Math Forum: Ask Dr. Math FAQ: Negative Times a Negative

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03 Aug 17

marcomx66
math mathematics matematica kids kid
21 May 17

Tom Wilcox
c_math
- Number Line
  
  Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y. Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps. So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.
- On the Road
  
  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.
    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).
    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 mph? 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.
- Here's a plausibility argument drawn from multiplication patterns:
  3 x -3 = -9 2 x -3 = -6 1 x -3 = -3 0 x -3 = 0 -1 x -3 = 3
- A Proof
  
  Let a and b be any two real numbers. Consider the number x defined by
  
  x = ab + (-a)(b) + (-a)(-b).
  
    We can write
  x = ab + (-a)[ (b) + (-b) ] (factor out -a) = ab + (-a)(0) = ab + 0 = ab.
    Also,
  x = [ a + (-a) ]b + (-a)(-b) (factor out b) = 0 * b + (-a)(-b) = 0 + (-a)(-b) = (-a)(-b).
    So we have
      x = ab
    and
      x = (-a)(-b)
  
    Hence, by the transitivity of equality, we have
      ab = (-a)(-b).
2 more annotations...
27 Nov 16

Marco Almeida
Why is a negative times a negative a positive? Some excellent explanations here on @themathforum https://t.co/bUIVHpFF9C

tweets
22 Jun 16

smarkowski
24 May 16

piercespicks
23 Nov 15

Kristina Barnaby
negative times a negative = positive

integers multiplication
18 Aug 14

Tim Long
Great examples of why a negative X a negative is a positive

negative IntegerRules
- Number Line
  
  Imagine a number line on which you walk. Multiplying x*y is taking x steps, each of size y. Negative steps require you to face the negative end of the line before you start walking and negative step sizes are backward (i.e., heel first) steps. So, -x*-y means to stand on zero, face in the negative direction, and then take x backward steps, each of size y.
12 Jul 11

grovek
cfs integers
30 Oct 10

Barry Lewis
math education resources algebra
13 Jul 10

Julie Reulbach
Here's a plausibility argument drawn from multiplication patterns:
3 x -3 = -9
2 x -3 = -6
1 x -3 = -3
0 x -3 = 0
-1 x -3 = 3

negative numbers
07 Mar 10

Debbie Gallagher
tries to explain why a negative number multiplied by a negative number is a positive number

mathematics resource lessons
16 Aug 09

William McNeary
Why is a negative times a negative a positive?

teacher math
10 Nov 08

John Faig
Why is a negative times a negative a positive?

teacher math
04 Apr 08

Andy Byerley
math
29 Apr 06

Bunyarat Rungtaweevoranit
Mathematics
03 Feb 05

inutile