homework 1 the real numbers answers

1.1 Real Numbers: Algebra Essentials

• ⓐ 11 1 11 1
• ⓒ − 4 1 − 4 1
• ⓐ 4 (or 4.0), terminating;
• ⓑ 0. 615384 ¯ , 0. 615384 ¯ , repeating;
• ⓒ –0.85, terminating
• ⓐ rational and repeating;
• ⓑ rational and terminating;
• ⓒ irrational;
• ⓓ rational and terminating;
• ⓔ irrational
• ⓐ positive, irrational; right
• ⓑ negative, rational; left
• ⓒ positive, rational; right
• ⓓ negative, irrational; left
• ⓔ positive, rational; right
• ⓐ 11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;
• ⓑ 33, distributive property;
• ⓒ 26, distributive property;
• ⓓ 4 9 , 4 9 , commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;
• ⓔ 0, distributive property, inverse property of addition, identity property of addition
• ⓒ 121 3 π 121 3 π ;
• ⓐ −2 y −2 z or  −2 ( y + z ) ; −2 y −2 z or  −2 ( y + z ) ;
• ⓑ 2 t −1 ; 2 t −1 ;
• ⓒ 3 p q −4 p + q ; 3 p q −4 p + q ;
• ⓓ 7 r −2 s + 6 7 r −2 s + 6

A = P ( 1 + r t ) A = P ( 1 + r t )

1.2 Exponents and Scientific Notation

• ⓐ k 15 k 15
• ⓑ ( 2 y ) 5 ( 2 y ) 5
• ⓒ t 14 t 14
• ⓑ ( −3 ) 5 ( −3 ) 5
• ⓒ ( e f 2 ) 2 ( e f 2 ) 2
• ⓐ ( 3 y ) 24 ( 3 y ) 24
• ⓑ t 35 t 35
• ⓒ ( − g ) 16 ( − g ) 16
• ⓐ 1 ( −3 t ) 6 1 ( −3 t ) 6
• ⓑ 1 f 3 1 f 3
• ⓒ 2 5 k 3 2 5 k 3
• ⓐ t −5 = 1 t 5 t −5 = 1 t 5
• ⓑ 1 25 1 25
• ⓐ g 10 h 15 g 10 h 15
• ⓑ 125 t 3 125 t 3
• ⓒ −27 y 15 −27 y 15
• ⓓ 1 a 18 b 21 1 a 18 b 21
• ⓔ r 12 s 8 r 12 s 8
• ⓐ b 15 c 3 b 15 c 3
• ⓑ 625 u 32 625 u 32
• ⓒ −1 w 105 −1 w 105
• ⓓ q 24 p 32 q 24 p 32
• ⓔ 1 c 20 d 12 1 c 20 d 12
• ⓐ v 6 8 u 3 v 6 8 u 3
• ⓑ 1 x 3 1 x 3
• ⓒ e 4 f 4 e 4 f 4
• ⓓ 27 r s 27 r s
• ⓕ 16 h 10 49 16 h 10 49
• ⓐ $ 1.52 × 10 5 $ 1.52 × 10 5
• ⓑ 7.158 × 10 9 7.158 × 10 9
• ⓒ $ 8.55 × 10 13 $ 8.55 × 10 13
• ⓓ 3.34 × 10 −9 3.34 × 10 −9
• ⓔ 7.15 × 10 −8 7.15 × 10 −8
• ⓐ 703 , 000 703 , 000
• ⓑ −816 , 000 , 000 , 000 −816 , 000 , 000 , 000
• ⓒ −0.000 000 000 000 39 −0.000 000 000 000 39
• ⓓ 0.000008 0.000008
• ⓐ − 8.475 × 10 6 − 8.475 × 10 6
• ⓑ 8 × 10 − 8 8 × 10 − 8
• ⓒ 2.976 × 10 13 2.976 × 10 13
• ⓓ − 4.3 × 10 6 − 4.3 × 10 6
• ⓔ ≈ 1.24 × 10 15 ≈ 1.24 × 10 15

Number of cells: 3 × 10 13 ; 3 × 10 13 ; length of a cell: 8 × 10 −6 8 × 10 −6 m; total length: 2.4 × 10 8 2.4 × 10 8 m or 240 , 000 , 000 240 , 000 , 000 m.

1.3 Radicals and Rational Exponents

5 | x | | y | 2 y z . 5 | x | | y | 2 y z . Notice the absolute value signs around x and y ? That’s because their value must be positive!

10 | x | 10 | x |

x 2 3 y 2 . x 2 3 y 2 . We do not need the absolute value signs for y 2 y 2 because that term will always be nonnegative.

b 4 3 a b b 4 3 a b

14 −7 3 14 −7 3

• ⓒ 88 9 3 88 9 3

( 9 ) 5 = 3 5 = 243 ( 9 ) 5 = 3 5 = 243

x ( 5 y ) 9 2 x ( 5 y ) 9 2

28 x 23 15 28 x 23 15

1.4 Polynomials

The degree is 6, the leading term is − x 6 , − x 6 , and the leading coefficient is −1. −1.

2 x 3 + 7 x 2 −4 x −3 2 x 3 + 7 x 2 −4 x −3

−11 x 3 − x 2 + 7 x −9 −11 x 3 − x 2 + 7 x −9

3 x 4 −10 x 3 −8 x 2 + 21 x + 14 3 x 4 −10 x 3 −8 x 2 + 21 x + 14

3 x 2 + 16 x −35 3 x 2 + 16 x −35

16 x 2 −8 x + 1 16 x 2 −8 x + 1

4 x 2 −49 4 x 2 −49

6 x 2 + 21 x y −29 x −7 y + 9 6 x 2 + 21 x y −29 x −7 y + 9

1.5 Factoring Polynomials

( b 2 − a ) ( x + 6 ) ( b 2 − a ) ( x + 6 )

( x −6 ) ( x −1 ) ( x −6 ) ( x −1 )

• ⓐ ( 2 x + 3 ) ( x + 3 ) ( 2 x + 3 ) ( x + 3 )
• ⓑ ( 3 x −1 ) ( 2 x + 1 ) ( 3 x −1 ) ( 2 x + 1 )

( 7 x −1 ) 2 ( 7 x −1 ) 2

( 9 y + 10 ) ( 9 y − 10 ) ( 9 y + 10 ) ( 9 y − 10 )

( 6 a + b ) ( 36 a 2 −6 a b + b 2 ) ( 6 a + b ) ( 36 a 2 −6 a b + b 2 )

( 10 x − 1 ) ( 100 x 2 + 10 x + 1 ) ( 10 x − 1 ) ( 100 x 2 + 10 x + 1 )

( 5 a −1 ) − 1 4 ( 17 a −2 ) ( 5 a −1 ) − 1 4 ( 17 a −2 )

1.6 Rational Expressions

1 x + 6 1 x + 6

( x + 5 ) ( x + 6 ) ( x + 2 ) ( x + 4 ) ( x + 5 ) ( x + 6 ) ( x + 2 ) ( x + 4 )

2 ( x −7 ) ( x + 5 ) ( x −3 ) 2 ( x −7 ) ( x + 5 ) ( x −3 )

x 2 − y 2 x y 2 x 2 − y 2 x y 2

1.1 Section Exercises

irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

−14 y − 11 −14 y − 11

−4 b + 1 −4 b + 1

43 z − 3 43 z − 3

9 y + 45 9 y + 45

−6 b + 6 −6 b + 6

16 x 3 16 x 3

1 2 ( 40 − 10 ) + 5 1 2 ( 40 − 10 ) + 5

irrational number

g + 400 − 2 ( 600 ) = 1200 g + 400 − 2 ( 600 ) = 1200

inverse property of addition

1.2 Section Exercises

No, the two expressions are not the same. An exponent tells how many times you multiply the base. So 2 3 2 3 is the same as 2 × 2 × 2 , 2 × 2 × 2 , which is 8. 3 2 3 2 is the same as 3 × 3 , 3 × 3 , which is 9.

