Multiplying Two Negative Numbers

Date: 01/16/97 at 12:25:15
From: InnisArt
Subject: Integers

Dear Dr. Math:

I know that when you multiply two negative numbers you get a positive 
answer. I want to know why the answer is positive. Can you give me a 
written example why it is that when you multiply two negative numbers 
you get a positive number?

Thanks!
-Sunny.

Date: 01/16/97 at 17:22:52
From: Doctor Tom
Subject: Re: Integers

Hi Sunny,

Here's a way to think of it. Suppose you're standing on a road, 
and you measure mileage to the east as positive, and to the west 
as negative.  So you are at zero, and a town one mile east is at 
+1 mile, while a town two miles to the west is at -2 miles.  
So far, so good?

A car travelling east will have a positive velocity, and a car 
travelling west will have a negative one.  So a car going east 
at 60 mph goes at +60 mph, and a car going west at the same 
speed goes at -60 mph.

This makes sense, since if they go for an hour (+1 hour), the
east-going car will be at (+1)(+60) = 60 miles, and the car
going west will be at (+1)(-60) = -60 miles (= 60 miles west).

Still okay?

Now suppose a car passes you going east at 60 mph.  Where
was it one hour ago?  Or at -1 hour?  Just multiply:

  (-1)(60) = -60 = 60 miles west.  

How about a car going west at 60 - where was it an hour ago?  
Its velocity is -60, the time is -1, so it was at (-1)(-60), 
and the answer should be 60 miles east, or +60.  

  So (-1)(-60) = +60.

I hope this helps.

Try making up a similar example with money - money you earn is 
positive, and money you spend is negative; if you earn or spend at
a constant rate, multiply to find out your change in worth.  But 
negative time refers to the past, et cetera.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/