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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.Objectives of this module: be able to multiply and divide signed numbers.

Overview

  • Multiplication of Signed Numbers
  • Division of Signed Numbers

Multiplication of signed numbers

Let us consider first the product of two positive numbers.

Multiply: 3 5 .
3 5 means 5 + 5 + 5 = 15 .

This suggests that

( positive number ) ( positive number ) = positive number .

More briefly, ( + ) ( + ) = + .

Now consider the product of a positive number and a negative number.

Multiply: ( 3 ) ( 5 ) .
( 3 ) ( 5 ) means ( 5 ) + ( 5 ) + ( 5 ) = 15 .

This suggests that

( positive number ) ( negative number ) = negative number

More briefly, ( + ) ( - ) = - .

By the commutative property of multiplication, we get

( negative number ) ( positive number ) = negative number

More briefly, ( - ) ( + ) = - .

The sign of the product of two negative numbers can be determined using the following illustration: Multiply 2 by, respectively, 4 , 3 , 2 , 1 , 0 , 1 , 2 , 3 , 4 . Notice that when the multiplier decreases by 1, the product increases by 2.

4 ( 2 ) = 8 3 ( 2 ) = 6 2 ( 2 ) = 4 1 ( 2 ) = 2 } As we know , ( + ) ( ) = . 0 ( 2 ) = 0 As we know , 0 ( any number ) = 0.

1 ( 2 ) = 2 2 ( 2 ) = 4 3 ( 2 ) = 6 4 ( 2 ) = 8 } This pattern suggests ( ) ( ) = + .

We have the following rules for multiplying signed numbers.

Rules for multiplying signed numbers

To multiply two real numbers that have

  1. the same sign , multiply their absolute values. The product is positive.
    ( + ) ( + ) = + ( ) ( ) = +
  2. opposite signs , multiply their absolute values. The product is negative.
    ( + ) ( ) = ( ) ( + ) =

Sample set a

Find the following products.

8 6

Multiply these absolute values . | 8 | = 8 | 6 | = 6 } 8 6 = 48 Since the numbers have the same sign, the product is positive . 8 6 = + 48 or 8 6 = 48

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( 8 ) ( 6 )

Multiply these absolute values . | 8 | = 8 | 6 | = 6 } 8 6 = 48 Since the numbers have the same sign, the product is positive . ( 8 ) ( 6 ) = + 48 or ( 8 ) ( 6 ) = 48

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( 4 ) ( 7 )

Multiply these absolute values . | 4 | = 4 | 7 | = 7 } 4 7 = 28 Since the numbers have opposite signs, the product is negative . ( 4 ) ( 7 ) = 28

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6 ( 3 )

Multiply these absolute values . | 6 | = 6 | 3 | = 3 } 6 3 = 18 Since the numbers have opposite signs, the product is negative . 6 ( 3 ) = 18

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Practice set a

Find the following products.

Division of signed numbers

We can determine the sign pattern for division by relating division to multiplication. Division is defined in terms of multiplication in the following way.

If b c = a , then a b = c , b 0 .

For example, since 3 4 = 12 , it follows that 12 3 = 4 .

Notice the pattern:

Since 3 4 b c = a = 12 , it follows that 12 3 a b = c = 4

The sign pattern for division follows from the sign pattern for multiplication.

  1. Since ( + ) ( + ) b c = a = + , it follows that ( + ) ( + ) a b = c = + , that is,

    ( positive number ) ( positive number ) = positive number

  2. Since ( ) ( ) b c = a = + , it follows that ( + ) ( ) a b = c = , that is,

    ( positive number ) ( negative number ) = negative number

  3. Since ( + ) ( ) b c = a = , it follows that ( ) ( + ) a b = c = , that is,

    ( negative number ) ( positive number ) = negative number

  4. Since ( ) ( + ) b c = a = , it follows that ( ) ( ) a b = c = + , that is

    ( negative number ) ( negative number ) = positive number

We have the following rules for dividing signed numbers.

Rules for dividing signed numbers

To divide two real numbers that have

  1. the same sign , divide their absolute values. The quotient is positive.
    ( + ) ( + ) = + ( ) ( ) = +
  2. opposite signs , divide their absolute values. The quotient is negative.
    ( ) ( + ) = ( + ) ( ) =

Sample set b

Find the following quotients.

