this number is, if we were to take all the atoms in the observable Universe, they would not even begin to approach a googol.  That’s rather startling.  There’s trillions upon trillions of atoms in the human body.  There’s 7 billion humans on this planet.  Of course the amount of atoms, added up for every human, pales dramatically in comparison to the amount of atoms on this Earth.  Then there’s Jupiter, where we can fit one thousand Earths into it easily.  Then there’s the Sun, far vaster still.  You get the point.

That’s just a surface glance at our solar system.  We know there’s roughly 1 billion stars in each galaxy, and roughly 100 billion galaxies in the observable Universe.  It seems hard to fathom that all these atoms don’t add up to a googol, but they don’t.  Not even close.

Then there’s a googolplex.  Such a number isn’t written out on a chalk board.  In fact, if you stretched the Universe into one giant chalk board, you can’t fit enough zeroes on it to write it.  Instead mathematicians would write 1010100.

At this point, let’s take a break to listen to Carl Sagan talk about these particular numbers.  I can guarantee he was more familiar with them than I.

There is a number that makes a googolplex appear Infinitesimal by comparison, as absurd as that might seem.  This is called Graham’s Number.  Graham’s number is named after the mathematician who used it to help solve a mathematical problem in the Ramsey Theory.  The math involved in this number is so far beyond my comprehension that I will not even attempt to tackle it.  I can tell you that the last ten digits of it are …2464195387, but the formulas necessary for that proof to be derived at leaves no simple way to write it out, or to express it with words.  If you want to know about this number, you’ll have to click the provided link above, and watch the below clip.

To Infinity and Beyond

Now onto infinity, which of course cannot be expressed with numbers.  Instead the symbol ∞ is used.  You can fit an infinite amount of Graham’s Numbers into infinity.  It boggles the mind, vexing the brightest mathematicians that have ever lived.  You can add one to infinity, and you still have infinity.  You can add an infinity to infinity, and again, you have infinity.  In fact you can have infinite infinities, and you remain with infinity still.

There are astrophysicists that believe our Universe may be infinite – we know about the observable Universe, but there are galaxies moving away from us faster than the speed of light, and thus are lost to us completely, and so we have no way of knowing just how far the bounds of our Universe traverse.  Our Universe, even if it is infinite, might be simply one infinite Universe in an infinite Multiverse.

In such a scenario there are an infinite number of you living out an infinite number of existences – all possible choices that would have lead you down varying paths are played out, infinite times.  If this seems wildly preposterous don’t worry, no one can really fathom the depth of such possibilities to the fullest extent.

Hilbert’s Paradox of the Grand Hotel

Imagine an infinite hotel with an infinite amount of rooms, and all of them are full.  You go to check in, and low and behold, they’re able to make space for you.  In order to do so they move the guest in room one to room two, the guest in room two to room three, and so on and so forth.

Stranger still, you can show up with an infinite amount of friends, and that’s no problem either.  They simply move the guests from the even numbered rooms over – guest in room two to room four, guest in room four to room six, and so on and so forth, and suddenly there’s enough space for all of your infinite friends.

I would hate to work at this hotel, it’s an administrative nightmare.  I’d hate to stay in it too, because what a pain it would be, always being asked to move rooms.  It would be a crappy hotel to have any association with, no matter how infinitely impressive.

The below clip is an hour long BBC special on the topic of infinity.  It will cover in depth many of the ideas expressed in this article.  If your attention and interest has been captured, please enjoy.  If your brain has had enough for the time being, I fully understand.

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