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Basic Properties of Real Numbers: Properties of the Real Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: 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 a and b b are real numbers, then a+b a+b is a unique real number, and a⋅b a⋅b is a unique real number.

For example, 3 and 11 are real numbers; 3+11=14 3+11=14 and 3⋅11=33, 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,

Example 1

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

Example 2

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

The Commutative Properties

Let a a and b b represent real numbers.

The Commutative Properties

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

Example 3

3+4=4+3 Both equal 7. 3+4=4+3 Both equal 7.

Example 4

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

Example 5

4⋅8=8⋅4 Both equal 32. 4⋅8=8⋅4 Both equal 32.

Example 6

y7=7y Both represent the same product. y7=7y Both represent the same product.

Example 7

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

Example 8

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

Practice Set A

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

Exercise 1

6+5=( )+6 6+5=( )+6

Solution

5 5

Exercise 2

m+12=12+( ) m+12=12+( )

Solution

m m

Exercise 3

9⋅7=( )⋅9 9⋅7=( )⋅9

Solution

7 7

Exercise 4

6a=a( ) 6a=a( )

Solution

6 6

Exercise 5

4(k−5)=( )4 4(k−5)=( )4

Solution

(k−5) (k−5)

Exercise 6

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

Solution

(2b+7) (2b+7)

The Associative Properties

Let a,b, a,b, and c c represent real numbers.

The Associative Properties

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

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.

Example 9

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

Example 10

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

Example 11

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

Example 12

(9y)4=9(y4) Both represent the same product. (9y)4=9(y4) Both represent the same product.

Practice Set B

Fill in the ( ) ( ) to make each statement true. Use the associative properties.

Exercise 7

(9+2)+5=9+( ) (9+2)+5=9+( )

Solution

2+5 2+5

Exercise 8

x+(5+y)=( )+y x+(5+y)=( )+y

Solution

x+5 x+5

Exercise 9

(11a)6=11( ) (11a)6=11( )

Solution

a⋅6 a⋅6

Exercise 10

[ (7m−2)(m+3) ](m+4)=(7m−2)[ ( )( ) ] [ (7m−2)(m+3) ](m+4)=(7m−2)[ ( )( ) ]

Solution

(m+3)(m+4) (m+3)(m+4)

Sample Set C

Example 13

Simplify (rearrange into a simpler form): 5x6b8ac4 5x6b8ac4 .

According to the commutative property of multiplication, we can make a series of consecutive switches and get all the numbers together and all the letters together.

5⋅6⋅8⋅4⋅x⋅b⋅a⋅c 960xbac Multiply the numbers. 960abcx By convention, we will, when possible, write all letters in alphabetical order. 5⋅6⋅8⋅4⋅x⋅b⋅a⋅c 960xbac Multiply the numbers. 960abcx By convention, we will, when possible, write all letters in alphabetical order.

Practice Set C

Simplify each of the following quantities.

Exercise 11

3a7y9d 3a7y9d

Solution

189ady 189ady

Exercise 12

6b8acz4⋅5 6b8acz4⋅5

Solution

960abcz 960abcz

Exercise 13

4p6qr3(a+b) 4p6qr3(a+b)

Solution

72pqr(a+b) 72pqr(a+b)

The Distributive Properties

When we were first introduced to multiplication we saw that it was developed as a description for repeated addition.

4+4+4=3⋅4 4+4+4=3⋅4

Notice that there are three 4’s, that is, 4 appears 3 times. Hence, 3 times 4.
We know that algebra is generalized arithmetic. We can now make an important generalization.

When a number a a is added repeatedly n n times, we have
 a+a+a+⋯+a ︸ a appears n times a+a+a+⋯+a ︸ a appears n times
Then, using multiplication as a description for repeated addition, we can replace
 a+a+a+⋯+a ︸ n times with na a+a+a+⋯+a ︸ n times with na

For example:

Example 14

x+x+x+x x+x+x+x can be written as 4x 4x since x x is repeatedly added 4 times.

x+x+x+x=4x x+x+x+x=4x

Example 15

r+r r+r can be written as 2r 2r since r r is repeatedly added 2 times.

r+r=2r r+r=2r

The distributive property involves both multiplication and addition. Let’s rewrite 4(a+b). 4(a+b). We proceed by reading 4(a+b) 4(a+b) as a multiplication: 4 times the quantity (a+b) (a+b) . This directs us to write

4(a+b) = (a+b)+(a+b)+(a+b)+(a+b) = a+b+a+b+a+b+a+b 4(a+b) = (a+b)+(a+b)+(a+b)+(a+b) = a+b+a+b+a+b+a+b

Now we use the commutative property of addition to collect all the a's a's together and all the b's b's together.

4(a+b) = a+a+a+a ︸ 4a's + b+b+b+b ︸ 4b's 4(a+b) = a+a+a+a ︸ 4a's + b+b+b+b ︸ 4b's

Now, using multiplication as a description for repeated addition, we have

4(a+b) = 4a+4b 4(a+b) = 4a+4b

We have distributed the 4 over the sum to both a a and b b .