It is a method of writing very small and very large numbers.

12 40 12 40

1 7 9 1 7 9

3.14 × 10 − 5 3.14 × 10 − 5

16,000,000,000

b 6 c 8 b 6 c 8

a b 2 d 3 a b 2 d 3

q 5 p 6 q 5 p 6

y 21 x 14 y 21 x 14

72 a 2 72 a 2

c 3 b 9 c 3 b 9

y 81 z 6 y 81 z 6

1.0995 × 10 12 1.0995 × 10 12

0.00000000003397 in.

12,230,590,464 m 66 m 66

a 14 1296 a 14 1296

n a 9 c n a 9 c

1 a 6 b 6 c 6 1 a 6 b 6 c 6

0.000000000000000000000000000000000662606957

1.3 Section Exercises

When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.

The principal square root is the nonnegative root of the number.

9 5 5 9 5 5

6 10 19 6 10 19

− 1 + 17 2 − 1 + 17 2

7 2 3 7 2 3

20 x 2 20 x 2

17 m 2 m 17 m 2 m

2 b a 2 b a

15 x 7 15 x 7

5 y 4 2 5 y 4 2

4 7 d 7 d 4 7 d 7 d

2 2 + 2 6 x 1 −3 x 2 2 + 2 6 x 1 −3 x

− w 2 w − w 2 w

3 x − 3 x 2 3 x − 3 x 2

5 n 5 5 5 n 5 5

9 m 19 m 9 m 19 m

2 3 d 2 3 d

3 2 x 2 4 2 3 2 x 2 4 2

6 z 2 3 6 z 2 3

−5 2 −6 7 −5 2 −6 7

m n c a 9 c m n m n c a 9 c m n

2 2 x + 2 4 2 2 x + 2 4

1.4 Section Exercises

The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.

Use the distributive property, multiply, combine like terms, and simplify.

4 x 2 + 3 x + 19 4 x 2 + 3 x + 19

3 w 2 + 30 w + 21 3 w 2 + 30 w + 21

11 b 4 −9 b 3 + 12 b 2 −7 b + 8 11 b 4 −9 b 3 + 12 b 2 −7 b + 8

24 x 2 −4 x −8 24 x 2 −4 x −8

24 b 4 −48 b 2 + 24 24 b 4 −48 b 2 + 24

99 v 2 −202 v + 99 99 v 2 −202 v + 99

8 n 3 −4 n 2 + 72 n −36 8 n 3 −4 n 2 + 72 n −36

9 y 2 −42 y + 49 9 y 2 −42 y + 49

16 p 2 + 72 p + 81 16 p 2 + 72 p + 81

9 y 2 −36 y + 36 9 y 2 −36 y + 36

16 c 2 −1 16 c 2 −1

225 n 2 −36 225 n 2 −36

−16 m 2 + 16 −16 m 2 + 16

121 q 2 −100 121 q 2 −100

16 t 4 + 4 t 3 −32 t 2 − t + 7 16 t 4 + 4 t 3 −32 t 2 − t + 7

y 3 −6 y 2 − y + 18 y 3 −6 y 2 − y + 18

3 p 3 − p 2 −12 p + 10 3 p 3 − p 2 −12 p + 10

a 2 − b 2 a 2 − b 2

16 t 2 −40 t u + 25 u 2 16 t 2 −40 t u + 25 u 2

4 t 2 + x 2 + 4 t −5 t x − x 4 t 2 + x 2 + 4 t −5 t x − x

24 r 2 + 22 r d −7 d 2 24 r 2 + 22 r d −7 d 2

32 x 2 −4 x −3 32 x 2 −4 x −3 m 2

32 t 3 − 100 t 2 + 40 t + 38 32 t 3 − 100 t 2 + 40 t + 38

a 4 + 4 a 3 c −16 a c 3 −16 c 4 a 4 + 4 a 3 c −16 a c 3 −16 c 4

1.5 Section Exercises

The terms of a polynomial do not have to have a common factor for the entire polynomial to be factorable. For example, 4 x 2 4 x 2 and −9 y 2 −9 y 2 don’t have a common factor, but the whole polynomial is still factorable: 4 x 2 −9 y 2 = ( 2 x + 3 y ) ( 2 x −3 y ) . 4 x 2 −9 y 2 = ( 2 x + 3 y ) ( 2 x −3 y ) .

Divide the x x term into the sum of two terms, factor each portion of the expression separately, and then factor out the GCF of the entire expression.

10 m 3 10 m 3

( 2 a −3 ) ( a + 6 ) ( 2 a −3 ) ( a + 6 )

( 3 n −11 ) ( 2 n + 1 ) ( 3 n −11 ) ( 2 n + 1 )

( p + 1 ) ( 2 p −7 ) ( p + 1 ) ( 2 p −7 )

( 5 h + 3 ) ( 2 h −3 ) ( 5 h + 3 ) ( 2 h −3 )

( 9 d −1 ) ( d −8 ) ( 9 d −1 ) ( d −8 )

( 12 t + 13 ) ( t −1 ) ( 12 t + 13 ) ( t −1 )

( 4 x + 10 ) ( 4 x − 10 ) ( 4 x + 10 ) ( 4 x − 10 )

( 11 p + 13 ) ( 11 p − 13 ) ( 11 p + 13 ) ( 11 p − 13 )

( 19 d + 9 ) ( 19 d − 9 ) ( 19 d + 9 ) ( 19 d − 9 )

( 12 b + 5 c ) ( 12 b − 5 c ) ( 12 b + 5 c ) ( 12 b − 5 c )

( 7 n + 12 ) 2 ( 7 n + 12 ) 2

( 15 y + 4 ) 2 ( 15 y + 4 ) 2

( 5 p − 12 ) 2 ( 5 p − 12 ) 2

( x + 6 ) ( x 2 − 6 x + 36 ) ( x + 6 ) ( x 2 − 6 x + 36 )

( 5 a + 7 ) ( 25 a 2 − 35 a + 49 ) ( 5 a + 7 ) ( 25 a 2 − 35 a + 49 )

( 4 x − 5 ) ( 16 x 2 + 20 x + 25 ) ( 4 x − 5 ) ( 16 x 2 + 20 x + 25 )

( 5 r + 12 s ) ( 25 r 2 − 60 r s + 144 s 2 ) ( 5 r + 12 s ) ( 25 r 2 − 60 r s + 144 s 2 )

( 2 c + 3 ) − 1 4 ( −7 c − 15 ) ( 2 c + 3 ) − 1 4 ( −7 c − 15 )

