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### Direct Conversion of Binary to Octal

```Date: 05/14/2002 at 13:49:23
From: greg
Subject: conversions

Given the base two number

110110100
2

give a brief explanation of how this number can be converted directly
to a base eight number (without first converting to base ten and
then back to a base eight).  Hint:

10110100  = 264
2      8

I just don't understand the problem at all!
```

```
Date: 05/14/2002 at 16:05:03
From: Doctor Peterson
Subject: Re: conversions

Hi, Greg.

See this page from our archives (which is about base 16, but the same
idea applies):

http://mathforum.org/dr.math/problems/hamilton12.8.98.html

To convert base 2 to base 8, you divide the digits into groups of 3,
starting at the right, and write each group as one octal digit:

10 110 100(base 2)
\ / \ / \ /
2   6   4 (base 8)

This works because 8 = 2^3, so that each group of three bits takes
values from 0 to 7, exactly what you need for octal. For example, the
middle group above is

1*2^5 + 1*2^4 + 0*2^3

as part of the binary expansion, but can be factored to give

(1*2^2 + 1*2^1 + 0*2^0)*2^3 = 6*8^1

which corresponds to the middle digit in octal.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Number Theory
High School Number Theory

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