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### Why Does a Negative Times a Negative Equal a Positive?

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Date: 03/01/98 at 21:12:30
From: Jessica
Subject: Multiplying Negative numbers

How does a negative number times another negative number equal
a positive number?
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Date: 03/02/98 at 09:15:10
From: Doctor Bruce
Subject: Re: Multiplying Negative numbers

Hello Jessica,

I detect in your question a measure of annoyance at having to learn
the rule for multiplying negative numbers. I'll bet it seems like
someone just made the rule up out of thin air, with no particular
reason why the answer should be positive. I want to reassure you that
this rule is not just "made up." There is a chain of reasoning -- a
mathematical "argument" -- that shows why the rule *has* to be that
negative times negative equals positive.

Mathematical argument takes a little getting used to. This might look
rather strange at first. Here's how the reasoning goes:

(1) Zero times anything equals zero.

(2) Every number has exactly one additive inverse. This means if N
is a positive number, then -N is its additive inverse, so that
N + (-N) = 0. Likewise, the additive inverse of -N is N.

(3) We want negative numbers to obey the distributive law.  This
says that

a*(b+c) = a*b + a*c.

(4) Now, we are forced to accept a new law, that negative times
positive equals negative. This is because we can use the
distributive law on an expression like

2*(3 + (-3)).

This equals 2*(0), which is zero. But by the distributive law,
it also equals

2*3 + 2*(-3).

So 2*(-3) does the job of the additive inverse of 2*3, and
therefore 2*(-3) is the additive inverse of 2*3. But the
additive inverse of 6 is just -6. So 2 times -3 equals -6.

(5) Next, we are forced to accept another new law, that negative
times negative equals positive. It's a lot like the example
in (4). We use the distributive law on, say,

-3*(5 + (-5)).

This is again equal to zero. But by the distributive law, it
also equals

-3*5 + (-3)*(-5).

We know the first thing, (-3*5) equals -15 because of the law
in (4). So (-3)*(-5) is doing the job of the additive inverse
of -15. We know -15 has exactly one additive inverse, namely 15.
Therefore,

(-3)*(-5) = 15.

I hope this doesn't frighten you! The main thing is, keep right on
questioning the things that don't make sense. In mathematics, you are
always entitled to an explanation of WHY things are the way your
teacher (or I) say they are.

Good luck,

-Doctor Bruce, The Math Forum
Check out our web site http://mathforum.org/dr.math/
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Associated Topics:
High School Negative Numbers
Middle School Negative Numbers

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