( x + 2 ) − 2 5 ( 19 x + 10 ) ( x + 2 ) − 2 5 ( 19 x + 10 )

( 2 z − 9 ) − 3 2 ( 27 z − 99 ) ( 2 z − 9 ) − 3 2 ( 27 z − 99 )

( 14 x −3 ) ( 7 x + 9 ) ( 14 x −3 ) ( 7 x + 9 )

( 3 x + 5 ) ( 3 x −5 ) ( 3 x + 5 ) ( 3 x −5 )

( 2 x + 5 ) 2 ( 2 x − 5 ) 2 ( 2 x + 5 ) 2 ( 2 x − 5 ) 2

( 4 z 2 + 49 a 2 ) ( 2 z + 7 a ) ( 2 z − 7 a ) ( 4 z 2 + 49 a 2 ) ( 2 z + 7 a ) ( 2 z − 7 a )

1 ( 4 x + 9 ) ( 4 x −9 ) ( 2 x + 3 ) 1 ( 4 x + 9 ) ( 4 x −9 ) ( 2 x + 3 )

1.6 Section Exercises

You can factor the numerator and denominator to see if any of the terms can cancel one another out.

True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.

y + 5 y + 6 y + 5 y + 6

3 b + 3 3 b + 3

x + 4 2 x + 2 x + 4 2 x + 2

a + 3 a − 3 a + 3 a − 3

3 n − 8 7 n − 3 3 n − 8 7 n − 3

c − 6 c + 6 c − 6 c + 6

d 2 − 25 25 d 2 − 1 d 2 − 25 25 d 2 − 1

t + 5 t + 3 t + 5 t + 3

6 x − 5 6 x + 5 6 x − 5 6 x + 5

p + 6 4 p + 3 p + 6 4 p + 3

2 d + 9 d + 11 2 d + 9 d + 11

12 b + 5 3 b −1 12 b + 5 3 b −1

4 y −1 y + 4 4 y −1 y + 4

10 x + 4 y x y 10 x + 4 y x y

9 a − 7 a 2 − 2 a − 3 9 a − 7 a 2 − 2 a − 3

2 y 2 − y + 9 y 2 − y − 2 2 y 2 − y + 9 y 2 − y − 2

5 z 2 + z + 5 z 2 − z − 2 5 z 2 + z + 5 z 2 − z − 2

x + 2 x y + y x + x y + y + 1 x + 2 x y + y x + x y + y + 1

2 b + 7 a a b 2 2 b + 7 a a b 2

18 + a b 4 b 18 + a b 4 b

a − b a − b

3 c 2 + 3 c − 2 2 c 2 + 5 c + 2 3 c 2 + 3 c − 2 2 c 2 + 5 c + 2

15 x + 7 x −1 15 x + 7 x −1

x + 9 x −9 x + 9 x −9

1 y + 2 1 y + 2

Review Exercises

y = 24 y = 24

3 a 6 3 a 6

x 3 32 y 3 x 3 32 y 3

1.634 × 10 7 1.634 × 10 7

4 2 5 4 2 5

7 2 50 7 2 50

3 x 3 + 4 x 2 + 6 3 x 3 + 4 x 2 + 6

5 x 2 − x + 3 5 x 2 − x + 3

k 2 − 3 k − 18 k 2 − 3 k − 18

x 3 + x 2 + x + 1 x 3 + x 2 + x + 1

3 a 2 + 5 a b − 2 b 2 3 a 2 + 5 a b − 2 b 2

4 a 2 4 a 2

( 4 a − 3 ) ( 2 a + 9 ) ( 4 a − 3 ) ( 2 a + 9 )

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

( 2 h − 3 k ) 2 ( 2 h − 3 k ) 2

( p + 6 ) ( p 2 − 6 p + 36 ) ( p + 6 ) ( p 2 − 6 p + 36 )

( 4 q − 3 p ) ( 16 q 2 + 12 p q + 9 p 2 ) ( 4 q − 3 p ) ( 16 q 2 + 12 p q + 9 p 2 )

( p + 3 ) 1 3 ( −5 p − 24 ) ( p + 3 ) 1 3 ( −5 p − 24 )

x + 3 x − 4 x + 3 x − 4

m + 2 m − 3 m + 2 m − 3

6 x + 10 y x y 6 x + 10 y x y

Practice Test

x = –2 x = –2

3 x 4 3 x 4

13 q 3 − 4 q 2 − 5 q 13 q 3 − 4 q 2 − 5 q

n 3 − 6 n 2 + 12 n − 8 n 3 − 6 n 2 + 12 n − 8

( 4 x + 9 ) ( 4 x − 9 ) ( 4 x + 9 ) ( 4 x − 9 )

( 3 c − 11 ) ( 9 c 2 + 33 c + 121 ) ( 3 c − 11 ) ( 9 c 2 + 33 c + 121 )

4 z − 3 2 z − 1 4 z − 3 2 z − 1

3 a + 2 b 3 b 3 a + 2 b 3 b

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Chapter 1 Prerequisites

1.1 Real Numbers: Algebra Essentials

Learning objectives.

In this section, you will:

• Classify a real number as a natural, whole, integer, rational, or irrational number.
• Perform calculations using order of operations.
• Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
• Evaluate algebraic expressions.
• Simplify algebraic expressions.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate, items. Farmers, cattlemen, and tradesmen used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century A.D. in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century A.D., negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers we use for counting or enumerating items are the natural numbers: 1, 2, 3, 4, 5, and so on. We describe them in set notation as[latex]\,\left\{1,2,3,...\right\}\,[/latex], where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers . Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero:[latex]\,\left\{0,1,2,3,...\right\}.[/latex]

The set of integers adds the opposites of the natural numbers to the set of whole numbers:[latex]\,\left\{...,-3,-2,-1,0,1,2,3,...\right\}.[/latex] It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

[latex]\begin{array}{lllll}\stackrel{\text{negative integers}}{\stackrel{}{\dots ,-3,-2,-1,}}\hfill & \hfill & \stackrel{\text{zero}}{\stackrel{}{0,}}\hfill & \hfill & \stackrel{\text{positive integers}}{\stackrel{}{1,2,3,\cdots }}\hfill \end{array}[/latex]

The set of rational numbers is written as[latex]\,\left\{\frac{m}{n}\,|m\text{ and }n\text{ are integers and }n\ne 0\right\}.\,[/latex]Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

• a terminating decimal:[latex]\,\frac{15}{8}=1.875,[/latex] or
• a repeating decimal:[latex]\,\frac{4}{11}=0.36363636\dots =0.\overline{36}[/latex]

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Writing Integers as Rational Numbers

Write each of the following as a rational number.

Write a fraction with the integer in the numerator and 1 in the denominator.

• [latex]7=\frac{7}{1}[/latex]
• [latex]0=\frac{0}{1}[/latex]
• [latex]-8=-\frac{8}{1}[/latex]
• [latex]\frac{11}{1}[/latex]
• [latex]\frac{3}{1}[/latex]
• [latex]-\frac{4}{1}[/latex]

Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

• [latex]-\frac{5}{7}[/latex]
• [latex]\frac{15}{5}[/latex]
• [latex]\frac{13}{25}[/latex]

Write each fraction as a decimal by dividing the numerator by the denominator.

