Base what? How do numbers work again?
You're actually already aware of number base systems, even if you think you've never heard of this before. You're used to working in a base-10 (decimal) system, meaning you count zero through nine, then increment the next digit over and repeat.
01, 02....09, 10, 11, 12...
To change the base, you do the same thing (mostly, we'll get into the complicated in a moment). IE Binary is counted as:
001, 010, 011, 100, 101...
I'm intentionally starting with extra zero's at the beginning of these sequences to make sure you recognize those are always there, but dropped in every-day math for clarity. It's importaint to recognize this though, it makes a lot of sense when you get outside of your normal counting system.
Why are alternate number bases important? Well, the real question is why is base-10 so important? The simply answer is that we have 10 fingers (usually), so we arbitrarily emphasize such a thing. Had we been octopi perhaps octals would be all the rage. This is an introspective post, after all. Get your brain working on how you've been conditioned to be special...
So we pulled ten "things" as a core to our general math understanding/teaching for, basically, no good reason. We could have chosen any number of "things" to base this on. There's a problem though, we're still putting emphasis on our most basic- how many apples, how many fingers, how many integer items. Rather than choosing an arbitrary "unit," why not choose something less specific, but equally well defined? These are called "non-integer" bases, and I'll give you an example of my favorite: Phinary.
Phinary, or base-Phi, is centered around "the golden ratio." The golden ratio is a relationship that occurs in a lot of simple shapes like pentagons and dodecahedron, plus a lot in nature. The number, in base 10, is arrived at by solving g=1/(1+g). It's about 1.618 to 1. Try writing out the obvious repeating nature of the g=1/(1+g) on a sheet of paper, each time substituting g on the right for g on the left. You'll get an idea of what we're talking about.
So how do you represent a ratio as a system of counting? First is to realize that "numbers" are a representation of a real thing- I think binary shows this pretty well in its "on" or "off" nature. Base-10 is like this too, but slightly more complicated (think switches with positions, or something). Math in this light seems very logical and, well, rational.
Phi-base works around a non-rational number ratio, and as such it needs to be represented in a pretty strange way. As a ratio all "digits" need to be separated to address a standard form (only one representation for a single number, similar to base-10), and counting is a little less obviously linear. To do this in a semi-logical way, you create a couple rules for the problems you'll inevitably face, for example let's try some simple incrementing:
001.00 Seems pretty simple so far. Add them together.
002.00, except that 2 itself is wrong, and to translate it becomes:
010.01. It's a ratio, see? Maybe? Let's try the next one, add those two together to get:
001.00 + 010.01 = 011.01, except now you've lost your ratio representation by putting two 1's next to each other; 011 translates to 100; therefore 3 = 100.01. You've pretty much hit all the rules you run into now, so....
Can you do 4 and 5?
4 is easy, 001.00 + 100.01 = 101.01. 5 is a little harder, but same rules: 102.01, translate 2 to become 110.02, again to become 1000.1001. If you can make it to 7 or more you're doing pretty good. You'll also find you can multiply in the same fashion and use the same limited rules.
It's pretty, it's strange, but mostly it's helpful for you to get out of your assumptions about how we measure things. Give it some introspection. Imagine that this approach was the way your brain naturally worked, as apposed to having to fight yourself to translate everything from base-10.
Also imagine this is why it's not a good idea to prod Spiv when he's in seemingly deep thought, expecting to get an honest, easy answer.