# 1.9 Properties of real numbers

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By the end of this section, you will be able to:
• Use the commutative and associative properties
• Use the identity and inverse properties of addition and multiplication
• Use the properties of zero
• Simplify expressions using the distributive property

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, The Properties of Real Numbers .

## Use the commutative and associative properties

Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?

$\begin{array}{cccc}\hfill 5+3\hfill & & & \hfill 3+5\hfill \\ \hfill 8\hfill & & & \hfill 8\hfill \end{array}$
$5+3=3+5$

The results are the same.

As we can see, the order in which we add does not matter!

What about multiplying $5\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3?$

$\begin{array}{cccc}\hfill 5·3\hfill & & & \hfill 3·5\hfill \\ \hfill 15\hfill & & & \hfill 15\hfill \end{array}$
$5·3=3·5$

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.

## Commutative property

$\begin{array}{ccccccccc}\mathbf{\text{of Addition}}\hfill & & & \text{If}\phantom{\rule{0.2em}{0ex}}a,b\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill a+b& =\hfill & b+a\hfill \\ \mathbf{\text{of Multiplication}}\hfill & & & \text{If}\phantom{\rule{0.2em}{0ex}}a,b\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\hfill & & & \hfill a·b& =\hfill & b·a\hfill \end{array}$

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 $7-3$ give the same result as $3-7?$

$\begin{array}{c}\hfill \begin{array}{cc}\hfill 7-3\hfill & \hfill \phantom{\rule{1em}{0ex}}3-7\hfill \\ \hfill 4\hfill & \hfill \phantom{\rule{1em}{0ex}}-4\hfill \end{array}\hfill \\ \\ \\ \hfill \phantom{\rule{0.5em}{0ex}}4\ne \text{−}4\hfill \\ \hfill 7-3\ne 3-7\hfill \end{array}$

The results are not the same.

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?

$\begin{array}{}\\ \hfill \begin{array}{cc}\hfill 12÷4\hfill & \hfill \phantom{\rule{1em}{0ex}}4÷12\hfill \\ \hfill \frac{12}{4}\hfill & \hfill \phantom{\rule{1em}{0ex}}\frac{4}{12}\hfill \\ \hfill 3\hfill & \hfill \phantom{\rule{1em}{0ex}}\frac{1}{3}\hfill \end{array}\hfill \\ \hfill 3\ne \frac{1}{3}\hfill \\ \hfill 12÷4\ne 4÷12\hfill \end{array}$

The results are not the same.

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!

• Addition and multiplication are commutative.
• Subtraction and Division are not commutative.

If you were asked to simplify this expression, how would you do it and what would your answer be?

$7+8+2$

Some people would think $7+8\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}15$ and then $15+2\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}17.$ Others might start with $8+2\phantom{\rule{0.2em}{0ex}}\text{makes}\phantom{\rule{0.2em}{0ex}}10$ and then $7+10\phantom{\rule{0.2em}{0ex}}\text{makes}\phantom{\rule{0.2em}{0ex}}17.$

Either way gives the same result. Remember, we use parentheses as grouping symbols to indicate which operation should be done first.

$\begin{array}{c}\begin{array}{ccc}& & \hfill \phantom{\rule{4em}{0ex}}\left(7+8\right)+2\hfill \\ \text{Add}\phantom{\rule{0.2em}{0ex}}7+8.\hfill & & \hfill \phantom{\rule{4em}{0ex}}15+2\hfill \\ \text{Add.}\hfill & & \hfill \phantom{\rule{4em}{0ex}}17\hfill \\ \\ \\ & & \hfill \phantom{\rule{4em}{0ex}}7+\left(8+2\right)\hfill \\ \text{Add}\phantom{\rule{0.2em}{0ex}}8+2.\hfill & & \hfill \phantom{\rule{4em}{0ex}}7+10\hfill \\ \text{Add.}\hfill & & \hfill \phantom{\rule{4em}{0ex}}17\hfill \end{array}\hfill \\ \\ \\ \left(7+8\right)+2=7+\left(8+2\right)\hfill \end{array}$

When adding three numbers, changing the grouping of the numbers gives the same result.

This is true for multiplication, too.

$\begin{array}{c}\begin{array}{ccc}& & \hfill \phantom{\rule{2em}{0ex}}\left(5·\frac{1}{3}\right)·3\hfill \\ \text{Multiply.}\phantom{\rule{1em}{0ex}}5·\frac{1}{3}\hfill & & \hfill \phantom{\rule{2em}{0ex}}\frac{5}{3}·3\hfill \\ \text{Multiply.}\hfill & & \hfill \phantom{\rule{2em}{0ex}}5\hfill \\ \\ \\ & & \hfill \phantom{\rule{2em}{0ex}}5·\left(\frac{1}{3}·3\right)\hfill \\ \text{Multiply.}\phantom{\rule{1em}{0ex}}\frac{1}{3}·3\hfill & & \hfill \phantom{\rule{2em}{0ex}}5·1\hfill \\ \text{Multiply.}\hfill & & \hfill \phantom{\rule{2em}{0ex}}5\hfill \end{array}\hfill \\ \\ \\ \left(5·\frac{1}{3}\right)·3=5·\left(\frac{1}{3}·3\right)\hfill \end{array}$

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 .