• [latex]-\frac{5}{7}=-0.\stackrel{\text{———}}{714285},[/latex] a repeating decimal
• [latex]\frac{15}{5}=3\,[/latex](or 3.0), a terminating decimal
• [latex]\frac{13}{25}=0.52,[/latex] a terminating decimal
• [latex]\frac{68}{17}[/latex]
• [latex]\frac{8}{13}[/latex]
• [latex]-\frac{17}{20}[/latex]
• 4 (or 4.0), terminating;
• [latex]0.\overline{615384},[/latex]repeating;
• –0.85, terminating

Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even[latex]\,\frac{3}{2},[/latex]but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

• [latex]\sqrt{25}[/latex]
• [latex]\frac{33}{9}[/latex]
• [latex]\sqrt{11}[/latex]
• [latex]\frac{17}{34}[/latex]
• [latex]0.3033033303333\dots[/latex]
• [latex]\sqrt{25}:\,[/latex]This can be simplified as[latex]\,\sqrt{25}=5.\,[/latex]Therefore,[latex]\sqrt{25}\,[/latex]is rational.

So,[latex]\,\frac{33}{9}\,[/latex]is rational and a repeating decimal.

• [latex]\sqrt{11}:\,[/latex]This cannot be simplified any further. Therefore,[latex]\,\sqrt{11}\,[/latex]is an irrational number.

So,[latex]\,\frac{17}{34}\,[/latex]is rational and a terminating decimal.

• [latex]0.3033033303333\dots \,[/latex]is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
• [latex]\frac{7}{77}[/latex]
• [latex]\sqrt{81}[/latex]
• [latex]4.27027002700027\dots[/latex]
• [latex]\frac{91}{13}[/latex]
• [latex]\sqrt{39}[/latex]
• rational and repeating;
• rational and terminating;
• irrational;

Real Numbers

Given any number n , we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line. The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line, as shown in (Figure 1) .

Classifying Real Numbers

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

• [latex]-\frac{10}{3}[/latex]
• [latex]\sqrt{5}[/latex]
• [latex]-\sqrt{289}[/latex]
• [latex]-6\pi[/latex]
• [latex]0.615384615384\dots[/latex]
• [latex]-\frac{10}{3}\,[/latex]is negative and rational. It lies to the left of 0 on the number line.
• [latex]\sqrt{5}\,[/latex]is positive and irrational. It lies to the right of 0.
• [latex]-\sqrt{289}=-\sqrt{{17}^{2}}=-17\,[/latex]is negative and rational. It lies to the left of 0.
• [latex]-6\pi \,[/latex]is negative and irrational. It lies to the left of 0.
• [latex]0.615384615384\dots \,[/latex]is a repeating decimal so it is rational and positive. It lies to the right of 0.
• [latex]\sqrt{73}[/latex]
• [latex]-11.411411411\dots[/latex]
• [latex]\frac{47}{19}[/latex]
• [latex]-\frac{\sqrt{5}}{2}[/latex]
• [latex]6.210735[/latex]
• positive, irrational; right
• negative, rational; left
• positive, rational; right
• negative, irrational; left

Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such as Figure 2.

Sets of Numbers

The set of natural numbers includes the numbers used for counting:[latex]\,\left\{1,2,3,...\right\}.[/latex]

The set of whole numbers is the set of natural numbers plus zero:[latex]\,\left\{0,1,2,3,...\right\}.[/latex]

The set of integers adds the negative natural numbers to the set of whole numbers:[latex]\,\left\{...,-3,-2,-1,0,1,2,3,...\right\}.[/latex]

The set of rational numbers includes fractions written as[latex]\,\left\{\frac{m}{n}\,|m\text{ and }n\text{ are integers and }n\ne 0\right\}.[/latex]

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating:[latex]\,\left\{h|h\text{ is not a rational number}\right\}.[/latex]

Differentiating the Sets of Numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

• [latex]\sqrt{36}[/latex]
• [latex]\frac{8}{3}[/latex]

[latex]-6[/latex]

• [latex]3.2121121112\dots[/latex]
• [latex]-\frac{35}{7}[/latex]
• [latex]0[/latex]
• [latex]\sqrt{169}[/latex]
• [latex]\sqrt{24}[/latex]
• [latex]4.763763763\dots[/latex]

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example,[latex]\,{4}^{2}=4\cdot 4=16.\,[/latex]We can raise any number to any power. In general, the exponential notation[latex]\,{a}^{n}\,[/latex]means that the number or variable[latex]\,a\,[/latex]is used as a factor[latex]\,n\,[/latex]times.

In this notation,[latex]\,{a}^{n}\,[/latex]is read as the n th power of[latex]\,a,\,[/latex]where[latex]\,a\,[/latex]is called the base and[latex]\,n\,[/latex]is called the exponent . A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example,[latex]\,24+6\cdot \frac{2}{3}-{4}^{2}\,[/latex]is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify[latex]\,{4}^{2}\,[/latex]as 16.

Next, perform multiplication or division, left to right.

Lastly, perform addition or subtraction, left to right.

Therefore,[latex]\,24+6\cdot \frac{2}{3}-{4}^{2}=12.[/latex]

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

Order of Operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :

P (arentheses)

E (xponents)

M (ultiplication) and D (ivision)

A (ddition) and S (ubtraction)

Given a mathematical expression, simplify it using the order of operations.

• Simplify any expressions within grouping symbols.
• Simplify any expressions containing exponents or radicals.
• Perform any multiplication and division in order, from left to right.
• Perform any addition and subtraction in order, from left to right.

Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

• [latex]{\left(3\cdot 2\right)}^{2}-4\left(6+2\right)[/latex]
• [latex]\frac{{5}^{2}-4}{7}-\sqrt{11-2}[/latex]
• [latex]6-|5-8|+3\left(4-1\right)[/latex]
• [latex]\frac{14-3\cdot 2}{2\cdot 5-{3}^{2}}[/latex]
• [latex]7\left(5\cdot 3\right)-2\left[\left(6-3\right)-{4}^{2}\right]+1[/latex]
• [latex]\begin{array}{cccc}\hfill{\left(3\cdot 2\right)}^{2}-4\left(6+2\right) =& {\left(6\right)}^{2}-4\left(8\right)\phantom{\rule{1em}{0ex}} \text{Simplify parentheses}\end{array}[/latex] [latex]\begin{array}\\ =& 36-4\left(8\right) & \phantom{\rule{2em}{0ex}}\text{Simplify exponent}\end{array}[/latex] [latex]\begin{array}\\ =& 36-32 & \phantom{\rule{2em}{0ex}}\text{Simplify multiplication}\end{array}[/latex] [latex]\begin{array}\\ =& 4 & \phantom{\rule{2em}{0ex}}\text{Simplify subtraction}\end{array}[/latex]

Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol, so the numerator is considered to be grouped.