The product of four and the expression, a plus b, is equal to four a plus four b. The distributive property is shown by the arrows from four to each term of expression a plus b in the product.

The Distributive Property

a(b+c)=a⋅b+a⋅c (b+c) a=a⋅b+a⋅c a(b+c)=a⋅b+a⋅c (b+c) a=a⋅b+a⋅c

The distributive property is useful when we cannot or do not wish to perform operations inside parentheses.

Sample Set D

Use the distributive property to rewrite each of the following quantities.

Example 16

The product of two and the expression, five plus seven, is equal to the sum of the products of two and five, and two and seven. The distributive property is shown by the arrows from two to each term of the expression five plus seven in the product. A comment 'Both equal twenty four' is written on the right side of the equation.

Example 17

The product of six and the expression, x plus three, is equal to sum of the products of six and x, and six and three . This is further equalt to six x plus eighteen. The distributive property is shown by the arrows from six to each term of the expression x plus three in the product. A comment 'Both represent the same number' is written on the right side of the equation.

Example 18

The product of the expression z plus five, and y, is equal to zy plus five y which is further rewritten as yz plus five y. The distributive property is shown by the arrows from the y to each term of the expression z plus five in the product.

Practice Set D

Exercise 14

What property of real numbers justifies
 a(b+c)=(b+c)a? a(b+c)=(b+c)a?

Solution

the commutative property of multiplication

Use the distributive property to rewrite each of the following quantities.

Exercise 15

3(2+1) 3(2+1)

Solution

6+3 6+3

Exercise 16

(x+6)7 (x+6)7

Solution

7x+42 7x+42

Exercise 17

4(a+y) 4(a+y)

Solution

4a+4y 4a+4y

Exercise 18

(9+2)a (9+2)a

Solution

9a+2a 9a+2a

Exercise 19

a(x+5) a(x+5)

Solution

ax+5a ax+5a

Exercise 20

1(x+y) 1(x+y)

Solution

x+y x+y

The Identity Properties

Additive Identity

The number 0 is called the additive identity since when it is added to any real number, it preserves the identity of that number. Zero is the only additive identity.
For example, 6+0=6 6+0=6 .

Multiplicative Identity

The number 1 is called the multiplicative identity since when it multiplies any real number, it preserves the identity of that number. One is the only multiplicative identity.
For example 6⋅1=6 6⋅1=6 .

We summarize the identity properties as follows.

ADDITIVE IDENTITY          PROPERTY MULTIPLICATIVE IDENTITY ​​​​​                   PROPERTY If a is a real number, then If a is a real number, then a+0=a         and        0+a=a a⋅1=a        and        1⋅a=a ADDITIVE IDENTITY          PROPERTY MULTIPLICATIVE IDENTITY ​​​​​                   PROPERTY If a is a real number, then If a is a real number, then a+0=a         and        0+a=a a⋅1=a        and        1⋅a=a

The Inverse Properties

Additive Inverses

When two numbers are added together and the result is the additive identity, 0, the numbers are called additive inverses of each other. For example, when 3 is added to −3 −3 the result is 0, that is, 3+(−3)=0 3+(−3)=0 . The numbers 3 and −3 −3 are additive inverses of each other.

Multiplicative Inverses

When two numbers are multiplied together and the result is the multiplicative identity, 1, the numbers are called multiplicative inverses of each other. For example, when 6 and 1 6 1 6 are multiplied together, the result is 1, that is, 6⋅ 1 6 =1 6⋅ 1 6 =1 . The numbers 6 and 1 6 1 6 are multiplicative inverses of each other.

We summarize the inverse properties as follows.

The Inverse Properties

  1. If a a is any real number, then there is a unique real number −a −a , such that
    a+(−a)=0 and −a+a=0 a+(−a)=0 and −a+a=0
    The numbers a a and −a −a are called additive inverses of each other.
  2. If a a is any nonzero real number, then there is a unique real number 1 a 1 a such that
    a⋅ 1 a =1 and 1 a ⋅a=1 a⋅ 1 a =1 and 1 a ⋅a=1
    The numbers a a and 1 a 1 a are called multiplicative inverses of each other.

Expanding Quantities

When we perform operations such as 6(a+3)=6a+18 6(a+3)=6a+18 , we say we are expanding the quantity 6(a+3) 6(a+3) .

Exercises

Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations.

Exercise 21

x+3 x+3

Solution

3+x 3+x

Exercise 22

5+y 5+y

Exercise 23

10x 10x

Solution

10x 10x

Exercise 24

18z 18z

Exercise 25

r6 r6

Solution

6r 6r

Exercise 26

ax ax

Exercise 27

xc xc

Solution

cx cx

Exercise 28

7(2+b) 7(2+b)

Exercise 29

6(s+1) 6(s+1)

Solution

( s+1 )6 ( s+1 )6

Exercise 30

(8+a)(x+6) (8+a)(x+6)

Exercise 31

(x+16)(a+7) (x+16)(a+7)

Solution

( a+7 )( x+16 ) ( a+7 )( x+16 )

Exercise 32

(x+y)(x−y) (x+y)(x−y)

Exercise 33

0.06m 0.06m

Solution

m( 0.06 ) m( 0.06 )

Exercise 34

Eight times a star.

