1.9 Properties of Real Numbers - Elementary Algebra | OpenStax
Elementary Algebra1.9 Properties of Real Numbers
A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, The Properties of Real Numbers.
Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?
5+33+5885+33+588
5+3=3+55+3=3+5
As we can see, the order in which we add does not matter!
What about multiplying 5and3?5and3?
5·33·515155·33·51515
5·3=3·55·3=3·5
Again, the results are the same!
The order in which we multiply does not matter!
These examples illustrate the commutative property. When adding or multiplying, changing the order gives the same result.
of AdditionIfa,bare real numbers, thena+b=b+aof MultiplicationIfa,bare real numbers, thena·b=b·aof AdditionIfa,bare real numbers, thena+b=b+aof MultiplicationIfa,bare real numbers, thena·b=b·a
When adding or multiplying, changing the order gives the same result.
The commutative property has to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.
What about subtraction? Does order matter when we subtract numbers? Does 7−37−3 give the same result as 3−7?3−7?
7−33−74−44≠−47−3≠3−77−33−74−44≠−47−3≠3−7
Since changing the order of the subtraction did not give the same result, we know that subtraction is not commutative.
Let’s see what happens when we divide two numbers. Is division commutative?
12÷44÷121244123133≠1312÷4≠4÷1212÷44÷121244123133≠1312÷4≠4÷12
Since changing the order of the division did not give the same result, division is not commutative. The commutative properties only apply to addition and multiplication!
If you were asked to simplify this expression, how would you do it and what would your answer be?
Some people would think 7+8is157+8is15 and then 15+2is17.15+2is17. Others might start with 8+2makes108+2makes10 and then 7+10makes17.7+10makes17.
Either way gives the same result. Remember, we use parentheses as grouping symbols to indicate which operation should be done first.
Add 7+87+8.
Add. (7+8)+2 15+2 17 (7+8)+2 15+2 17
Add. 7+(8+2) 7+10 17 7+(8+2) 7+10 17
(7+8)+2=7+(8+2)(7+8)+2=7+(8+2)
When adding three numbers, changing the grouping of the numbers gives the same result.
This is true for multiplication, too.
Multiply. 5·135·13
Multiply. (5·13)·3 53·3 5 (5·13)·3 53·3 5
Multiply. 13·313·3.
Multiply. 5·(13·3) 5·1 5 5·(13·3) 5·1 5
(5·13)·3=5·(13·3)(5·13)·3=5·(13·3)
When multiplying three numbers, changing the grouping of the numbers gives the same result.
You probably know this, but the terminology may be new to you. These examples illustrate the associative property.
of AdditionIfa,b,care real numbers, then(a+b)+c=a+(b+c)of MultiplicationIfa,b,care real numbers, then(a·b)·c=a·(b·c)of AdditionIfa,b,care real numbers, then(a+b)+c=a+(b+c)of MultiplicationIfa,b,care real numbers, then(a·b)·c=a·(b·c)
When adding or multiplying, changing the grouping gives the same result.
Let’s think again about multiplying 5·13·3.5·13·3. We got the same result both ways, but which way was easier? Multiplying 1313 and 33 first, as shown above on the right side, eliminates the fraction in the first step. Using the associative property can make the math easier!
The associative property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.
We saw that subtraction and division were not commutative. They are not associative either.
When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the commutative property of addition to write the like terms together.
Simplify: 18p+6q+15p+5q.18p+6q+15p+5q.
18p+6q+15p+5q18p+6q+15p+5q
Use the commutative property of addition to re-order so that like terms are together. 18p+15p+6q+5q18p+15p+6q+5q
Add like terms. 33p+11q33p+11q
Simplify: 23r+14s+9r+15s.23r+14s+9r+15s.
Simplify: 37m+21n+4m−15n.37m+21n+4m−15n.
When we have to simplify algebraic expressions, we can often make the work easier by applying the commutative or associative property first, instead of automatically following the order of operations. When adding or subtracting fractions, combine those with a common denominator first.
Simplify: (513+34)+14.(513+34)+14.
(513+34)+14(513+34)+14
Notice that the last 2 terms have a common denominator, so change the grouping. 513+(34+14)513+(34+14)
Add in parentheses first. 513+(44)513+(44)
Simplify the fraction. 513+1513+1
Add. 15131513
Convert to an improper fraction. 18131813
Simplify: (29+712)+512.(29+712)+512.
Use the associative property to simplify 6(3x).6(3x).
6(3x)6(3x)
Change the grouping. (6·3)x(6·3)x
Multiply in the parentheses. 18x18x
Notice that we can multiply 6·36·3 but we could not multiply 3x without having a value for x.
