Properties Of Real Numbers Practice

Learning Objectives

By the end of this section, you will be able to:

• Use the commutative and associative properties
• Use the properties of identity, inverse, and zero
• Simplify expressions using the Distributive Holding

Be Prepared ane.5

A more thorough introduction to the topics covered in this section tin can be found in the Elementary Algebra 2e chapter, Foundations.

Utilize the Commutative and Associative Properties

The order we add two numbers doesn't touch on the result. If we add $\mathrm{viii}+9$ or $9+8,$ the results are the same—they both equal 17. And then, $8+9=9+8.$ The order in which we add together does not matter!

Similarly, when multiplying ii numbers, the society does not affect the upshot. If we multiply $\mathrm{ix}·8$ or $8·9$ the results are the same—they both equal 72. And then, $9·8=8·9.$ The club in which nosotros multiply does non matter!

These examples illustrate the Commutative Property.

Commutative Property

$$\begin{array}{ccccccc}\text{of Addition}\hfill & & & \text{If}a\text{and}b\text{are real numbers, and so}\hfill & & & a+b=b+a.\hfill \\ \text{of Multiplication}\hfill & & & \text{If}a\text{and}b\text{are real numbers, then}\hfill & & & a·b=b·a.\hfill \end{array}$$

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

The Commutative Property has to do with order. We decrease $\mathrm{ix}-8$ and $\mathrm{viii}-9$ , and see that $\mathrm{nine}-\mathrm{viii}\ne 8-9.$ Since irresolute the order of the subtraction does not give the same upshot, we know that subtraction is not commutative.

Partition is not commutative either. Since $12÷3\ne \mathrm{three}÷12,$ changing the order of the partitioning did not give the aforementioned result. The commutative properties use only to addition and multiplication!

  Addition and multiplication are commutative.

  Subtraction and division are not commutative.

When adding three numbers, changing the grouping of the numbers gives the aforementioned issue. For instance, $\left(\mathrm{seven}+8\right)+\mathrm{two}=7+\left(8+2\right),$ since each side of the equation equals 17.

This is true for multiplication, too. For example, $\left(5·\frac{1}{3}\right)·3=5·\left(\frac{1}{\mathrm{three}}·3\right),$ since each side of the equation equals 5.

These examples illustrate the Associative Holding.

Associative Belongings

$$\begin{array}{ccccccc}\text{of Addition}\hfill & & & \text{If}a,b,\text{and}c\text{are real numbers, then}\hfill & & & \left(a+b\right)+c=a+\left(b+c\right).\hfill \\ \text{of Multiplication}\hfill & & & \text{If}\text{a},b,\text{and}c\text{are real numbers, then}\hfill & & & \left(a·b\right)·c=a·\left(b·c\right).\hfill \end{array}$$

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

The Associative Property has to do with group. If we modify how the numbers are grouped, the result will exist the aforementioned. Notice it is the same iii numbers in the same club—the only difference is the grouping.

Nosotros saw that subtraction and division were non commutative. They are not associative either.

$$\begin{array}{cccc}\hfill \left(10-\mathrm{three}\right)-\mathrm{ii}\ne \mathrm{ten}-\left(3-\mathrm{ii}\right)\hfill & & & \hfill \left(24÷4\right)÷2\ne 24÷\left(4÷2\right)\hfill \\ \hfill 7-2\ne \mathrm{x}-1\hfill & & & \hfill \mathrm{six}÷\mathrm{two}\ne 24÷2\hfill \\ \hfill \mathrm{five}\ne 9\hfill & & & \hfill 3\ne 12\hfill \end{array}$$

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 adjacent instance, we will use the Commutative Belongings of addition to write the similar terms together.

Example 1.45

Simplify: $18p+\mathrm{half\; dozen}q+15p+\mathrm{five}q.$

Try It 1.89

Simplify: $23r+14s+9r+\mathrm{fifteen}\mathrm{south}.$

Effort It one.90

Simplify: $37\mathrm{chiliad}+21n+4\mathrm{grand}-15n.$

When nosotros have to simplify algebraic expressions, we can often make the work easier by applying the Commutative Property or Associative Property get-go.

