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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.Objectives of this module: understand the closure, commutative, associative, and distributive properties, understand the identity and inverse properties.

Overview

  • The Closure Properties
  • The Commutative Properties
  • The Associative Properties
  • The Distributive Properties
  • The Identity Properties
  • The Inverse Properties

Property

A property of a collection of objects is a characteristic that describes the collection. We shall now examine some of the properties of the collection of real numbers. The properties we will examine are expressed in terms of addition and multiplication.

The closure properties

The closure properties

If a and b are real numbers, then a + b is a unique real number, and a b is a unique real number.

For example, 3 and 11 are real numbers; 3 + 11 = 14 and 3 11 = 33 , and both 14 and 33 are real numbers. Although this property seems obvious, some collections are not closed under certain operations. For example,

The real numbers are not closed under division since, although 5 and 0 are real numbers, 5 / 0 and 0 / 0 are not real numbers.

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The natural numbers are not closed under subtraction since, although 8 is a natural number, 8 8 is not. ( 8 8 = 0 and 0 is not a natural number.)

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The commutative properties

Let a and b represent real numbers.

The commutative properties

COMMUTATIVE PROPERTY OF ADDITION COMMUTATIVE PROPERTY OF MULTIPLICATION a + b = b + a a b = b a

The commutative properties tell us that two numbers can be added or multiplied in any order without affecting the result.

Sample set a

The following are examples of the commutative properties.

3 + 4 = 4 + 3 Both equal 7.

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5 + x = x + 5 Both represent the same sum .

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4 8 = 8 4 Both equal 32 .

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y 7 = 7 y Both represent the same product .

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5 ( a + 1 ) = ( a + 1 ) 5 Both represent the same product .

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( x + 4 ) ( y + 2 ) = ( y + 2 ) ( x + 4 ) Both represent the same product .

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Practice set a

Fill in the ( ) with the proper number or letter so as to make the statement true. Use the commutative properties.

4 ( k 5 ) = ( ) 4

( k 5 )

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( 9 a 1 ) ( ) = ( 2 b + 7 ) ( 9 a 1 )

( 2 b + 7 )

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The associative properties

Let a , b , and c represent real numbers.

The associative properties

ASSOCIATIVE PROPERTY OF ADDITION ASSOCIATIVE PROPERTY OF MULTIPLICATION ( a + b ) + c = a + ( b + c ) ( a b ) c = a ( b c )

The associative properties tell us that we may group together the quantities as we please without affecting the result.

Sample set b

The following examples show how the associative properties can be used.

( 2 + 6 ) + 1 = 2 + ( 6 + 1 ) 8 + 1 = 2 + 7 9 = 9 Both equal 9 .

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( 3 + x ) + 17 = 3 + ( x + 17 ) Both represent the same sum .

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( 2 3 ) 5 = 2 ( 3 5 ) 6 5 = 2 15 30 = 30 Both equal 30.

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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
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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
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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
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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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