Base 10 and other Bases in Mathematics

Maybe you know the answer to those questions, but if you are like most people, there has been a time in your life when you didn't, and found the whole "base" idea frustrating.

If you have children, or if you are a teacher, your children or students are definitely going to go through a period of frustration about bases if they haven't already.

The mission of this lens is to make that period as short and painless as possible for your child, your students and yourself.

First let's try to define bases in normal English. A base is a way to express numbers using place value (that means like using columns). The typical system that we use, and that you are familiar with, is called the base 10 system. In base 10, each column is worth 10 times the amount of the column in the place to the right of it.

The column furthest to the right, is always the ones column. (I must point out that we are talking about whole numbers here, not decimals or fractions, or negative numbers. The whole numbers are the numbers 0,1,2,3...)

So the column to the left of that would be the 10s column because 10 is 10 x 1.

The next highest column to the left would be 10 times the 10s column, which of course would make it the hundreds column.

I imagine you already know what comes after the hundreds column. It's the thousands column of course because 100 x 10 = 1,000. There is no end to how high you can count when you use the base system.

This seems all matter-of-fact, until you realize that humanity didn't start using base systems until very late in its development. Think about Roman numerals - they consisted of letters like I, V, X, L, C, M., and, um, what came after that? See, with other systems you typically run out of letters or symbols, because each symbol stands for a different amount of numbers. And if you had large amounts of numbers, you'd have to memorize lots and lots of symbols.

Some ancient "programmer" must've figured out that there was a better way. To that nameless programmer, we owe a great debt of gratitude. I know some of you may not feel that way, because some people just "hate math." But imagine how much more someone might hate math if instead of learning how to multiply, say 14 x 8, they'd have to multiply XIV x VIII !

There are plenty of other bases. there can be a base of just about any number you like, like base two, base three, base four, etc. There are many reasons you might want to use a base other than 10. One major reason will be explained in the next section.

A typical reason to use other bases, is to solve problems concerning specific amounts. For example, some items are sold in dozens, and grosses. A dozen is 12 x 1. A gross is 12 x 12. I'll bet you can guess what base we are dealing with there. Computer programs that are required to inventory merchandise that is sold in dozens can be written to solve problems using base 12, which makes them more streamlined and efficient, than calculating in base 10.

Well of course we don't have to. But we don't want to walk around and be ignorant of things that can be useful, do we? You may not have a use for them now, just like you don't have any use for a car right now, but if you never learn to drive, you're limiting yourself for no reason.

Learning bases is easy - you already use base 10 quite well, and believe it or not, you already use base 2, whether you know it or not. As a matter of fact, you're using it right now.

That's right. Are you aware that almost all computers and circuits operate on the base 2 system? What do you think all those zeros and ones are about? You've seen a number like 100010011, haven't you? Numbers like that, only much longer, he turn up in lots of science-fiction movies like The Matrix all the time. I'm sure you're aware that a number like that usually doesn't mean, "10 billion - is something," it's actually a number in base two.

The reason computers and circuits use base 2, is because based to only consists of ones and zeros. That conveniently can represent two states of a circuit - on, or off. There's more to it than I can explain in this simple lesson, but that's the basic deal.

So in a nutshell, no base 2, no computer games for you!

This is a very basic explanation of bases, and I'm pretty sure you followed all of it.
To learn a little more about bases, and how you can start using them and manipulating them, please check out my series of posts about base 10 and other bases at my website at:

Bases - What are They?

That series of lessons will help you easily learn and understand things like:

• How can we change a number from base 10 to base 2?


• What bases are commonly used for ?


• Which bases are commonly used?


• How different bases are written.


• Do bases have anything to do with exponents (powers)?


• Operations ( addition, subtraction, division, application, etc.) with other bases besides base 10.


• Can there be bases higher than base 10?

There will also be some trivia about bases, and much much more.