Exercise 35

5(6h+1) 5(6h+1)

Solution

( 6h+1 )5 ( 6h+1 )5

Exercise 36

m(a+2b) m(a+2b)

Exercise 37

k(10a−b) k(10a−b)

Solution

( 10a−b )k ( 10a−b )k

Exercise 38

(21c)(0.008) (21c)(0.008)

Exercise 39

(−16)(4) (−16)(4)

Solution

( 4 )( −16 ) ( 4 )( −16 )

Exercise 40

(5)(b−6) (5)(b−6)

Exercise 41

□ ⋅ ○ □ ⋅ ○

Solution

○ ⋅ □ ○ ⋅ □

Exercise 42

The product of a star and a rhombus.

Simplify using the commutative property of multiplication for the following problems. You need not use the distributive property.

Exercise 43

9x2y 9x2y

Solution

18xy 18xy

Exercise 44

5a6b 5a6b

Exercise 45

2a3b4c 2a3b4c

Solution

24abc 24abc

Exercise 46

5x10y5z 5x10y5z

Exercise 47

1u3r2z5m1n 1u3r2z5m1n

Solution

30mnruz 30mnruz

Exercise 48

6d4e1f2(g+2h) 6d4e1f2(g+2h)

Exercise 49

( 1 2 )d( 1 4 )e( 1 2 )a ( 1 2 )d( 1 4 )e( 1 2 )a

Solution

1 16 ade 1 16 ade

Exercise 50

3(a+6)2(a−9)6b 3(a+6)2(a−9)6b

Exercise 51

1(x+2y)(6+z)9(3x+5y) 1(x+2y)(6+z)9(3x+5y)

Solution

9( x+2y )( 6+z )( 3x+5y ) 9( x+2y )( 6+z )( 3x+5y )

For the following problems, use the distributive property to expand the quantities.

Exercise 52

2(y+9) 2(y+9)

Exercise 53

b(r+5) b(r+5)

Solution

br+5b br+5b

Exercise 54

m(u+a) m(u+a)

Exercise 55

k(j+1) k(j+1)

Solution

jk+k jk+k

Exercise 56

x(2y+5) x(2y+5)

Exercise 57

z(x+9w) z(x+9w)

Solution

xz+9wz xz+9wz

Exercise 58

(1+ d)e (1+ d)e

Exercise 59

(8+ 2f)g (8+ 2f)g

Solution

8g+2fg 8g+2fg

Exercise 60

c(2a+ 10b) c(2a+ 10b)

Exercise 61

15x(2y+ 3z) 15x(2y+ 3z)

Solution

30xy+45xz 30xy+45xz

Exercise 62

8y(12a+b) 8y(12a+b)

Exercise 63

z(x+y+m) z(x+y+m)

Solution

xz+yz+mz xz+yz+mz

Exercise 64

(a+6)(x+y) (a+6)(x+y)

Exercise 65

(x+10)(a+b+c) (x+10)(a+b+c)

Solution

ax+bx+cx+10a+10b+10c ax+bx+cx+10a+10b+10c

Exercise 66

1(x+y) 1(x+y)

Exercise 67

1(a+16) 1(a+16)

Solution

a+16 a+16

Exercise 68

Use a calculator. 0.48(0.34a+0.61) 0.48(0.34a+0.61)

Exercise 69

Use a calculator. 21.5(16.2a+3.8b+0.7c) 21.5(16.2a+3.8b+0.7c)

Solution

348.3a+81.7b+15.05c 348.3a+81.7b+15.05c

Exercise 70

The product of five times a star, and the sum of two times a square and three times a rhombus.

Exercise 71

2 z t ( L m +8k) 2 z t ( L m +8k)

Solution

2 L m z t +16k z t 2 L m z t +16k z t

Exercises for Review

Exercise 72

((Reference)) Find the value of 4⋅2+5(2⋅4−6÷3)−2⋅5 4⋅2+5(2⋅4−6÷3)−2⋅5 .

Exercise 73

((Reference)) Is the statement 3(5⋅3−3⋅5)+6⋅2−3⋅4<0 3(5⋅3−3⋅5)+6⋅2−3⋅4<0 true or false?

Solution

false

Exercise 74

((Reference)) Draw a number line that extends from −2 −2 to 2 and place points at all integers between and including −2 −2 and 3.

Exercise 75

((Reference)) Replace the ∗ ∗ with the appropriate relation symbol (<,>).−7∗−3 (<,>).−7∗−3 .

Solution

< <

Exercise 76

((Reference)) What whole numbers can replace x x so that the statement −2≤x<2 −2≤x<2 is true?

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