## Associative property

$\begin{array}{cccc}\mathbf{\text{of Addition}}\hfill & & & \text{If}\phantom{\rule{0.2em}{0ex}}a,b,c\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\phantom{\rule{0.2em}{0ex}}\left(a+b\right)+c=a+\left(b+c\right)\hfill \\ \mathbf{\text{of Multiplication}}\hfill & & & \text{If}\phantom{\rule{0.2em}{0ex}}a,b,c\phantom{\rule{0.2em}{0ex}}\text{are real numbers, then}\phantom{\rule{0.2em}{0ex}}\left(a·b\right)·c=a·\left(b·c\right)\hfill \end{array}$

When adding or multiplying, changing the grouping gives the same result.

Let’s think again about multiplying $5·\frac{1}{3}·3.$ We got the same result both ways, but which way was easier? Multiplying $\frac{1}{3}$ and $3$ first, as shown above on the right side, eliminates the fraction in the first step. Using the associative property can make the math easier!

-65r to the 4th power-50r cubed-15r squared+8r+23 ÷ 5r
write in this form a/b answer should be in the simplest form 5%
convert to decimal 9/11
August
Equation in the form of a pending point y+2=1/6(×-4)
write in simplest form 3 4/2
August
From Google: The quadratic formula, , is used in algebra to solve quadratic equations (polynomial equations of the second degree). The general form of a quadratic equation is , where x represents a variable, and a, b, and c are constants, with . A quadratic equation has two solutions, called roots.
Melissa
what is the answer of w-2.6=7.55
10.15
Michael
w = 10.15 You add 2.6 to both sides and then solve for w (-2.6 zeros out on the left and leaves you with w= 7.55 + 2.6)
Korin
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her$66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
x > $110,000 bruce greater than$110,000
Michael
Estelle is making 30 pounds of fruit salad from strawberries and blueberries. Strawberries cost $1.80 per pound, and blueberries cost$4.50 per pound. If Estelle wants the fruit salad to cost her $2.52 per pound, how many pounds of each berry should she use? nawal Reply$1.38 worth of strawberries + $1.14 worth of blueberries which=$2.52
Leitha
how
Zaione
is it right😊
Leitha
lol maybe
Robinson
8 pound of blueberries and 22 pounds of strawberries
Melissa
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
8 pounds x 4.5 equal 36 22 pounds x 1.80 equal 39.60 36 + 39.60 equal 75.60 75.60 / 30 equal average 2.52 per pound
Melissa
hmmmm...... ?
Robinson
8 pounds x 4.5 = 36 22 pounds x 1.80 = 39.60 36 + 39.60 = 75.60 75.60 / 30 = average 2.52 per pound
Melissa
The question asks how many pounds of each in order for her to have an average cost of $2.52. She needs 30 lb in all so 30 pounds times$2.52 equals $75.60. that's how much money she is spending on the fruit. That means she would need 8 pounds of blueberries and 22 lbs of strawberries to equal 75.60 Melissa good Robinson 👍 Leitha thanks Melissa. Leitha nawal let's do another😊 Leitha we can't use emojis...I see now Leitha Sorry for the multi post. My phone glitches. Melissa Vina has$4.70 in quarters, dimes and nickels in her purse. She has eight more dimes than quarters and six more nickels than quarters. How many of each coin does she have?
10 quarters 16 dimes 12 nickels
Leitha
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet.
wtf. is a tail wind or headwind?
Robert
48 miles per hour with headwind and 68 miles per hour with tailwind
Leitha
average speed is 58 mph
Leitha
Into the wind (headwind), 125 mph; with wind (tailwind), 175 mph. Use time (t) = distance (d) ÷ rate (r). since t is equal both problems, then 1210/(x-25) = 1694/(×+25). solve for x gives x=150.
bruce
the jet will fly 9.68 hours to cover either distance
bruce
Riley is planning to plant a lawn in his yard. He will need 9 pounds of grass seed. He wants to mix Bermuda seed that costs $4.80 per pound with Fescue seed that costs$3.50 per pound. How much of each seed should he buy so that the overall cost will be $4.02 per pound? Vonna Reply 33.336 Robinson Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost$8 per square foot and decorator tiles that cost $20 per square foot. How many square feet of each tile should she use so that the overall cost of the backsplash will be$10 per square foot?
Ivan has $8.75 in nickels and quarters in his desk drawer. The number of nickels is twice the number of quarters. How many coins of each type does he have? mikayla Reply 2q=n ((2q).05) + ((q).25) = 8.75 .1q + .25q = 8.75 .35q = 8.75 q = 25 quarters 2(q) 2 (25) = 50 nickles Answer check 25 x .25 = 6.25 50 x .05 = 2.50 6.25 + 2.50 = 8.75 Melissa John has$175 in $5 and$10 bills in his drawer. The number of $5 bills is three times the number of$10 bills. How many of each are in the drawer?
7-$10 21-$5
Robert
Enrique borrowed $23,500 to buy a car. He pays his uncle 2% interest on the$4,500 he borrowed from him, and he pays the bank 11.5% interest on the rest. What average interest rate does he pay on the total \$23,500? (Round your answer to the nearest tenth of a percent.)
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hour longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
8mph
michele
16mph
Robert
3.8 mph
Ped
16 goes into 80 5times while 20 goes into 80 4times and is 4mph faster
Robert