• [latex]\begin{array}{cccc}{ 6-|5-8|+3\left(4-1\right)} \end{array}[/latex] [latex]\begin{array}= 6-|-3|+3\left(3\right) \phantom{\rule{2em}{0ex}}\text{Simplify inside grouping symbols} \end{array}[/latex] [latex]\begin{array}\\&= 6-3+3\left(3\right)\phantom{\rule{2em}{0ex}}\text{Simplify absolute value} \end{array}[/latex] [latex]\begin{array}\\&= 6-3+9\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify multiplication}\end{array}[/latex] [latex]\begin{array}\\&= 3+9\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify subtraction}\end{array}[/latex] [latex]\begin{array}\\&= 12\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify addition} \end{array}[/latex]

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

• [latex]\begin{array}{cccc}\hfill 7\left(5\cdot 3\right)-2\left[\left(6-3\right)-{4}^{2}\right]+1& =& 7\left(15\right)-2\left[\left(3\right)-{4}^{2}\right]+1\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify inside parentheses}\hfill \\ & =& 7\left(15\right)-2\left(3-16\right)+1\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify exponent}\hfill \\ & =& 7\left(15\right)-2\left(-13\right)+1\hfill & \phantom{\rule{2em}{0ex}}\text{Subtract}\hfill \\ & =& 105+26+1\hfill & \phantom{\rule{2em}{0ex}}\text{Multiply}\hfill \\ & =& 132\hfill & \phantom{\rule{2em}{0ex}}\text{Add}\hfill \end{array}[/latex]
• [latex]\sqrt{{5}^{2}-{4}^{2}}+7{\left(5-4\right)}^{2}[/latex]
• [latex]1+\frac{7\cdot 5-8\cdot 4}{9-6}[/latex]
• [latex]|1.8-4.3|+0.4\sqrt{15+10}[/latex]
• [latex]\frac{1}{2}\left[5\cdot {3}^{2}-{7}^{2}\right]+\frac{1}{3}\cdot {9}^{2}[/latex]
• [latex]\left[{\left(3-8\right)}^{2}-4\right]-\left(3-8\right)[/latex]

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

It is important to note that neither subtraction nor division is commutative. For example,[latex]\,17-5\,[/latex]is not the same as[latex]\,5-17.\,[/latex]Similarly,[latex]\,20÷5\ne 5÷20.[/latex]

Associative Properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

Consider this example.

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

This property can be especially helpful when dealing with negative integers. Consider this example.

Are subtraction and division associative? Review these examples.

As we can see, neither subtraction nor division is associative.

Distributive Property

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum.

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

A special case of the distributive property occurs when a sum of terms is subtracted.

For example, consider the difference[latex]\,12-\left(5+3\right).\,[/latex]We can rewrite the difference of the two terms 12 and[latex]\,\left(5+3\right)\,[/latex]by turning the subtraction expression into addition of the opposite. So instead of subtracting[latex]\,\left(5+3\right),[/latex]we add the opposite.

Now, distribute[latex]\,-1\,[/latex]and simplify the result.

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

For example, we have[latex]\,\left(-6\right)+0=-6\,[/latex]and[latex]\,23\cdot 1=23.\,[/latex]There are no exceptions for these properties; they work for every real number, including 0 and 1.

Inverse Properties

The inverse property of addition states that for every real number a , there is a unique number, called the additive inverse (or opposite), denoted− a , that, when added to the original number, results in the additive identity, 0.

For example, if[latex]\,a=-8,[/latex]the additive inverse is 8, since[latex]\,\left(-8\right)+8=0.[/latex]

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that for every real number a , there is a unique number, called the multiplicative inverse (or reciprocal), denoted[latex]\,\frac{1}{a},[/latex] that, when multiplied by the original number, results in the multiplicative identity, 1.

For example, if[latex]\,a=-\frac{2}{3},[/latex] the reciprocal, denoted[latex]\,\frac{1}{a},[/latex] is[latex]\,-\frac{3}{2}\,[/latex] because

Properties of Real Numbers

The following properties hold for real numbers a , b , and c .

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

• [latex]\,3\cdot 6+3\cdot 4[/latex]
• [latex]\,\left(5+8\right)+\left(-8\right)[/latex]
• [latex]\,6-\left(15+9\right)[/latex]
• [latex]\,\frac{4}{7}\cdot \left(\frac{2}{3}\cdot \frac{7}{4}\right)[/latex]
• [latex]\,100\cdot \left[0.75+\left(-2.38\right)\right][/latex]
• [latex]\begin{array}{cccc}\hfill 3\cdot 6+3\cdot 4& =& 3\cdot \left(6+4\right)\hfill & \phantom{\rule{7em}{0ex}}\text{Distributive property}\hfill \\ & =& 3\cdot 10\hfill & \phantom{\rule{7em}{0ex}}\text{Simplify}\hfill \\ & =& 30\hfill & \phantom{\rule{7em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \left(5+8\right)+\left(-8\right)& =& 5+\left[8+\left(-8\right)\right]\hfill & \phantom{\rule{3em}{0ex}}\text{Associative property of addition}\\ & =& 5+0\hfill & \phantom{\rule{3em}{0ex}}\text{Inverse property of addition}\hfill \\ & =& 5\hfill & \phantom{\rule{3em}{0ex}}\text{Identity property of addition}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill 6-\left(15+9\right)\hfill & =& 6+\left[\left(-15\right)+\left(-9\right)\right]\hfill & \phantom{\rule{2em}{0ex}}\text{Distributive property}\hfill \\ & =& 6+\left(-24\right)\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}\hfill \\ & =& -18\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \frac{4}{7}\cdot \left(\frac{2}{3}\cdot \frac{7}{4}\right)& =& \frac{4}{7}\cdot \left(\frac{7}{4}\cdot \frac{2}{3}\right)\hfill & \phantom{\rule{6em}{0ex}}\text{Commutative property of multiplication}\hfill \\ & =& \left(\frac{4}{7}\cdot \frac{7}{4}\right)\cdot \frac{2}{3}\hfill & \phantom{\rule{6em}{0ex}}\text{Associative property of multiplication}\hfill \\ & =& 1\cdot \frac{2}{3}\hfill & \phantom{\rule{6em}{0ex}}\text{Inverse property of multiplication}\hfill \\ & =& \frac{2}{3}\hfill & \phantom{\rule{6em}{0ex}}\text{Identity property of multiplication}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill 100\cdot \left[0.75+\left(-2.38\right)\right]& =& 100\cdot 0.75+100\cdot \left(-2.38\right)\hfill & \phantom{\rule{2em}{0ex}}\text{Distributive property}\\ & =& 75+\left(-238\right)\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}\hfill \\ & =& -163\hfill & \phantom{\rule{2em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\,\left(-\frac{23}{5}\right)\cdot \left[11\cdot \left(-\frac{5}{23}\right)\right][/latex]
• [latex]\,5\cdot \left(6.2+0.4\right)[/latex]
• [latex]\,18-\left(7-15\right)[/latex]
• [latex]\,\frac{17}{18}+\left[\frac{4}{9}+\left(-\frac{17}{18}\right)\right][/latex]
• [latex]\,6\cdot \left(-3\right)+6\cdot 3[/latex]
• 11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;
• 33, distributive property;
• 26, distributive property;
• [latex]\,\frac{4}{9},[/latex] commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;
• 0, distributive property, inverse property of addition, identity property of addition

Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as[latex]\,x+5,\frac{4}{3}\pi {r}^{3},[/latex] or[latex]\,\sqrt{2{m}^{3}{n}^{2}}.\,[/latex]In the expression[latex]\,x+5,[/latex] 5 is called a constant because it does not vary, and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

• [latex]\frac{4}{3}\pi {r}^{3}[/latex]
• [latex]\sqrt{2{m}^{3}{n}^{2}}[/latex]
• [latex]2\pi r\left(r+h\right)[/latex]
• [latex]4{y}^{3}+y[/latex]

Evaluating an Algebraic Expression at Different Values

Evaluate the expression[latex]\,2x-7\,[/latex]for each value for x.