Example one.46

Simplify: $\left(\frac{5}{\mathrm{thirteen}}+\frac{3}{4}\right)+\frac{1}{\mathrm{iv}}.$

Try It 1.91

Simplify: $\left(\frac{7}{15}+\frac{5}{\mathrm{eight}}\right)+\frac{3}{8}.$

Endeavor It i.92

Simplify: $\left(\frac{2}{9}+\frac{7}{12}\right)+\frac{5}{12}.$

Utilize the Properties of Identity, Inverse, and Zero

What happens when we add 0 to whatever number? Adding 0 doesn't change the value. For this reason, we telephone call 0 the additive identity. The Identity Property of Add-on that states that for whatsoever real number $a,a+0=a$ and $0+a=a.$

What happens when we multiply whatsoever number by one? Multiplying by 1 doesn't modify the value. And so we call 1 the multiplicative identity. The Identity Belongings of Multiplication that states that for any real number $a,a·1=a$ and $1·a=a.$

Nosotros summarize the Identity Properties here.

Identity Holding

$$\begin{array}{cccc}\text{of Addition}\text{For any existent number}a:a+0=a\hfill & & & 0+a=a\hfill \\ 0\text{is the}\text{condiment identity}\hfill & & & \\ \text{of Multiplication}\text{For any existent number}a:a·\mathrm{ane}=a\hfill & & & 1·a=a\hfill \\ \mathrm{ane}\text{is the}\text{multiplicative identity}\hfill \end{array}$$

What number added to 5 gives the condiment identity, 0? We know

The missing number was the opposite of the number!

We call $\text{−}a$ 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 $a,a+\left(\text{−}a\right)=0.$

What number multiplied by $\frac{\mathrm{ii}}{\mathrm{iii}}$ gives the multiplicative identity, 1? In other words, $\frac{2}{3}$ times what results in 1? Nosotros know

The missing number was the reciprocal of the number!

We call $\frac{\mathrm{i}}{a}$ the multiplicative changed of a. The reciprocal of a number is its multiplicative inverse. This leads to the Changed Property of Multiplication that states that for whatever existent number $a,a\ne 0,a·\frac{1}{a}=\mathrm{one}.$

Nosotros'll formally country the changed properties here.

Changed Property

$$\begin{array}{cccc}\text{of Add-on}\hfill & & & \text{For whatsoever real number}a,a+\left(\text{−}a\right)=0\hfill \\ & & & \text{−}a\text{is the}\text{additive inverse}\text{of}a\hfill \\ & & & \text{A number and its}opposite\text{add to zero.}\hfill \\ \\ \\ \\ \\ \text{of Multiplication}\hfill & & & \text{For any real number}a,a\ne 0,a·\frac{1}{a}=\mathrm{ane}.\hfill \\ & & & \frac{1}{a}\text{is the}\text{multiplicative inverse}\text{of}a.\hfill \\ & & & \text{A number and its}r\mathrm{eastward}ciprocal\text{multiply to 1.}\hfill \end{array}$$

The Identity Property of add-on says that when nosotros 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.

What about division involving null? What is $0÷\mathrm{iii}?$ Retrieve about a existent instance: 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, $0÷3=0.$

We can check division with the related multiplication fact. So we know $0÷3=0$ because $0·3=0.$

Now retrieve most dividing by zero. What is the result of dividing 4 by $0?$ Recollect almost the related multiplication fact:

Is at that place a number that multiplied by 0 gives $4?$ Since any existent number multiplied by 0 gives 0, there is no real number that tin be multiplied by 0 to obtain 4. We conclude that there is no answer to $4÷0$ and so we say that segmentation by 0 is undefined.

We summarize the properties of zero here.

