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### An Algorithm for Converting Bases

```
Date: 2/2/96 at 16:19:14
From: Paul Frieden
Subject: base conversion

I would like a formula for base conversions.  My math teacher
said that he had one, but now he can't find it.  Thank you.
Paul Frieden
```

```
Date: 2/6/96 at 14:52:18
From: Doctor Ethan
Subject: Re: base conversion

Hello Paul,

Here it is, butI wouldn't call it a formula. It is more like an algorithm.
That means it is a method that always works, kind of like long division.

Here we go.

Let's say you have a number

10121 in base 3

and you want to know what it is in base 10.  Well, in base three the
place values are

1,3,9,27,81

so we have 1 *81 + 0 * 27 + 1 * 9 + 2* 3 + 1* 1

81 +0+ 9 +6 +1 = 97

that is how it works.  You just take the number, figure out the place
values, and then add them together to get the answer.  Now, let's do one
the other way.

45 in base ten (that is the normal one.) Let's convert it to base five.

Well, in base five the place values will be 1,5,25,125

We won't have any 125's but we will have one 25. Then we will have
20 left.  That is four 5's, so in base five our number will be 140.

Hope that makes sense.  If you don't see a formula, try to work
out a bunch more examples and they should get easier.

-Doctor Ethan,  The Math Forum
```
Associated Topics:
Middle School Number Sense/About Numbers

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