• [latex]\,x=0[/latex]
• [latex]\,x=1[/latex]
• [latex]\,x=\frac{1}{2}[/latex]
• [latex]\,x=-4[/latex]
• Substitute 0 for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 2x-7& =& 2\left(0\right)-7\\ & =& 0-7\hfill \\ & =& -7\hfill \end{array}[/latex]
• Substitute 1 for[latex]\,x.[/latex] [latex]\begin{array}{ccc}2x-7& =& 2\left(1\right)-7\hfill \\ & =& 2-7\hfill \\ & =& -5\hfill \end{array}[/latex]
• Substitute[latex]\,\frac{1}{2}\,[/latex]for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 2x-7& =& 2\left(\frac{1}{2}\right)-7\hfill \\ & =& 1-7\hfill \\ & =& -6\hfill \end{array}[/latex]
• Substitute[latex]\,-4\,[/latex]for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill 2x-7& =& 2\left(-4\right)-7\\ & =& -8-7\hfill \\ & =& -15\hfill \end{array}[/latex]

Evaluate the expression[latex]\,11-3y\,[/latex]for each value for y.

• [latex]\,y=2[/latex]
• [latex]\,y=0[/latex]
• [latex]\,y=\frac{2}{3}[/latex]
• [latex]\,y=-5[/latex]

Evaluate each expression for the given values.

• [latex]\,x+5\,[/latex]for[latex]\,x=-5[/latex]
• [latex]\,\frac{t}{2t-1}\,[/latex]for[latex]\,t=10[/latex]
• [latex]\,\frac{4}{3}\pi {r}^{3}\,[/latex]for[latex]\,r=5[/latex]
• [latex]\,a+ab+b\,[/latex]for[latex]a=11,b=-8[/latex]
• [latex]\,\sqrt{2{m}^{3}{n}^{2}}\,[/latex]for[latex]\,m=2,n=3[/latex]
• Substitute[latex]\,-5\,[/latex]for[latex]\,x.[/latex] [latex]\begin{array}{ccc}\hfill x+5& =& \left(-5\right)+5\hfill \\ & =& 0\hfill \end{array}[/latex]
• Substitute 10 for[latex]\,t.[/latex] [latex]\begin{array}{ccc}\hfill \frac{t}{2t-1}& =& \frac{\left(10\right)}{2\left(10\right)-1}\hfill \\ & =& \frac{10}{20-1}\hfill \\ & =& \frac{10}{19}\hfill \end{array}[/latex]
• Substitute 5 for[latex]\,r.[/latex] [latex]\begin{array}{ccc}\hfill \frac{4}{3}\pi {r}^{3}& =& \frac{4}{3}\pi {\left(5\right)}^{3}\\ & =& \frac{4}{3}\pi \left(125\right)\hfill \\ & =& \frac{500}{3}\pi \hfill \end{array}[/latex]
• Substitute 11 for[latex]\,a\,[/latex]and –8 for[latex]\,b.[/latex] [latex]\begin{array}{ccc}\hfill a+ab+b& =& \left(11\right)+\left(11\right)\left(-8\right)+\left(-8\right)\\ & =& 11-88-8\hfill \\ & =& -85\hfill \end{array}[/latex]
• Substitute 2 for[latex]\,m\,[/latex]and 3 for[latex]\,n.[/latex] [latex]\begin{array}{ccc}\hfill \sqrt{2{m}^{3}{n}^{2}}& =& \sqrt{2{\left(2\right)}^{3}{\left(3\right)}^{2}}\hfill \\ & =& \sqrt{2\left(8\right)\left(9\right)}\hfill \\ & =& \sqrt{144}\hfill \\ & =& 12\hfill \end{array}[/latex]
• [latex]\,\frac{y+3}{y-3}\,[/latex]for[latex]\,y=5[/latex]
• [latex]\,7-2t\,[/latex]for[latex]\,t=-2[/latex]
• [latex]\,\frac{1}{3}\pi {r}^{2}\,[/latex]for[latex]\,r=11[/latex]
• [latex]\,{\left({p}^{2}q\right)}^{3}\,[/latex]for[latex]\,p=-2,q=3[/latex]
• [latex]\,4\left(m-n\right)-5\left(n-m\right)\,[/latex]for[latex]\,m=\frac{2}{3},n=\frac{1}{3}[/latex]
• [latex]\,\frac{121}{3}\pi[/latex];

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation[latex]\,2x+1=7\,[/latex]has the unique solution of 3[latex][/latex] because when we substitute 3 for[latex]\,x\,[/latex]in the equation, we obtain the true statement[latex]\2\left(3\right)+1=7.[/latex]

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area[latex]\,A\,[/latex]of a circle in terms of the radius[latex]\,r\,[/latex]of the circle:[latex]\,A=\pi {r}^{2}.\,[/latex]For any value of[latex]\,r,[/latex] the area[latex]\,A\,[/latex]can be found by evaluating the expression[latex]\,\pi {r}^{2}.[/latex]

Using a Formula

A right circular cylinder with radius[latex]\,r\,[/latex]and height[latex]\,h\,[/latex]has the surface area[latex]\,S\,[/latex](in square units) given by the formula[latex]\,S=2\pi r\left(r+h\right).\,[/latex]See Figure 3. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of[latex]\,\pi .[/latex]

Evaluate the expression[latex]\,2\pi r\left(r+h\right)\,[/latex]for[latex]\,r=6\,[/latex]and[latex]\,h=9.[/latex]

The surface area is[latex]\,180\pi \,[/latex]square inches.

A photograph with length L and width W is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm 2 ) is found to be[latex]\,A=\left(L+16\right)\left(W+16\right)-L\cdot W.\,[/latex]See Figure 4. Find the area of a matte for a photograph with length 32 cm and width 24 cm.

Given that [latex]L=32[/latex] and [latex]W=24[/latex], plug the numbers into the formula [latex]\,A=\left(L+16\right)\left(W+16\right)-L\cdot W.\,[/latex]

[latex]\,A=\left(32+16\right)\left(24+16\right)-32\cdot 16.\,[/latex]

[latex]\,A=\left(48\right)\left(40\right)-512.\,[/latex]

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Simplify each algebraic expression.

