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### Converting Bases Without Going through Base 10

```Date: 09/08/2002 at 12:02:31
From: Sarah Naegele
Subject: Base conversions

Is there a way to convert between bases without going through base 10?
```

```
Date: 09/08/2002 at 16:14:42
From: Doctor Jerry
Subject: Re: Base conversions

Hi Sarah,

Yes, but it's a bit awkward. To keep the discussion simple, I'll
assume that the number in question is an integer.

Suppose x is a number in base b and we want to convert it to a number
in base a. So, we originally have

x = p_0 + p_1*b + p_2*b^2 + ... + p_n*b^n,

where p_0 means p sub 0 and so on. The integers p_0, p_1,...,p_n are
chosen from 0,1,...,b-1.  This means that the base b representation
of x is p_n p_{n-1}...p_1 p_0.

We want to express x as

x = q_0 + q_1*a + q_2*a^2 + ... + q_m*a^m,

where the integers q_0, q_1,...,q_m are chosen from 0,1,...,a-1.

We know that

q_0 + q_1*a + q_2*a^2 + ... + q_m*a^m

= p_0 + p_1*b + p_2*b^2 + ... + p_n*b^n

We divide both sides by a (doing this in, say, base b arithmetic).
The left side will be

q_0/a + (q_1 + q_2*a^1 + ... + q_m*a^{m-1}).

The right side (on which we will do the arithmetic in base b) will be,
after division, a quotient  Q_1 and a remainder f_1. The remainder
f_1 is q_0. We repeat this process with the quotient, thus peeling off
the "digits" one by one.

Suppose we have 102 in base 5 (this is 27 in base 10) and want to
convert it to base 8 (base 10), that is, to base 13 (base 5).

We divide 102 by 13 and find quotient 3 and remainder 3. So, we're
finished (in this particular case) and we find 33 (which is 27 in
base 8).

- Doctor Jerry, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 09/09/2002 at 12:34:51
From: Sarah Naegele
Subject: Thank you (base conversions)

This is exactly what I wanted to know and I followed it completely!
Thank you so much for your time!

- Sarah
```
Associated Topics:
High School Number Theory

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