Subtraction Using Nine's and Ten's Complements

Date: 05/27/2000 at 01:24:37
From: Katie Artus
Subject: Method of complements

I am trying to figure out why the method below, called the "method of 
complements" by my university professor, will give me the correct 
answer all of the time. It is used in place of regrouping in 
subtraction problems.  

      623     999
     -465    -465
     -----   -----
      158     534
             +623
             -----
             1157

             /157     (Cross off the 1 in the thousands column and add 
 
               +1      this 1 to the ones column.  
             -----
              158

Nines must always be used, and the order of operations depicted above 
must be followed. If these conditions are met, the problem will always 
work no matter how many numbers are in the problem. This will also 
work with standard subtraction when regrouping is not necessary. I 
would like to know why this method works. What mathematics are lurking 
behind the scenes? I wish to explain this to a third grade child I am 
student teaching.

Thank you,
Katie Artus

Date: 05/29/2000 at 14:44:11
From: Doctor TWE
Subject: Re: Method of complements

Hi Katie - thanks for writing to Dr. Math.

The method of complements is perhaps most famous in its binary 
variations (called one's complement and two's complement) because 
those are the methods that most computers use to subtract. What your 
professor showed you is the decimal equivalent of one's complement, 
sometimes called nine's complement.

As to why it works, let's examine what we're doing algebraically. 
Let's call the minuend (623 in your example) X and the subtrahend (465 
in your example) Y. We wish to find X-Y.

In the first step, we subtract Y from 99...9 (the number of 9's 
equaling the number of digits in the larger of X and Y). In your 
example, 999. So we have:

     999-Y

Next, we add X to that to get:

     999-Y+X

As long as X > Y, we can see that the result will be greater than or 
equal to 1000. When we cross out the leading digit (the 1000's digit 
in this example), we are in fact subtracting 1000 from our result. 
Thus we have:

     999-Y+X-1000

Finally, we add 1 to the units digit. So we have:

     999-Y+X-1000+1

Can you see how algebraically this equals X-Y?

If I wanted to explain this to a third grader, I'd probably explain 
the ten's complement method instead. Ten's complement works similar to 
nine's complement with two slight differences. First, instead of 
subtracting the subtrahend (Y) from 99...9, you subtract it from 
100...0 instead. [I write this as 99...(10) so I don't have to 
borrow.] The second difference is that you don't have to add the 1 at 
the end. Can you see why this method produces the same results as 
nine's complement?

Here's how I'd explain why it works:

Imagine you want to make some money by selling widgets (pick the 
student's favorite thing-a-ma-bob). First, you have to buy the 
materials to make widgets. This costs $465. Then you can sell them for 
$623. Now you want to figure out how much money you'll make.

But there's one problem - you have to buy the materials BEFORE you can 
sell the widgets. So you borrow $1000 from Mom and Dad. From that 
$1000 you spend $465 to buy the materials. So you have:

     $1000
     - 465
      ----
     $ 535

Now you sell the widgets for $623, you now have:

     $1000
     - 465
      ----
     $ 535
     + 623
      ----
     $1158

Finally, you have to pay back Mom and Dad the $1000 you borrowed to 
start up. So your final profit is:

     $1000
     - 465
      ----
     $ 535
     + 623
      ----
     $/158
      
Which is the selling price ($623) less the cost of materials ($465).


A final word of caution: While nine's complement and ten's complement 
produce the same results when X > Y, they need to be handled 
differently when X <= Y. In the cases where X < Y (and where X = Y for 
nine's complement), there will not be a "final carry" to cross out. 
For example, to subtract 235-687:

     9'S Comp.     10'S Comp.
       999           1000
     - 687         -  687
       ---           ----
       312            313
     + 235         +  235
       ---           ----
       547            548

How do we interpret this? With nine's complement, we subtract the 
result from 99...9 and add a negative sign. With ten's complement, we 
subtract the result from 100...0 and add a negative sign, like so:

     9'S Comp.     10'S Comp.
       999           1000
     - 547         -  548
       ---           ----
    (-)452         (-)452

I hope this helps. If you have any more questions, write back.

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