<< Chapter < Page Chapter >> Page >
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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Practice set c

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

1

Got questions? Get instant answers now!

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

1

Got questions? Get instant answers now!

Exercises

Find the value of each of the following expressions.

4 ( 1 8 ) + 3 ( 10 3 )

49

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

140

Got questions? Get instant answers now!

3 ( 4 + 1 ) 2 ( 5 ) 2

Got questions? Get instant answers now!

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

7

Got questions? Get instant answers now!

1 ( 3 + 2 ) + 5 1

Got questions? Get instant answers now!

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

3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

2 [ ( 4 8 ) ( 5 11 ) ]

4

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

[ ( 4 9 ) + ( 2 8 ) ]

15

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

1458

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

120

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

40

Got questions? Get instant answers now!

Exercises for review

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

Got questions? Get instant answers now!

( [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 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

11

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask