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/