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4

MULTIPLYING AND DIVIDING
SIGNED NUMBERS


We can only do arithmetic in the usual way.  To calculate 5(−2), we have to do 5· 2 = 10 -- and then decide on the sign.  Is it +10 or −10?  To decide, we have the followng Rule of Signs.


1.  What is the Rule of Signs for multiplying, dividing, and

fractions?

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Do the problem yourself first!

Like signs produce a positive number;
unlike signs, a negative number.


   Example 1. −5(−2)   =   10.   Like signs.
 
  5(−2)   =   −10.   Unlike signs.
 
  −12
−4 
  =   3.   Like signs.
 
   12 
−4 
  =   −3.   Unlike signs.

For an explaination of these rules, see below.


2.  Write the formal Rule of Signs as it applies to fractions.

a
b
  =   a
b
  a
  b
  =  −  a
b
    a 
b
  =  −  a
b
   Examples. −2
−6
  =   1
3
 
  −2
6
  =   −  1
3
 
    1 
−3
  =   −  1
3

Problem 1.   Calculate the following.

   a)   7(−8) = −56   b)   (−7)8 = −56   c)   8(−7)  = −56   d)   −8(−7) = 56
 
   e)   (−3)7 = −21   f)   5(−9) = −45   g)   −6(−9) = 54   h)   −8(−4) = 32

Problem 2.   Evaluate the following.  (Be careful to distinguish the operations.)

   a)   4 − 6 = −2   b)   4(−6) = −24
 
   c)   (−4) − 6 = −10   d)   (−4)(−6) = 24
 
   e)   5 − 8· 2 = −11   f)   (5 − 8)· 2 = −6
 
   g)   5 − 8 + 2 = −1   h)   5 − (8 + 2) = −5
 
   i)   2 − 3(−6) = 20   j)   (2 − 3)(−6) = 6

Example 2.  The form  ab(−c).   Consider a problem in this form:

3 − 5(−2).

We are to subtract  5 times −2:

3 − 5(−2)   =   3 − (−10)
 
    =   3 + 10
 
    =   13.

And so even though the problem means to subtract  (5 times −2), we may interpret it to mean:  −5 times −2 = +10.  We may simply write

3 − 5(−2)   =   3 + 10
 
    =   13.

In other words, any problem that looks like this --

ab(−c)

-- we may evaluate like this:

a + bc.

Problem 3.   Evaluate the following.

   a)   8 − 2(−4) = 16   b)   9 − 5(−2) = 19
 
   c)   −20 − 3(−5) = −5   d)   −70 − 9(−7) = −7
 
   e)   3 + 4(−9) = −33   f)   −6 + 5(−2) = −16
 
   g)   −10 − 2(4 −8) = −2   h)   (−10 − 2)(4 −8) = (−12)(−4) = 48

Problem 4.  Two variables.   Let the value of y depend on the value
of x as follows:

y = 3x − 6.

Calculate the value of y that corresponds to each value of x:

When x  =  0,   y  =  3· 0 − 6  =  0 − 6  =  −6.

When x  =  1,   y  =  3· 1 − 6  =  3 − 6  =  −3 .

When x  =  −1,   y  =  3· −1 − 6  =  −3 − 6  =  −9.

When x  =  2,   y  =  3· 2 − 6  =  6 − 6  =  0.

When x  =  −2,   y  =  3· −2 − 6  =  −6 − 6  =  −12.

When x  =  3,   y  =  3· 3 − 6  =  9 − 6 = 3.

When x  =  −3,   y  =  3· −3 − 6  =  −9 − 6  =  −15.


Problem 5.  Negative factors.

   a)   (−2)(−2) =    b)   (−2)(−2)(−2) = −8
 
   c)   (−2)(−2)(−2)(−2) = 16   d)   (−2)(−2)(−2)(−2)(−2) = −32

Problem 6.   According to the previous problem:

An even number of negative factors produces a positive number.   While an odd number of negative factors produces a negative number.

Problem 7.   Calculate.

   a)   2(−3)4  = −24   b)   (−2)3(−4)  = 24   c)   2(−3 −4)  = −14
 
   d)   (−3)(−4)(−5)  = −60   e)   (−1)(−2)(−3)(−4) = 24
 
   f)    (−2)13(−5)2 = 260   g)   25(−8)(−3)(−4) = −100· 24 = −2400
 
   h)   (−1)(−1)(−1) = −1   i)   (−1)(−1)(−1)(−1) = 1

Problem 8.   Evaluate each of the following as a positive or negative fraction in lowest terms, or as an integer.

   a)   −24
  6 
  =   4      b)    24
−6
  =   4      c)   −24
 −6
  =  
 
   d)      3  
−12
  =   1
4
     e)    −8
−20
  =   2
5
     f)   −18
 42
  =   3
7
 
   g)   −2
  =   2
3
     h)     2 
−3
  =   2
3
     i)   −2
−3
  =   2
3
 
   j)   −12
   3
  =   4      k)    −5 
−20
  =   1
4
     l)     3 
−4
  =  3
4
  Example 3.  Multiplying fractions.   

To multiply fractions, multiply the numerators, and
multiply the denominators.

Problem 9.   Multiply.

  a)    −3
  5
·  7
8
  =   −21
 40
  =   21
40
  b)    1
2
·  x
  4
  =   x
 8
  =   x
8
  c)    −2
  3
·    x 
−8
  =   −2x
−24
  =    x
12
  d)      x 
−6
·    3 
−5
  =   3x
30
  =    x
10

An explanation of the Rule of Signs

The inclusion of a negative factor changes the sign of a product.  That is, if ab is positive, then (−a)b cannot also be positive.  It must be negative.

For, suppose  (−a)b  were also positive. Then  ab + (−a)b  would be positive.  But that sum is actually 0

ab + (−a)b = [a + (−a)]b = 0· b = 0

Therefore, (−a)b cannot be positive. It must be negative.

In fact, since

ab + (−a)b = 0,

(−a)b must be the negative of ab.

(−a)b = −ab.

Similarly, the inclusion of two negative factors will change the sign twice -- the sign will return to positive

(−a)(−b) = ab.


Next Lesson:  Some rules of algebra


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