Properties of Goose egg

Multiplication by Zero: For any existent number a,

$a·0=00·a=0\text{The product of any number and 0 is 0.}$

Division past Nil: For whatever real number a, $a\ne 0$

$$\begin{array}{cccc}\hfill \frac{0}{a}=0\hfill & & & \text{Nada divided by any real number, except itself, is zero.}\hfill \\ \hfill \frac{a}{0}\text{is undefined}\hfill & & & \text{Division past zero is undefined.}\hfill \end{array}$$

We will at present do using the properties of identities, inverses, and nil to simplify expressions.

Instance 1.47

Simplify: $\mathrm{-84}\mathrm{due\; north}+\left(\mathrm{-73}n\right)+84n.$

Effort It i.93

Simplify: $\mathrm{-27}a+\left(\mathrm{-48}a\right)+27a.$

Endeavour It i.94

Simplify: $39x+\left(\mathrm{-92}\mathrm{10}\right)+\left(\mathrm{-39}\mathrm{ten}\right).$

Now we will see how recognizing reciprocals is helpful. Before multiplying left to correct, await for reciprocals—their production is 1.

Example 1.48

Simplify: $\frac{7}{15}·\frac{\mathrm{viii}}{23}·\frac{15}{7}.$

Attempt It 1.95

Simplify: $\frac{9}{\mathrm{xvi}}·\frac{5}{49}·\frac{16}{\mathrm{nine}}.$

Try Information technology i.96

Simplify: $\frac{6}{17}·\frac{\mathrm{eleven}}{25}·\frac{17}{\mathrm{half\; dozen}}.$

The next example makes united states enlightened of the distinction betwixt dividing 0 by some number or some number beingness divided past 0.

Example 1.49

Simplify: ⓐ $\frac{0}{n+5},$ where $n\ne \text{−}\mathrm{v}$ ⓑ $\frac{\mathrm{ten}-\mathrm{iii}p}{0},$ where $\mathrm{ten}-\mathrm{three}p\ne 0.$

Try It i.97

Simplify: ⓐ $\frac{0}{m+\mathrm{vii}},$ where $m\ne \text{−}7$ ⓑ $\frac{18-\mathrm{vi}c}{0},$ where $18-\mathrm{six}c\ne 0.$

Attempt It 1.98

Simplify: ⓐ $\frac{0}{d-4},$ where $d\ne 4$ ⓑ $\frac{\mathrm{xv}-\mathrm{four}q}{0},$ where $\mathrm{fifteen}-4q\ne 0.$

Simplify Expressions Using the Distributive Property

Suppose that three friends are going to the movies. They each need $9.25—that'southward 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?

Yous tin can remember about the dollars separately from the quarters. They demand iii times $ix so $27 and three times 1 quarter, and so 75 cents. In total, they need $27.75. If yous think about doing the math in this way, you are using the Distributive Property.

Distributive Property

$\begin{array}{cccc}\text{If}a,b,\text{and}c\text{are existent numbers, then}\hfill & & & a\left(b+c\right)=ab+ac\hfill \\ & & & \left(b+c\right)a=ba+ca\hfill \\ & & & a\left(b-c\right)=ab-ac\hfill \\ & & & \left(b-c\right)a=ba-ca\hfill \end{array}$

In algebra, we use the Distributive Belongings to remove parentheses as nosotros simplify expressions.

Instance 1.50

Try It ane.99

Try It one.100

Some students observe it helpful to draw in arrows to remind them how to use the Distributive Property. And then the get-go stride in Instance 1.50 would look similar this:

Example 1.51

Simplify: $8\left(\frac{3}{8}\mathrm{ten}+\frac{1}{4}\right).$

Try It one.101

Simplify: $6\left(\frac{5}{\mathrm{half\; dozen}}y+\frac{1}{2}\right).$

Endeavour Information technology 1.102

Simplify: $12\left(\frac{\mathrm{i}}{\mathrm{three}}n+\frac{3}{4}\right).$

Using the Distributive Belongings as shown in the next example volition exist very useful when nosotros solve money applications in later chapters.

Example 1.52

Simplify: $100\left(0.3+0.25q\right).$

Try Information technology i.103

Simplify: $100\left(0.7+0.15p\right).$

Effort It 1.104

Simplify: $100\left(0.04+0.35d\right).$

When we distribute a negative number, we need to be extra careful to get the signs right!