• [latex]3x-2y+x-3y-7[/latex]
• [latex]2r-5\left(3-r\right)+4[/latex]
• [latex]\left(4t-\frac{5}{4}s\right)-\left(\frac{2}{3}t+2s\right)[/latex]
• [latex]2mn-5m+3mn+n[/latex]
• [latex]\begin{array}{cccc}\hfill 3x-2y+x-3y-7& =& 3x+x-2y-3y-7\hfill & \phantom{\rule{6.5em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& 4x-5y-7\hfill & \phantom{\rule{6.5em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill 2r-5\left(3-r\right)+4& =& 2r-15+5r+4\hfill & \phantom{\rule{10em}{0ex}}\text{Distributive property}\hfill \\ & =& 2r+5r-15+4\hfill & \phantom{\rule{10em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& 7r-11\hfill & \phantom{\rule{10em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}\hfill \left(4t-\frac{5}{4}s\right)-\left(\frac{2}{3}t+2s\right)& =& 4t-\frac{5}{4}s-\frac{2}{3}t-2s\hfill & \phantom{\rule{4em}{0ex}}\text{Distributive property}\hfill \\ & =& 4t-\frac{2}{3}t-\frac{5}{4}s-2s\hfill & \phantom{\rule{4em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& \frac{10}{3}t-\frac{13}{4}s\hfill & \phantom{\rule{4em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\begin{array}{cccc}2mn-5m+3mn+n& =& 2mn+3mn-5m+n& \phantom{\rule{5em}{0ex}}\text{Commutative property of addition}\hfill \\ & =& \text{ }5mn-5m+n\hfill & \phantom{\rule{5em}{0ex}}\text{Simplify}\hfill \end{array}[/latex]
• [latex]\frac{2}{3}y-2\left(\frac{4}{3}y+z\right)[/latex]
• [latex]\frac{5}{t}-2-\frac{3}{t}+1[/latex]
• [latex]4p\left(q-1\right)+q\left(1-p\right)[/latex]
• [latex]9r-\left(s+2r\right)+\left(6-s\right)[/latex]
• [latex]\,-2y-2z\text{ or }-2\left(y+z\right);[/latex]
• [latex]\,\frac{2}{t}-1;[/latex]
• [latex]\,3pq-4p+q;[/latex]
• [latex]\,7r-2s+6[/latex]

Simplifying a Formula

A rectangle with length[latex]\,L\,[/latex]and width[latex]\,W\,[/latex]has a perimeter[latex]\,P\,[/latex]given by[latex]\,P=L+W+L+W.\,[/latex]Simplify this expression.

If the amount[latex]\,P\,[/latex]is deposited into an account paying simple interest[latex]\,r\,[/latex]for time[latex]\,t,[/latex] the total value of the deposit[latex]\,A\,[/latex]is given by[latex]\,A=P+Prt.\,[/latex]Simplify the expression. (This formula will be explored in more detail later in the course.)

[latex]A=P\left(1+rt\right)[/latex]

Key Concepts

• Rational numbers may be written as fractions or terminating or repeating decimals.
• Determine whether a number is rational or irrational by writing it as a decimal.
• The rational numbers and irrational numbers make up the set of real numbers. A number can be classified as natural, whole, integer, rational, or irrational.
• The order of operations is used to evaluate expressions.
• The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties.
• Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. They take on a numerical value when evaluated by replacing variables with constants.
• Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression.
• Is[latex]\,\sqrt{2}\,[/latex]an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

• What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?
• What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

For the following exercises, simplify the given expression.

• [latex]10+2\,×\,\left(5-3\right)[/latex]
• [latex]6÷2-\left(81÷{3}^{2}\right)[/latex]
• [latex]18+{\left(6-8\right)}^{3}[/latex]
• [latex]-2\,×\,{\left[16÷{\left(8-4\right)}^{2}\right]}^{2}[/latex]

[latex]-2[/latex]

• [latex]4-6+2\,×\,7[/latex]
• [latex]3\left(5-8\right)[/latex]

[latex]-9[/latex]

• [latex]4+6-10÷2[/latex]
• [latex]12÷\left(36÷9\right)+6[/latex]
• [latex]{\left(4+5\right)}^{2}÷3[/latex]
• [latex]3-12\,×\,2+19[/latex]
• [latex]2+8\,×\,7÷4[/latex]
• [latex]5+\left(6+4\right)-11[/latex]
• [latex]9-18÷{3}^{2}[/latex]
• [latex]14\,×\,3÷7-6[/latex]
• [latex]9-\left(3+11\right)\,×\,2[/latex]
• [latex]6+2\,×\,2-1[/latex]
• [latex]64÷\left(8+4\,×\,2\right)[/latex]
• [latex]9+4\left({2}^{2}\right)[/latex]
• [latex]{\left(12÷3\,×\,3\right)}^{2}[/latex]
• [latex]25÷{5}^{2}-7[/latex]
• [latex]\left(15-7\right)\,×\,\left(3-7\right)[/latex]
• [latex]2\,×\,4-9\left(-1\right)[/latex]
• [latex]{4}^{2}-25\,×\,\frac{1}{5}[/latex]
• [latex]12\left(3-1\right)÷6[/latex]

For the following exercises, solve for the variable.

• [latex]8\left(x+3\right)=64[/latex]
• [latex]4y+8=2y[/latex]

[latex]-4[/latex]

• [latex]\left(11a+3\right)-18a=-4[/latex]
• [latex]4z-2z\left(1+4\right)=36[/latex]
• [latex]4y{\left(7-2\right)}^{2}=-200[/latex]
• [latex]-{\left(2x\right)}^{2}+1=-3[/latex]

[latex]±1[/latex]

• [latex]8\left(2+4\right)-15b=b[/latex]
• [latex]2\left(11c-4\right)=36[/latex]
• [latex]4\left(3-1\right)x=4[/latex]
• [latex]\frac{1}{4}\left(8w-{4}^{2}\right)=0[/latex]

For the following exercises, simplify the expression.

• [latex]4x+x\left(13-7\right)[/latex]
• [latex]2y-{\left(4\right)}^{2}y-11[/latex]

[latex]-14y-11[/latex]

• [latex]\frac{a}{{2}^{3}}\left(64\right)-12a÷6[/latex]
• [latex]8b-4b\left(3\right)+1[/latex]

[latex]-4b+1[/latex]

• [latex]5l÷3l\,×\,\left(9-6\right)[/latex]
• [latex]7z-3+z\,×\,{6}^{2}[/latex]

[latex]43z-3[/latex]

• [latex]4\,×\,3+18x÷9-12[/latex]
• [latex]9\left(y+8\right)-27[/latex]

[latex]9y+45[/latex]

• [latex]\left(\frac{9}{6}t-4\right)2[/latex]
• [latex]6+12b-3\,×\,6b[/latex]

[latex]-6b+6[/latex]

• [latex]18y-2\left(1+7y\right)[/latex]
• [latex]{\left(\frac{4}{9}\right)}^{2}\,×\,27x[/latex]

[latex]\frac{16x}{3}[/latex]

• [latex]8\left(3-m\right)+1\left(-8\right)[/latex]
• [latex]9x+4x\left(2+3\right)-4\left(2x+3x\right)[/latex]

[latex]9x[/latex]

• [latex]{5}^{2}-4\left(3x\right)[/latex]

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 mowing lawns. He spends $10 on mp3s, puts half of what is left in a savings account, and gets another $5 for washing his neighbor’s car.

• Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

[latex]\frac{1}{2}\left(40-10\right)+5[/latex]

• How much money does Fred keep?

For the following exercises, solve the given problem.

• According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by[latex]\,\pi .\,[/latex]Is the circumference of a quarter a whole number, a rational number, or an irrational number?

irrational number

• Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, consider this scenario: There is a mound of[latex]\,g\,[/latex]pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

• Write the equation that describes the situation.

[latex]g+400-2\left(600\right)=1200[/latex]

• Solve for g .

For the following exercise, solve the given problem.