10 2

| - 10 | = 10 | 2 | = 2 } Divide these absolute values . 10 2 = 5 - 10 2 = - 5 Since the numbers have opposite signs, the quotient is negative .

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35 7

| - 35 | = 35 | - 7 | = 7 } Divide these absolute values . 35 7 = 5 - 35 - 7 = 5 Since the numbers have same signs, the quotient is positive .

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18 9

| 18 | = 18 | - 9 | = 9 } Divide these absolute values . 18 9 = 2 18 - 9 = - 2 Since the numbers have opposite signs, the quotient is negative .

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Practice set b

Find the following quotients.

Sample set c

Find the value of 6 ( 4 7 ) 2 ( 8 9 ) ( 4 + 1 ) + 1 .

Using the order of operations and what we know about signed numbers, we get

6 ( 4 7 ) 2 ( 8 9 ) ( 4 + 1 ) + 1 = 6 ( 3 ) 2 ( 1 ) ( 5 ) + 1 = 18 + 2 5 + 1 = 20 4 = 5

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Find the value of z = x u s if x = 57 , u = 51 , and s = 2 .

Substituting these values we get

z = 57 51 2 = 6 2 = 3

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Practice set c

Find the value of 7 ( 4 8 ) + 2 ( 1 11 ) 5 ( 1 6 ) 17 .

1

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Find the value of P = n ( n 3 ) 2 n , if n = 5 .

1

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Exercises

Find the value of each of the following expressions.

4 ( 1 8 ) + 3 ( 10 3 )

49

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9 ( 0 2 ) + 4 ( 8 9 ) + 0 ( 3 )

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6 ( 2 9 ) 6 ( 2 + 9 ) + 4 ( 1 1 )

140

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3 ( 4 + 1 ) 2 ( 5 ) 2

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4 ( 8 + 1 ) 3 ( 2 ) 4 2

7

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1 ( 3 + 2 ) + 5 1

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3 ( 4 2 ) + ( 3 ) ( 6 ) 4

3

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3 [ ( 1 + 6 ) ( 2 7 ) ]

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2 [ ( 4 8 ) ( 5 11 ) ]

4

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5 [ ( 1 + 5 ) + ( 6 8 ) ]

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[ ( 4 9 ) + ( 2 8 ) ]

15

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3 [ 2 ( 1 5 ) 3 ( 2 + 6 ) ]

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2 [ 5 ( 10 + 11 ) 2 ( 5 7 ) ]

2

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P = R C . Find P if R = 2000 and C = 2500 .

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z = x u s . Find z if x = 23 , u = 25 , and s = 1.

2

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z = x u s . Find z if x = 410 , u = 430 , and s = 2.5.

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m = 2 s + 1 T . Find m if s = 8 and T = 5.

3

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m = 2 s + 1 T . Find m if s = 10 and T = 5.

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Use a calculator. F = ( p 1 p 2 ) r 4 9. Find F if p 1 = 10 , p 2 = 8 , r = 3.

1458

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Use a calculator. F = ( p 1 p 2 ) r 4 9. Find F if p 1 = 12 , p 2 = 7 , r = 2.

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P = n ( n 1 ) ( n 2 ) . Find P if n = 4.

120

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P = n ( n 1 ) ( n 2 ) ( n 3 ) . Find P if n = 5.

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P = n ( n 2 ) ( n 4 ) 2 n . Find P if n = 6.

40

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Exercises for review

( [link] ) What natural numbers can replace x so that the statement 4 < x 3 is true?

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( [link] ) Simplify ( x + 2 y ) 5 ( 3 x 1 ) 7 ( x + 2 y ) 3 ( 3 x 1 ) 6 .

( x + 2 y ) 2 ( 3 x 1 )

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( [link] ) Simplify ( x n y 3 t ) 5 .

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( [link] ) Find the sum. 6 + ( 5 ) .

11

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( [link] ) Find the difference. 2 ( 8 ) .

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Questions & Answers

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
hello
BenJay
hi
Method
I am eliacin, I need your help in maths
Rood
how can I help
Sir
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Amoon
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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