For those who aren't math geeks, those who don't want to do this stuff "just because it's cool," the big question becomes: "WHY? Why would we even bother with bases?"

Well, first of all, to be clear, we can't avoid using bases - we use base ten for just about everything. So I think you meant to ask, "Why use bases other than base ten?"

There are a few short answers for this.

For many students, having to generalize the number system into various bases helps them to more fully understand what base ten is, and how it works. The meaning and use of place value often becomes more clear after studying bases.

Computers are designed to work in base two (also known as binary). How many digits are there in base two? There are just two! Zero and one. This is perfect for computers, because zero represents "no electrical current" and one represents "electrical current." Thus, computers can store large numbers as a series of on and off switches. For example, the number 10010_binary is 2⁴ + 2 = 18.

Of course, binary is not very readable for people, because the numbers get to be very large very quickly: 11111111_binary is only 255 - imagine how many digits it would take to write the number 1,000,000,000 (it would take about 30 digits!).

So computer people decided to use base sixteen (also known as hexadecimal) because 16 is a power of 2, and therefore, it can be used to compress a base two number. Four digits of binary can always be written as one digit of hexadecimal. This became a convenient way to write larger binary numbers without taking up a lot of space, or risking leaving out a digit from a very large number.

For example, 1111111111111111_binary = FFFF_hexadecimal

The hexadecimal value is not only quicker to write, but it's also easier to wrap your brain around; most computer programmers will immediately recognize that value, but would have to stop and think about the binary version, and count the digits. To shorten it even more, programmers will use a short-hand to avoid writing the word "hexadecimal" - if you see something like &FFFF, or 0xFFFF, or FFFFh, those are all ways of indicating the number should be understood as hexadecimal, or base sixteen.

Computer programmers occasionally use base eight, or octal, because it's another way of shortening binary values. This gives rise to the awful math joke: Why is Halloween the same as Christmas? Because Oct 31 = Dec 25 (31 Octal = 25 Decimal).

Finally, it's worth mentioning that you - on a regular basis - use a sort of "hybrid" base created by the ancient babylonians. They didn't have sixty different symbols, but they did work in base 60. And we do too, every time we look at a clock.

2:15:35 means 2 x 60² + 15 x 60 + 35 = 8,135 seconds.

And for those who have done some work with trigonometry, the system of degrees, minutes, and seconds works the same way.