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### Subtracting and Borrowing in a Column

```Date: 09/05/2002 at 10:23:33
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??

```

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

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?

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

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
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
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,
```

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

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/
```
Associated Topics:
Elementary Place Value
Elementary Subtraction
High School Negative Numbers
Middle School Negative Numbers

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