Use the associative property to simplify 8(4x).
Use the associative property to simplify −9(7y).−9(7y).
What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity.
13+0−14+00+(−8)13−14−813+0−14+00+(−8)13−14−8
These examples illustrate the Identity Property of Addition that states that for any real number a,a, a+0=aa+0=a and 0+a=a.0+a=a.
What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity.
43·1−27·11·3543−273543·1−27·11·3543−2735
These examples illustrate the Identity Property of Multiplication that states that for any real number a,a, a·1=aa·1=a and 1·a=a.1·a=a.
We summarize the Identity Properties below.
of additionFor any real numbera:a+0=a0+a=a0is theadditive identityof multiplicationFor any real numbera:a·1=a1·a=a1is themultiplicative identityof additionFor any real numbera:a+0=a0+a=a0is theadditive identityof multiplicationFor any real numbera:a·1=a1·a=a1is themultiplicative identity
Notice that in each case, the missing number was the opposite of the number!
We call −a.−a. the additive inverse of a. The opposite of a number is its additive inverse. A number and its opposite add to zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number a,a+(−a)=0.a,a+(−a)=0. Remember, a number and its opposite add to zero.
What number multiplied by 2323 gives the multiplicative identity, 1? In other words, 2323 times what results in 1?
What number multiplied by 2 gives the multiplicative identity, 1? In other words 2 times what results in 1?
Notice that in each case, the missing number was the reciprocal of the number!
We call 1a1a the multiplicative inverse of a. The reciprocal of aa number is its multiplicative inverse. A number and its reciprocal multiply to one, which is the multiplicative identity. This leads to the Inverse Property of Multiplication that states that for any real number a,a≠0,a·1a=1.a,a≠0,a·1a=1.
We’ll formally state the inverse properties here:
of addition For any real number aa,
−a−a is the additive inverse of aa.
A number and its opposite add to zero. a+(−a)=0a+(−a)=0
of multiplication For any real number aa,
1a1a is the multiplicative inverse of aa.
A number and its reciprocal multiply to one. a·1a=1a·1a=1
Find the additive inverse of ⓐ 5858 ⓑ 0.60.6 ⓒ −8−8 ⓓ −43.−43.
To find the additive inverse, we find the opposite.
ⓐ The additive inverse of 5858 is the opposite of 58.58. The additive inverse of 5858 is −58.−58.
ⓑ The additive inverse of 0.6 is the opposite of 0.6. The additive inverse of 0.6 is −0.6.−0.6.
ⓒ The additive inverse of −8−8 is the opposite of −8.−8. We write the opposite of −8−8 as −(−8),−(−8), and then simplify it to 8. Therefore, the additive inverse of −8−8 is 8.
ⓓ The additive inverse of −43−43 is the opposite of −43.−43. We write this as −(−43),−(−43), and then simplify to 43.43. Thus, the additive inverse of −43−43 is 43.43.
Find the additive inverse of: ⓐ 7979 ⓑ 1.21.2 ⓒ −14−14 ⓓ −94.−94.
Find the additive inverse of: ⓐ 713713 ⓑ 8.48.4 ⓒ −46−46 ⓓ −52.−52.
Find the multiplicative inverse of ⓐ 99 ⓑ −19−19 ⓒ 0.9.0.9.
To find the multiplicative inverse, we find the reciprocal.
ⓐ The multiplicative inverse of 9 is the reciprocal of 9, which is 19.19. Therefore, the multiplicative inverse of 9 is 19.19.
ⓑ The multiplicative inverse of −19−19 is the reciprocal of −19,−19, which is −9.−9. Thus, the multiplicative inverse of −19−19 is −9.−9.
ⓒ To find the multiplicative inverse of 0.9, we first convert 0.9 to a fraction, 910.910. Then we find the reciprocal of the fraction. The reciprocal of 910910 is 109.109. So the multiplicative inverse of 0.9 is 109.109.
Find the multiplicative inverse of ⓐ 44 ⓑ −17−17 ⓒ 0.30.3
Find the multiplicative inverse of ⓐ 1818 ⓑ −45−45 ⓒ 0.6.0.6.
The identity property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.
For any real number a.
The product of any real number and 0 is 0.
What about division involving zero? What is 0÷3?0÷3? Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So,
0÷3=00÷3=0
We can check division with the related multiplication fact.
12÷6=2because2·6=12.12÷6=2because2·6=12.
So we know 0÷3=00÷3=0 because 0·3=0.0·3=0.
Division of Zero
For any real number a, except 0,0, 0a=00a=0 and 0÷a=0.0÷a=0.
