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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

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
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how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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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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