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What are Bases?

Reference > Mathematics > Number Theory > Bases

What are bases? Believe it or not, any time you do math, you're using bases (more precisely, you're using a base), even if you have never heard the term before, and don't really understand what a base is. The base you use is called "base ten." Notice that we don't call it "base 10" - we spell the word "ten" out.  There's a reason we do that, which we'll get into a little later.

It's generally assumed that the reason we do math in base ten is because we have ten fingers, and so we based our number system around the number ten.

How do you count? Well, you have a digit (a symbol) for each finger (did you know that "digit" is another word for "finger?"). Those symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. If you call your first finger 0, by the time you use up all your fingers, you've used up all your symbols, too!

So what comes after the symbol 9? After the symbol 9, we have the number 10. But what, exactly does that mean?

Split it up into its digits. One and zero. It means "ONE person's fingers, plus ZERO more fingers."

Try a different number. What does the number 47 mean? It means "FOUR peoples' fingers, plus SEVEN more fingers."

Does that make sense? Four people is forty fingers, plus seven more fingers makes 47 fingers.

Want to try one more? 62. What does the number 62 mean? Think it through for yourself, then check the next paragraph to make sure you got it right.

62 means "SIX peoples' fingers, plus TWO more fingers." Six people have a total of sixty fingers, plus two more makes 62.

So far, hopefully, you're following this pretty well. But now we come to three digit numbers. What does 423 mean?

The four means "FOUR groups of 10 people," the 2 means "TWO more people," and the 3 means "THREE more fingers."

4 groups of ten people is 40 people, which is 400 fingers, two people is 20 fingers, and three more fingers makes 423.

That's how bases got started - by counting fingers, and trying to make sure we never had to use more symbols than we have fingers. It's a pretty nice system, because if we weren't willing to break quantities into groups of ten, and ten tens, and so on, we'd have to have a thousand symbols just to write the number "one thousand!"

Notice that the second to last digit represents groups of ten, and the third to last digit represents groups of 100. You should be able to guess that the next digit before that would represent groups of 1000, and then 10,000, and so on.

So you can write any number using a longhand method like this:

4,329 = 4 x 1000 + 3 x 100 + 2 x 10 + 9

You can write that yet another way by using some exponents. This way of writing numbers will be helpful when we start using other bases:

4,329 = 4 x 103 + 3 x 102 + 2 x 10 + 9

Why do we use base ten? Because we have ten fingers. So what if we all had eight fingers instead? Well, we probably would have developed a system of counting called "base eight." And we'll explore that in the next section.


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Working in other BasesWorking in other Bases

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