Zero divided by any real number except zero is zero.
Now think about dividing by zero. What is the result of dividing 4 by 0? Think about the related multiplication fact: 4÷0=?4÷0=? means ?·0=4.?·0=4. Is there a number that multiplied by 0 gives 4? Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4.
We conclude that there is no answer to 4÷04÷0 and so we say that division by 0 is undefined.
For any real number a, except 0, a0a0 and a÷0a÷0 are undefined.
We summarize the properties of zero below.
Multiplication by Zero: For any real number a,
a·0=00·a=0a·0=00·a=0 The product of any number and 0 is 0.
Division of Zero, Division by Zero: For any real number a,a≠0a,a≠0
0a=00a=0 Zero divided by any real number except itself is zero.
a0is undefineda0is undefined Division by zero is undefined.
Simplify: ⓐ −8·0−8·0 ⓑ 0−20−2 ⓒ −320.−320.
The product of any real number and 0 is 0. −8·0 0 −8·0 0
The product of any real number and 0 is 0. 0−2 0 0−2 0
Division by 0 is undefined. −320 Undefined −320 Undefined
Simplify: ⓐ −14·0−14·0 ⓑ 0−60−6 ⓒ −20.−20.
Simplify: ⓐ 0(−17)0(−17) ⓑ 0−100−10 ⓒ −50.−50.
We will now practice using the properties of identities, inverses, and zero to simplify expressions.
Simplify: ⓐ 0n+5,0n+5, where n≠−5n≠−5 ⓑ 10−3p0,10−3p0, where 10−3p≠0.10−3p≠0.
Zero divided by any real number except itself is 0. 0n+5 0 0n+5 0
Division by 0 is undefined. 10−3p0 Undefined 10−3p0 Undefined
Simplify: −84n+(−73n)+84n.−84n+(−73n)+84n.
−84n+(−73n)+84n−84n+(−73n)+84n
Notice that the first and third terms are opposites; use the
commutative property of addition to re-order the terms. −84n+84n+(−73n)−84n+84n+(−73n)
Add left to right. 0+(−73)0+(−73)
Add. −73n−73n
Simplify: −27a+(−48a)+27a.−27a+(−48a)+27a.
Simplify: 39x+(−92x)+(−39x).39x+(−92x)+(−39x).
Simplify: 715·823·157.715·823·157.
715·823·157715·823·157
Notice that the first and third terms are reciprocals, so use the
commutative property of multiplication to re-order the factors. 715·157·823715·157·823
Multiply left to right. 1·8231·823
Simplify: 617·1125·176.617·1125·176.
Simplify: ⓐ 0m+7,0m+7, where m≠−7m≠−7 ⓑ 18−6c0,18−6c0, where 18−6c≠0.18−6c≠0.
Simplify: ⓐ 0d−4,whered≠40d−4,whered≠4 ⓑ 15−4q0,where15−4q≠0.15−4q0,where15−4q≠0.
Simplify: 34·43(6x+12).34·43(6x+12).
34·43(6x+12)34·43(6x+12)
There is nothing to do in the parentheses, so multiply the
two fractions first—notice, they are reciprocals. 1(6x+12)1(6x+12)
Simplify by recognizing the multiplicative identity. 6x+126x+12
Simplify: 25·52(20y+50).25·52(20y+50).
Simplify: 38·83(12z+16).38·83(12z+16).
Suppose that three friends are going to the movies. They each need $9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?
You can think about the dollars separately from the quarters. They need 3 times $9 so $27, and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the distributive property.
Ifa,b,care real numbers, thena(b+c)=ab+acAlso,(b+c)a=ba+caa(b−c)=ab−ac(b−c)a=ba−caIfa,b,care real numbers, thena(b+c)=ab+acAlso,(b+c)a=ba+caa(b−c)=ab−ac(b−c)a=ba−ca
Back to our friends at the movies, we could find the total amount of money they need like this:
3(9.25)3(9+0.25)3(9)+3(0.25)27+0.7527.753(9.25)3(9+0.25)3(9)+3(0.25)27+0.7527.75
In algebra, we use the distributive property to remove parentheses as we simplify expressions.
For example, if we are asked to simplify the expression 3(x+4),3(x+4), the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the distributive property, as shown in Example 1.132.
Simplify: 3(x+4).3(x+4).
3(x+4)3(x+4)
Distribute. 3·x+3·43·x+3·4
Multiply. 3x+123x+12
Some students find it helpful to draw in arrows to remind them how to use the distributive property. Then the first step in Example 1.132 would look like this:
Simplify: 8(38x+14).8(38x+14).
Simplify: 6(56y+12).6(56y+12).