Example one.53

Simplify: $\mathrm{-11}\left(4-\mathrm{three}a\right).$

Effort It 1.105

Simplify: $\mathrm{-v}\left(\mathrm{ii}-\mathrm{three}a\right).$

Try Information technology 1.106

Simplify: $\mathrm{-7}\left(\mathrm{viii}-\mathrm{fifteen}y\right).$

In the next case, we will show how to use the Distributive Property to detect the reverse of an expression.

Example ane.54

Try It 1.107

Simplify: $\text{−}\left(z-11\right).$

Effort Information technology ane.108

There will exist times when nosotros'll demand to use the Distributive Property equally 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 2 examples will illustrate this.

Example 1.55

Simplify: $8-2\left(x+3\right)$

Effort It ane.109

Simplify: $\mathrm{ix}-3\left(\mathrm{10}+2\right).$

Try It 1.110

Simplify: $7x-5\left(x+4\right).$

Example i.56

Simplify: $4\left(x-8\right)-\left(x+3\right).$

Try It 1.111

Simplify: $\mathrm{half-dozen}\left(\mathrm{ten}-9\right)-\left(x+12\right).$

Endeavour It 1.112

Simplify: $\mathrm{eight}\left(x-1\right)-\left(x+5\right).$

All the properties of real numbers we have used in this chapter are summarized hither.

Commutative Property When calculation or multiplying, changing the lodge gives the same event $\begin{array}{cccccc}\text{of add-on}\text{If}a,b\text{are real numbers, then}\hfill & & & \hfill a+b& =\hfill & b+a\hfill \\ \text{of multiplication}\text{If}a,b\text{are existent numbers, then}\hfill & & & \hfill a·b& =\hfill & b·a\hfill \end{array}$

Associative Property When adding or multiplying, irresolute the grouping gives the same issue. $\begin{array}{cccccc}\text{of addition}\text{If}a,b,\text{and}c\text{are real numbers, then}\hfill & & & \hfill \left(a+b\right)+c& =\hfill & a+\left(b+c\right)\hfill \\ \text{of multiplication}\text{If}a,b,\text{and}c\text{are real numbers, and so}\hfill & & & \hfill \left(a·b\right)·c& =\hfill & a·\left(b·c\right)\hfill \end{array}$

Distributive Property $\begin{array}{cccccc}\text{If}a,b,\text{and}c\text{are real numbers, so}\hfill & & & \hfill a\left(b+c\right)& =\hfill & ab+ac\hfill \\ \\ & & & \hfill \left(b+c\right)a& =\hfill & ba+ca\hfill \\ \\ & & & \hfill a\left(b-c\right)& =\hfill & ab-ac\hfill \\ \\ & & & \hfill \left(b-c\right)a& =\hfill & ba-ca\hfill \end{array}$

Identity Belongings $\begin{array}{cccc}\text{of addition}\text{For any real number}a\text{:}\hfill & & & a+0=a\hfill \\ 0\text{is the}\text{condiment identity}\hfill & & & 0+a=a\hfill \\ \text{of multiplication}\text{For any existent number}a\text{:}\hfill & & & a·1=a\hfill \\ 1\text{is the}\text{multiplicative identity}\hfill & & & 1·a=a\hfill \end{array}$

Inverse Property $\begin{array}{cccc}\text{of add-on}\text{For any real number}a,\hfill & & & \hfill a+\left(\text{−}a\right)=0\\ \text{−}a\text{is the}\text{condiment inverse}\text{of}a\hfill & & & \\ \text{A number and its}opposit\mathrm{due\; east}\text{add to zero.}\hfill & & & \\ \text{of multiplication}\text{For any real number}a,a\ne 0\hfill & & & \hfill a·\frac{1}{a}=\mathrm{i}\\ \\ \frac{1}{a}\text{is the}\text{multiplicative changed}\text{of}a\hfill & & & \\ \text{A number and its}reciproca\mathrm{50}\text{multiply to one.}\hfill & & & \end{array}$

Properties of Zilch $\begin{array}{cccc}\text{For any real number}a,\hfill & & & a·0=0\hfill \\ & & & 0·a=0\hfill \\ \text{For any real number}a,a\ne 0,\hfill & & & \frac{0}{a}=0\hfill \\ \text{For any real number}a,\hfill & & & \frac{a}{0}\text{is undefined}\hfill \end{array}$

Department 1.5 Exercises

Exercise Makes Perfect

Utilize the Commutative and Associative Properties

In the following exercises, simplify.

