## Bases

Did you know you could count in completely different ways?

Clocks, for example, are a type of number system where we count in terms of 12s and 60s!

What are different bases?

A base (sometimes called a radix) is a set of digits (and sometimes letters) that represent numbers. The most common base is base 10, consisting of digits 0-9. That’s the base which Alice (and perhaps the reader!) would think of when we think of numbers. The premise is simple:

If you have no dots, you have nothing.

When you have one dot, you write 1.

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When you have two dots, you write 2.

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Here’s nine dots.

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In base 10, when you reach ten dots, you start back at 0 again, but now with 1 on the left.

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Eleven dots, or 10 and 1 dots.

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Twenty-Five dots, or two sets of 10, so 2 on the left, and one set of 5, on the right.

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Here’s ninety-nine dots. It’s the same premise: 9 in the 10’s place, 9 in the 1s place.

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Here’s one hundred dots. Since we have exhausted the digits furthest left, we add one to the next digit right. But we’ve exhausted those as well, so we start back at 0 again, with another 0 to its left, and finally a 1 on its right, just as we did for 10.

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One-hundred thirty-six dots has the same idea.

Some other bases include binary (which your computer runs on!), or base two T and hexadecimal (which is used with colours!), or base sixteen1

In Binary, when you have no dots, you have 0.

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When you have one dot, you have 1.

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When you have two dots, you add 1 to 1—but you only have two digits, 1 and 0!—so it becomes 10: 1 on the left, 0 on the right.

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When you have three dots, you have 1 on the left and 1 on the right—11.

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And at four dots, you tally up again! 100!

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What do you do when you have twenty-five dots? (That’s a lot of dots to keep track of.)

Easy! To find a resulting number in any base system, first divide the number (in base 10, presumably) by the number of digits in the base, in this case 2, until we cannot anymore (ie we get 0)

25/2 is 12 remainder 1. We keep that 1 and put it furthest right.

12/2 is 6, no remainder. That gives us a 0. We put it left of the 1 before.

6/2 is 3, no remainder. 0.

3/2 is 1 remainder 1. We put the 1 left of the 0.

1/2 is 0 remainder 1. We keep that 1 and we’re done.

So 25 in binary is 11001.

What about bigger bases? Hexadecimal? Hexadecimal has 16 digits, 0-9 plus A,B,C,D,E and F.

Once again, no dots is 0…

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…and one dot is 1.

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Nine dots is still 9…

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… but ten dots is A!

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Thirteen dots is D.

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And sixteen dots is 10!

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And twenty-five dots is 19—1 set of 16 and a remainder of 9.

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A bigger number like seventy-one dots would be 47 in hexadecimal.

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Here’s an even larger number: in base 10, 292 dots. (Two hundred ninety-two!)

292/16 is 18, remainder 4. so 4 is furthest right.

18/16 is 1 remainder 2, so a 2 is the next left.

1/16 is 0, remainder 1, so a 1 starts the number

Thus, in base 16, 292 is 124!

Bases and Powers

There is a connection between bases and the power of a number. For example, 100 is 1—in the 1-s place. 101 is 10, with a 1 in the ones place. 102 is 100, and so on. This works in binary and hexadecimal as well.

That’s because the positions of numbers (when you add another number to the left) is dependent on the power of the base.

Imagine it somewhat like this, with the # being what the base number is.

1    0   0    0   0   0   0   0   0

######3 #2 #1 #0

Thus, a number like 16 is also 24 and thus is written in binary as 1000. Similarly, 1000 in hexadecimal is 164, or 65536.

Bases and Culture

Other cultures had different ideas about bases. The Babylonians used base 60! They had a different writing system to accommodate this:

The Babylonians are why our clocks have 60 seconds in a minute and 60 minutes in an hour!

Even today you see remnants of different bases—for example, there are 16 fluid ounces in a US pint (and 20 fluid ounces in a UK pint, just to make things a little more exciting!) Similarly, there are 14 pounds in a stone, 2 stones in a quarter (28 pounds), 4 quarters in a hundredweight, and 20 hundredweight in a ton. That’s not even the same base consistently!

Author
Julie Hatfield

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