Simplify: 12(13n+34).12(13n+34).
Using the distributive property as shown in Example 1.134 will be very useful when we solve money applications in later chapters.
Simplify: 100(0.3+0.25q).100(0.3+0.25q).
Simplify: 100(0.7+0.15p).100(0.7+0.15p).
Simplify: 100(0.04+0.35d).100(0.04+0.35d).
When we distribute a negative number, we need to be extra careful to get the signs correct!
Simplify: −2(4y+1).−2(4y+1).
Simplify: −3(6m+5).−3(6m+5).
Simplify: −6(8n+11).−6(8n+11).
Simplify: −11(4−3a).−11(4−3a).
Notice that you could also write the result as 33a−44.33a−44. Do you know why?
Simplify: −5(2−3a).−5(2−3a).
Simplify: −7(8−15y).−7(8−15y).
Example 1.137 will show how to use the distributive property to find the opposite of an expression.
Simplify: −(y+5).−(y+5).
Multiplying by −1 results in the opposite. −1(y+5)−1(y+5)
Distribute. −1·y+(−1)·5−1·y+(−1)·5
Simplify. −y+(−5)−y+(−5)
−y−5−y−5
Simplify: −(z−11).−(z−11).
There will be times when we’ll need to use the distributive property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.
Simplify: 8−2(x+3).8−2(x+3).
Be sure to follow the order of operations. Multiplication comes before subtraction, so we will distribute the 2 first and then subtract.
8−2(x+3)8−2(x+3)
Distribute. 8−2·x−2·38−2·x−2·3
Multiply. 8−2x−68−2x−6
Combine like terms. −2x+2−2x+2
Simplify: 7x−5(x+4).7x−5(x+4).
Simplify: 4(x−8)−(x+3).4(x−8)−(x+3).
4(x−8)−(x+3)4(x−8)−(x+3)
Distribute. 4x−32−x−34x−32−x−3
Combine like terms. 3x−353x−35
Simplify: 6(x−9)−(x+12).6(x−9)−(x+12).
All the properties of real numbers we have used in this chapter are summarized in Table 1.57.
of addition If a,ba,b are real numbers, then
of multiplication If a,ba,b are real numbers, then a+b=b+aa+b=b+a
of addition If a,b,ca,b,c are real numbers, then
of multiplication If a,b,ca,b,c are real numbers, then (a+b)+c=a+(b+c)(a+b)+c=a+(b+c)
(a·b)·c=a·(b·c)(a·b)·c=a·(b·c)
If a,b,ca,b,c are real numbers, then a(b+c)=ab+aca(b+c)=ab+ac
of addition For any real number a:a:
0 is the additive identity
of multiplication For any real number a:a:
11 is the multiplicative identity a+0=a0+a=aa+0=a0+a=a
a·1=a1·a=aa·1=a1·a=a
of addition For any real number a,a,
−a−a is the additive inverse of aa
of multiplication For any real number a,a≠0a,a≠0
1a1a is the multiplicative inverse of a.a. a+(−a)=0a+(−a)=0
a·1a=1a·1a=1
For any real number a,a≠0a,a≠0
For any real number a,a≠0a,a≠0 a·0=00·a=0a·0=00·a=0
0a=00a=0
a0a0 is undefined
4(7m)
(y+12)+28(y+12)+28
(n+17)+33(n+17)+33
12+78+(−12)12+78+(−12)
25+512+(−25)25+512+(−25)
320·4911·203320·4911·203
1318·257·18131318·257·1813
−24.7·38−24.7·38
−36·11·49−36·11·49
(56+815)+715(56+815)+715
(1112+49)+59(1112+49)+59
−15(5m)−15(5m)
−23(2n)−23(2n)
12(56p)12(56p)
20(35q)20(35q)
43m+(−12n)+(−16m)+(−9n)43m+(−12n)+(−16m)+(−9n)
−22p+17q+(−35p)+(−27q)−22p+17q+(−35p)+(−27q)
38g+112h+78g+512h38g+112h+78g+512h
56a+310b+16a+910b56a+310b+16a+910b
6.8p+9.14q+(−4.37p)+(−0.88q)6.8p+9.14q+(−4.37p)+(−0.88q)
9.6m+7.22n+(−2.19m)+(−0.65n)9.6m+7.22n+(−2.19m)+(−0.65n)
In the following exercises, find the additive inverse of each number.