313.

$43\mathrm{grand}+\left(\mathrm{-12}\mathrm{northward}\right)+\left(\mathrm{-16}m\right)+\left(\mathrm{-9}\mathrm{north}\right)$

314 .

$\mathrm{-22}p+17q+\left(\mathrm{-35}p\right)+\left(\mathrm{-27}q\right)$

315.

$\frac{\mathrm{three}}{8}g+\frac{1}{12}h+\frac{\mathrm{seven}}{8}\mathrm{one\; thousand}+\frac{\mathrm{v}}{12}h$

316 .

$\frac{5}{\mathrm{six}}a+\frac{\mathrm{three}}{\mathrm{ten}}b+\frac{1}{6}a+\frac{\mathrm{nine}}{10}b$

317.

$6.8p+9.14q+\left(\mathrm{-4.37}p\right)+\left(\mathrm{-0.88}q\right)$

318 .

$9.6g+7.22\mathrm{due\; north}+\left(\mathrm{-2.19}m\right)+\left(\mathrm{-0.65}n\right)$

319.

$\mathrm{-24}·7·\frac{\mathrm{iii}}{8}$

320 .

$\mathrm{-36}·11·\frac{4}{\mathrm{ix}}$

321.

$\left(\frac{5}{\mathrm{six}}+\frac{\mathrm{eight}}{15}\right)+\frac{7}{15}$

322 .

$\left(\frac{11}{12}+\frac{4}{9}\right)+\frac{5}{\mathrm{ix}}$

323.

$17\left(0.25\right)\left(4\right)$

324 .

$36\left(0.2\right)\left(\mathrm{five}\right)$

325.

$\left[\mathrm{ii.48}\left(12\right)\right]\left(0.5\right)$

326 .

$\left[9.731\left(\mathrm{iv}\right)\right]\left(0.75\right)$

327.

$12\left(\frac{5}{\mathrm{half\; dozen}}p\right)$

328 .

$\mathrm{xx}\left(\frac{\mathrm{three}}{\mathrm{v}}q\right)$

Employ the Properties of Identity, Inverse and Zero

In the post-obit exercises, simplify.

329.

$19a+44-19a$

330 .

$27c+\mathrm{xvi}-27c$

331.

$\frac{1}{2}+\frac{\mathrm{vii}}{\mathrm{eight}}+\left(-\frac{1}{2}\right)$

332 .

$\frac{\mathrm{two}}{\mathrm{v}}+\frac{5}{12}+\left(-\frac{2}{5}\right)$

333.

$\mathrm{ten}\left(0.1d\right)$

334 .

$100\left(0.01p\right)$

335.

$\frac{3}{20}·\frac{49}{11}·\frac{\mathrm{xx}}{3}$

336 .

$\frac{\mathrm{thirteen}}{18}·\frac{25}{\mathrm{vii}}·\frac{18}{13}$

337.

$\frac{0}{u-\mathrm{four.99}},$ where $u\ne \mathrm{four.99}$

338 .

$0÷\left(y-\frac{1}{6}\right),$ where $x\ne \frac{1}{\mathrm{half\; dozen}}$

339.

$\frac{32-\mathrm{five}a}{0},$ where $32-5a\ne 0$

340 .

$\frac{28-9b}{0},$ where $28-9b\ne 0$

341.

$\left(\frac{3}{4}+\frac{9}{\mathrm{x}}\mathrm{grand}\right)÷0,$ where $\frac{3}{4}+\frac{9}{10}\mathrm{yard}\ne 0$

342 .

