Working in Base 7

Date: 08/02/2007 at 07:12:00
From: Anne
Subject: Could you please explain base 7 to me?

How can I solve the four basic operations (addition, subtraction,
multiplication and division) using base 7?  Can you give me an example
on the formula on how to do it?

I tried solving and converting numbers to base 6 using blocks, but am 
still confused in using base 7.

Date: 08/02/2007 at 08:41:53
From: Doctor Ian
Subject: Re: Could you please explain base 7 to me?

Hi Anne,

The easiest way to see how it works is to do some operations in base
10 and base 7 side by side.  For example, 

   Base 10             Base 7

     56                 34
   + 78               + 65
   ----               ----

In base 10, we start with the units column, and add 6+8.  That gives
us 14, which is too large to represent with one digit.  So we keep 4
of that, and carry a group of 10 over to the tens column:

     1
     56                 34
   + 78               + 65
   ----               ----
      4

What about in base 7?  It's the same thing: 4+5 gives us 9, which is
too big to represent as a single digit in base 7 (because we have only
the digits 0-6).  In base 7, 9 is 12 (1*7 + 2), so we keep the 2 and
carry a group of 7 over to the 'sevens' column:

     1                  1
     56                 34
   + 78               + 65
   ----               ----
      4                  2

Now we move to the next column.  In base 10, we add 5 and 7, plus the
carry, to get 13.  That's too big for a single digit, so we carry again:

    11                  1
     56                 34
   + 78               + 65
   ----               ----
     34                  2

And the carry gets added to implied zeros:

     11                  1
     056                 34
   + 078               + 65
   -----               ----
     134                  2

Same thing in base 7:  we add 3 and 6, plus the carry, to get 10... or
13 in base 7... and we carry the same way:

     11                  11
     056                 034
   + 078               + 065
   -----               -----
     134                 132

So the point of all this is, the algorithms are exactly the same!  The
only thing that changes is that when you count in base 7, you run out
of digits more quickly, so you have to regroup more often. 

Try this with some other operations, and see if you can make sense of
those, too. 

Let me know if you need more help.

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

Date: 08/04/2007 at 05:24:14
From: Anne
Subject: Thank you (Could you please explain base 7 to me?)

Thank you very much for helping me this site is really very helpful.

Anne