ⓑ 4.3
ⓒ −8−8
ⓓ −103−103
ⓑ −0.075−0.075
ⓐ 6 ⓑ −34−34 ⓒ 0.7
ⓐ 12 ⓑ −92−92 ⓒ 0.13
ⓐ 11121112 ⓑ −1.1−1.1 ⓒ −4−4
0÷11120÷1112
(−3.14)(0)(−3.14)(0)
19a+44−19a19a+44−19a
27c+16−27c27c+16−27c
10(0.1d)10(0.1d)
100(0.01p)100(0.01p)
0u−4.99,0u−4.99, where u≠4.99u≠4.99
0v−65.1,0v−65.1, where v≠65.1v≠65.1
0÷(x−12),0÷(x−12), where x≠12x≠12
0÷(y−16),0÷(y−16), where x≠16x≠16
32−5a0,32−5a0, where
32−5a≠032−5a≠0
28−9b0,28−9b0, where
28−9b≠028−9b≠0
(34+910m)÷0(34+910m)÷0 where
34+910m≠034+910m≠0
(516n−37)÷0(516n−37)÷0 where
516n−37≠0516n−37≠0
15·35(4d+10)15·35(4d+10)
18·56(15h+24)18·56(15h+24)
8(4y+9)8(4y+9)
9(3w+7)9(3w+7)
6(c−13)6(c−13)
7(y−13)7(y−13)
14(3q+12)14(3q+12)
15(4m+20)15(4m+20)
9(59y−13)9(59y−13)
10(310x−25)10(310x−25)
12(14+23r)12(14+23r)
12(16+34s)12(16+34s)
r(s−18)r(s−18)
u(v−10)u(v−10)
(y+4)p(y+4)p
(a+7)x(a+7)x
−7(4p+1)−7(4p+1)
−9(9a+4)−9(9a+4)
−4(q−7)−4(q−7)
−(3x−7)−(3x−7)
−(5p−4)−(5p−4)
16−3(y+8)16−3(y+8)
18−4(x+2)18−4(x+2)
4−11(3c−2)4−11(3c−2)
9−6(7n−5)9−6(7n−5)
22−(a+3)22−(a+3)
8−(r−7)8−(r−7)
(5m−3)−(m+7)(5m−3)−(m+7)
(4y−1)−(y−2)(4y−1)−(y−2)
5(2n+9)+12(n−3)5(2n+9)+12(n−3)
9(5u+8)+2(u−6)9(5u+8)+2(u−6)
9(8x−3)−(−2)9(8x−3)−(−2)
14(c−1)−8(c−6)14(c−1)−8(c−6)
11(n−7)−5(n−1)11(n−7)−5(n−1)
6(7y+8)−(30y−15)6(7y+8)−(30y−15)
7(3n+9)−(4n−13)7(3n+9)−(4n−13)
ⓐ First, by multiplying 0.20 by 80 to find her copay for each filling and then multiplying your answer by 5 to find her total copay for 5 fillings.
ⓑ Next, by multiplying [5(0.20)](80)
ⓒ Which of the properties of real numbers says that your answers to parts (a), where you multiplied 5[(0.20)(80)] and (b), where you multiplied [5(0.20)](80), should be equal?
ⓐ First, by multiplying 24·2024·20 to find the total number of minutes and then multiplying the answer by 160160 to convert minutes into hours.
ⓑ Next, by multiplying 24(20·160).24(20·160).
ⓒ Which of the properties of real numbers says that your answers to parts (a), where you multiplied (24·20)160,(24·20)160, and (b), where you multiplied 24(20·160),24(20·160), should be equal?
Buying by the case Trader Joe’s grocery stores sold a bottle of wine they called “Two Buck Chuck” for $1.99. They sold a case of 12 bottles for $23.88. To find the cost of 12 bottles at $1.99, notice that 1.99 is 2−0.01.2−0.01.
ⓐ Multiply 12(1.99) by using the distributive property to multiply 12(2−0.01).12(2−0.01).
ⓑ Was it a bargain to buy “Two Buck Chuck” by the case?
Multi-pack purchase Adele’s shampoo sells for $3.99 per bottle at the grocery store. At the warehouse store, the same shampoo is sold as a 3 pack for $10.49. To find the cost of 3 bottles at $3.99, notice that 3.99 is 4−0.01.4−0.01.
ⓐ Multiply 3(3.99) by using the distributive property to multiply 3(4−0.01).3(4−0.01).
ⓑ How much would Adele save by buying 3 bottles at the warehouse store instead of at the grocery store?
Simplify 8(x−14)8(x−14) using the distributive property and explain each step.
Explain how you can multiply 4($5.97) without paper or calculator by thinking of $5.97 as 6−0.036−0.03 and then using the distributive property.
Section URL: https://openstax.org/books/elementary-algebra/pages/1-9-properties-of-real-numbers