$\left(\frac{5}{\mathrm{xvi}}\mathrm{north}-\frac{3}{7}\right)÷0,$ where $\frac{5}{16}\mathrm{north}-\frac{3}{7}\ne 0$

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the Distributive Property.

343.

$8\left(4y+9\right)$

344 .

$9\left(3w+\mathrm{vii}\right)$

345.

$6\left(c-13\right)$

346 .

$7\left(y-13\right)$

347.

$\frac{1}{4}\left(3q+12\right)$

348 .

$\frac{1}{\mathrm{v}}\left(4m+20\right)$

349.

$9\left(\frac{5}{\mathrm{ix}}y-\frac{1}{\mathrm{iii}}\right)$

350 .

$10\left(\frac{3}{10}\mathrm{ten}-\frac{2}{\mathrm{v}}\right)$

351.

$12\left(\frac{1}{4}+\frac{2}{3}r\right)$

352 .

$12\left(\frac{1}{\mathrm{vi}}+\frac{3}{4}s\right)$

353.

$15·\frac{\mathrm{three}}{5}\left(4d+10\right)$

354 .

$18·\frac{\mathrm{v}}{\mathrm{vi}}\left(15h+24\right)$

355.

$r\left(s-18\right)$

356 .

$u\left(\mathrm{five}-\mathrm{x}\right)$

357.

$\left(y+4\right)p$

358 .

$\left(a+\mathrm{vii}\right)x$

359.

$\mathrm{-7}\left(\mathrm{four}p+1\right)$

360 .

$\mathrm{-9}\left(9a+4\right)$

361.

$\mathrm{-three}\left(x-6\right)$

362 .

$\mathrm{-iv}\left(q-7\right)$

363.

$\text{−}\left(\mathrm{iii}x-\mathrm{seven}\right)$

364 .

$\text{−}\left(\mathrm{five}p-4\right)$

365.

$16-3\left(y+8\right)$

366 .

$\mathrm{xviii}-\mathrm{four}\left(x+2\right)$

367.

$\mathrm{four}-\mathrm{eleven}\left(\mathrm{three}c-\mathrm{two}\right)$

368 .

$9-6\left(\mathrm{vii}n-5\right)$

369.

$22-\left(a+3\right)$

370 .

$\mathrm{viii}-\left(r-\mathrm{seven}\right)$

371.

$\left(\mathrm{five}m-3\right)-\left(m+\mathrm{vii}\right)$

372 .

$\left(4y-1\right)-\left(y-2\right)$

373.

$9\left(8x-3\right)-\left(\mathrm{-ii}\right)$

374 .

$4\left(\mathrm{half-dozen}\mathrm{10}-\mathrm{ane}\right)-\left(\mathrm{-8}\right)$

375.

$\mathrm{v}\left(2\mathrm{due\; north}+9\right)+12\left(\mathrm{north}-3\right)$

376 .

$9\left(5u+8\right)+2\left(u-6\right)$

377.

$14\left(c-1\right)-\mathrm{eight}\left(c-\mathrm{vi}\right)$

378 .

$11\left(\mathrm{north}-7\right)-\mathrm{five}\left(\mathrm{due\; north}-\mathrm{i}\right)$

379.

$6\left(7y+8\right)-\left(30y-\mathrm{xv}\right)$

380 .

$7\left(3n+\mathrm{ix}\right)-\left(4n-13\right)$

Writing Exercises

381.

In your own words, country the Associative Belongings of addition.

382 .

What is the deviation betwixt the additive inverse and the multiplicative inverse of a number?

383.

Simplify $\mathrm{viii}\left(\mathrm{10}-\frac{1}{\mathrm{iv}}\right)$ using the Distributive Property and explain each step.

384 .

Explain how you can multiply $4\left(\text{\$}\mathrm{five.97}\right)$ without paper or reckoner by thinking of $\text{\$}5.97$ as $6-0.03$ and and so using the Distributive Holding.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what volition you do to become confident for all objectives?

Properties Of Real Numbers Practice,

Source: https://openstax.org/books/intermediate-algebra-2e/pages/1-5-properties-of-real-numbers

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