Subtracting and Borrowing in a Column

Date: 09/05/2002 at 10:23:33
From: Brad Hendrickson
Subject: Subtraction borrowing.

Hi,

I teach high school chemistry and teach significant figures.  This 
was the problem.

   6.9 - 7.92.  

I worked the problem as usual, getting -1.02, reported to -1.0 with 
significant figures.

A student of mine brought up a point that I had never thought about 
before, and boggled me.  He borrowed as follows

          6.9
       - 7.92
         ----
to get - 2.98

Why won't borrowing work with this number??

Thanks for your help.

Date: 09/05/2002 at 12:35:42
From: Doctor Peterson
Subject: Re: Subtraction borrowing

Hi, Brad.

It's really not the borrowing that doesn't work, but the whole idea 
of subtracting in a column. Let's take a case where there's no 
borrowing:

      6.58
    - 7.43
    ------
     -1.15

But 6.58 - 7.43 = -(7.43 - 6.58) = -.85. So this is wrong. Why?

The problem is that the minus sign in our answer really applies only 
to the first digit, the 1. What we've done is this:

    (6 + .58) - (7 + .43) = (6 - 7) + (.58 - .43)
                          = -1 + .15

This is the correct value, -.85; but it can't be written as -1.15, as 
we did by writing a digit in each column, because that would mean 
-(1 + .15) rather than -1 + .15.

So when you subtract this way (which I never do) you have to do one 
more step: subtract the .15 from the 1, giving .85, and apply the 
negative sign to the whole thing.

Let's try this on your problem. Your student got "-2.98"; this is 
really -2 + .98, so you have to subtract .98 from 2 and get 1.02, so 
that the answer is -1.02, the correct answer.

I should note that it is only the leftmost digit that has the 
negative sign on it, not the whole part of the number as you might be 
imagining. If we subtracted

      658
    - 743
    -----
     -115

we would finish by subtracting 15 from 100, and get -85.

These ideas are closely related to the 'twos-complement' method used 
by computers to handle negative numbers. That method allows the 
machine to add or subtract 'in a column' without worrying about 
whether the answer will be positive or negative.

If you have any further questions, feel free to write back. This was 
an interesting one!

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

Date: 09/05/2002 at 17:21:58
From: Brad Hendrickson
Subject: Thank you (Subtraction borrowing)

Hi!

Thanks so much for your response! That certainly clarifies the 
situation, and my students and I thank you tremendously. It's so neat 
when students bring up points you've never thought about. Thanks for 
the help!

All the best,
Brad Hendrickson

Date: 09/06/2002 at 08:31:43
From: Doctor Peterson
Subject: Re: Thank you (Subtraction borrowing)

Hi, Brad.

One of the things I enjoy as a Math Doctor is the chance to answer 
these off-the-wall questions that I'd never think of on my own, the 
wrong ideas that clarify the right ideas. 

Thanks for writing.

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