Thursday, November 7, 2013

Mathematical notation

One problem that I've had with mathematics is trying to figure out what all of the jargon and notation means.  You might be able to brush it off as merely shorthand, but that doesn't quite do it justice.  I don't think I'm ready to talk about math notation and vocabulary at length, but I did think of an interesting idea last week that I feel at least highlights why notation is useful.


Exponents are straightforward enough; it's just a notation for repeated multiplication.

103 = 10 ⋅ 10 ⋅ 10 = 1000

The rules you can use to manipulate exponents are also pretty simple.

xy ⋅ xz = xy+z
(xy)z = xy ⋅ z

So given an exponent we can continue to manipulate it as a number.  It doesn't become some magical box that can no longer interact with the rest of mathematics.  You can just treat it like a normal number for the most part.

Big Exponents

Now consider the following number:  10100

Let's try writing out this number by hand.

So this is pretty easy.  Although a bit cumbersome.  One followed by one hundred zeros.  It's very convenient to use 10100 instead of the monstrosity above, but I think that I could get along without using the exponent notation if I really had to. 

This number actually has a special name:  Googol.

Bigger Exponents 

Before we go even bigger, let's consider the number of atoms in the universe.  Supposedly, there are 1080 atoms in the universe.  I didn't research this number very carefully, it's just what I saw repeated once or twice with some internet searches.  The number itself doesn't matter too much; it's the principle I'm going for.  So for now let's just assume that 1080 is about right.

I'm going to stack exponents now.  It might look scary, but it's pretty simple when you think about it.  It's just giving our exponent an exponent.


Like the number before, this number also has a special name:  Googolplex.

So this might look similar to the rule given earlier ( (xy)z = xy ⋅ z ), but it's not.  The rule earlier was just decomposing (or recomposing as the case may be) factors of your exponent.  This number is an exponent that is represented as another exponent.  So what does something like this look like.  Before I was able to write out the actual number, but how do you write out the actual number when your exponent has an exponent?  I already wrote out the exponent 10100 earlier, and this time that's what our exponent is.  So I'll start by expanding the exponent part.

Okay.  Again it would be kind of pain if we had to write it this way all the time.  But not too terrible I guess.  So if googol is just a one followed by one hundred zeros (ten to the power of one hundred), then a googolplex would be a one followed by a googol zeros (ten to the power of googol).  Let's start writing then.

Or maybe not.

Earlier I mentioned that there's only 1080 atoms in the universe.  That's significantly less than a googol.  And that means that if I wrote one zero on every atom in the universe in order to write out googolplex with all of the zeros that it should have (ie googol zeros), I would need more atoms than are in the universe.

This is one of the powers that mathematical notations give us.  Exponents allow us to manipulate numbers which would otherwise be too large to be able to otherwise write down even if we had the entire universe at our disposal.