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Subtracting Two Numbers of Like Base

Date: 10/21/2004 at 13:18:42
From: Cheryl
Subject: Subtracting numbers of like bases

I have been trying to subtract 3-digit numbers of like bases, such as

  212 (base 4)
 - 33 (base 4)

The problem that I am having is the borrowing principle.  Help!!!  I 
just can't seem to get the right answer without changing both numbers 
to base 10 and that is not what my teacher wants.  I've searched the 
book for the way to do this, but all they talk about is how to 
subtract with numbers that don't need to be borrowed from.

Date: 10/21/2004 at 14:00:51
From: Doctor Mike
Subject: Re: Subtracting numbers of like bases

Hi Cheryl,

I'm glad you are trying to do it this way.  When you change to base
ten you are just avoiding dealing with base 4.  I'll guide you through
this problem.  That's probably all you need to get you started on
understanding it all.
Right off you run into a problem, in the "units" place.  You can't 
take 3 from 2.  What to do?  In a base ten problem, you would "borrow" 
one from the next place, which gives you 12 in the units place.  
Strangely enough, that's EXACTLY what you do here!  The only big 
difference is that in a base ten problem you get "12" meaning a base
ten number, whereas here in base 4 it means "12" as a base 4 number
(which is 6 base ten).
What I tell my students is to do what they would normally do in a base
ten problem, BUT to remember that the digits in the numbers they are
working with mean powers of four instead of powers of ten.  See?  
So, with this introduction, here are all the steps.
1.   Original problem (with the digits "spread out" so I
     will have plenty of room for when I get to borrowing):
      2  1  2 
    -    3  3 
2.   Borrow 1 from the four's place, so I can do the
     subtraction in the units place.  Note that if I
     borrow from the 4's place I have to change 1 to 0.
     (Also note that even though "12" is not a digit, I
     have 12 in the units place, because what I borrowed
     from the four's place got dumped in there.)
      2  0 12
    -    3  3
3.   Now, I can do the subtraction in the unit's place.
      2  0 12
    -    3  3
3a.  Just a comment here:  That is "base 4 number 12" minus
     "base 4 number 3".  That's the same as the base ten
     subtraction 6 minus 3.  This is really Really REALLY
     important for you to understand here.
4.   I can't take 3 from zero, so I borrow AGAIN, from the
     next higher place (just like you do in base ten, right?).
     Don't forget to change 2 down to 1 in the "4 squared"
     place.  You are taking one thing from the "4 squared"
     place and putting it into the fours place, where it
     is written as "10".  Always, when you borrow from one
     place, and move it into the next lower place, it
     becomes "10".  If you think about it, that's exactly
     what happens in base ten borrowing.  If you borrow
     1 from the hundreds place, say, that hundred is 10
     tens, so when you re-locate it down into the tens
     place you write it as 10.  Similarly, borrowing a 1
     from the thousands place, that thousand is 10 hundreds.
      1 10 12
    -    3  3
5.   Now go ahead and do the subtraction in the fours place.
      1 10 12
    -    3  3
         1  3
6.   Finally, bring down the 1 in the "four squared" place,
     because there is nothing to subtract from it.
      1 10 12
    -    3  3
      1  1  3 
7.   You can always check your work by converting to base 10.
     We have found that "212 in base 4" minus "33 in base 4"
     equals "113 in base 4".  Does that hold water?  Let's
     change all of those numbers to base ten.  We get ....
       38 - 15  =  23 
8.    That's correct, so we have verified the base 4 result.
Now.  Go after those other problems.  Thanks for writing. 

- Doctor Mike, The Math Forum 

Date: 10/21/2004 at 14:56:53
From: Cheryl
Subject: Thank you (Subtracting numbers of like bases)

Thanks for the help on the subtracting of bases.  I finally figured 
out a way to do it, but your insight really helps to pull it all 
Associated Topics:
High School Number Theory

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