• Ramon runs the marketing department at his company. His department gets a budget every year, and every year, he must spend the entire budget without going over. If he spends less than the budget, then his department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. He must spend the budget such that[latex]\,2,500,000-x=0.\,[/latex]What property of addition tells us what the value of x must be?

inverse property of addition

For the following exercises, use a graphing calculator to solve for x . Round the answers to the nearest hundredth.

• [latex]0.5{\left(12.3\right)}^{2}-48x=\frac{3}{5}[/latex]
• [latex]{\left(0.25-0.75\right)}^{2}x-7.2=9.9[/latex]
• If a whole number is not a natural number, what must the number be?
• Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.
• Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.
• Determine whether the simplified expression is rational or irrational:[latex]\,\sqrt{-18-4\left(5\right)\left(-1\right)}.[/latex]
• Determine whether the simplified expression is rational or irrational:[latex]\,\sqrt{-16+4\left(5\right)+5}.[/latex]
• The division of two whole numbers will always result in what type of number?
• What property of real numbers would simplify the following expression:[latex]\,4+7\left(x-1\right)?[/latex]

Media Attributions

• 1.1 Figure 1 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 1.1 Figure 2 © OpenStax Algebra and Trignometry is licensed under a CC BY (Attribution) license
• 1.1 Distributive Property Graphic © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 1.1 Figure 3 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
• 1.1 Figure 4 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license

College Algebra Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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1.1: Real numbers and the Number Line

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SKILLS TO DEVELOP

• Construct a number line and graph points on it.
• Use a number line to determine the order of real numbers.
• Determine the opposite of a real number.
• Determine the absolute value of a real number.

Definitions

A set is a collection of objects, typically grouped within braces \(\{\) \(\}\), where each object is called an element . For example, \(\{\text{red, green, blue}\}\) is a set of colors. A subset is a set consisting of elements that belong to a given set. For example, \(\{\text{green, blue}\}\) is a subset of the color set above. A set with no elements is called the empty set and has its own special notation, \(\{\) \(\}\) or \(\varnothing\).

When studying mathematics, we focus on special sets of numbers. The set of natural (or counting) numbers , denoted \(\mathbb{N}\), is

\( \{1,2,3,4,5 , \dots \} \quad \color{Cerulean}{Natural\: Numbers} \)

The three periods \((\dots)\) is called an ellipsis and indicates that the numbers continue without bound. The set of whole numbers , denoted \(\mathbb{W}\), is the set of natural numbers combined with zero.

\( \{0,1,2,3,4,5 , \dots\} \quad \color{Cerulean}{Whole\: Numbers} \)

The set of integers , denoted \(\mathbb{Z}\), consists of both positive and negative whole numbers, as well as zero.

\( \{\dots, -3,-2,-1,0,1,2,3 , \dots\} \quad \color{Cerulean}{Integers} \)

Notice that the sets of natural and whole numbers are both subsets of the set of integers.

Rational numbers , denoted \(\mathbb{Q}\), are defined as any number of the form \(\dfrac{a}{b}\), where \(a\) and \(b\) are integers and \(b\) is nonzero. Decimals that repeat or terminate are rational. For example,

\(0.7= \frac{7}{10} \quad \text{and} \quad 0. \overline{3} =0.3333 \dots = \frac{1}{3}\)

The set of integers is a subset of the set of rational numbers because every integer can be expressed as a ratio of the integer and \(1\). In other words, any integer can be written over \(1\) and can be considered a rational number. For example,

\(5= \frac{5}{1}\)

Irrational numbers are defined as any number that cannot be written as a ratio of two integers. Nonterminating decimals that do not repeat are irrational. For example,

\(\pi =3.14159 \dots \quad \text{and} \quad \sqrt{2} = 1.41421 \dots\)

The set of real numbers , denoted \(\mathbb{R}\), is defined as the set of all rational numbers combined with the set of all irrational numbers. Therefore, all the numbers defined so far are subsets of the set of real numbers. In summary,

Number Line

A real number line , or simply number line, allows us to visually display real numbers by associating them with unique points on a line. The real number associated with a point is called a coordinate . A point on the real number line that is associated with a coordinate is called its graph .

To construct a number line, draw a horizontal line with arrows on both ends to indicate that it continues without bound. Next, choose any point to represent the number zero; this point is called the origin .

Mark off consistent lengths on both sides of the origin and label each tick mark to define the scale. Positive real numbers lie to the right of the origin and negative real numbers lie to the left. The number zero \((0)\) is neither positive nor negative. Typically, each tick represents one unit.

As illustrated below, the scale need not always be one unit. In the first number line, each tick mark represents two units. In the second, each tick mark represents \(\frac{1}{7}\) of a unit.

The graph of each real number is shown as a dot at the appropriate point on the number line. A partial graph of the set of integers \(\mathbb{Z}\) follows:

Example \(\PageIndex{1}\)

Graph the following set of real numbers:

Graph the numbers on a number line with a scale where each tick mark represents \(\frac{1}{2}\) of a unit.

Ordering Real Numbers

When comparing real numbers on a number line, the larger number will always lie to the right of the smaller one. It is clear that \(15\) is greater than \(5\), but it may not be so clear to see that \(−1\) is greater than \(−5\) until we graph each number on a number line.

We use symbols to help us efficiently communicate relationships between numbers on the number line. The symbols used to describe an equality relationship between numbers follow:

\[\begin{align*} &= \quad \color{Cerulean}{is\ equal\ to} \\ &\neq \quad \color{Cerulean}{is\ not\ equal\ to} \\ &\approx \quad \color{Cerulean}{is\ approximately\ equal\ to} \end{align*}\]

These symbols are used and interpreted in the following manner:

\[\begin{align*} &5=5 \qquad &&\color{Cerulean}{5\ is\ equal\ to\ 5} \\ &0 \neq 5 \qquad &&\color{Cerulean}{0\ is\ not\ equal\ to\ 5} \\ &\pi \approx 3.14 \quad &&\color{Cerulean}{pi\ is\ approximately\ equal\ to\ 3.14} \end{align*}\]

We next define symbols that denote an order relationship between real numbers.

\[\begin{align*} &< \quad \color{Cerulean}{Less\ than} \\ &> \quad \color{Cerulean}{Greater\ than} \\ &\leq \quad \color{Cerulean}{Less\ than\ or\ equal\ to} \\ &\geq \quad \color{Cerulean}{Greater\ than\ or\ equal\ to} \end{align*}\]

These symbols allow us to compare two numbers. For example,

Since the graph of \(−120\) is to the left of the graph of \(–10\) on the number line, that number is less than \(−10\). We could write an equivalent statement as follows:

Similarly, since the graph of zero is to the right of the graph of any negative number on the number line, zero is greater than any negative number.

The symbols \(<\) and \(>\) are used to denote strict inequalities , and the symbols and are used to denote inclusive inequalities . In some situations, more than one symbol can be correctly applied. For example, the following two statements are both true:

In addition, the “or equal to” component of an inclusive inequality allows us to correctly write the following:

The logical use of the word “or” requires that only one of the conditions need be true: the “less than” or the “equal to.”

Example \(\PageIndex{2}\)

Fill in the blank with \(<, =\), or \(>: −2\) ____ \(−12\).

Use > because the graph of \(−2\) is to the right of the graph of \(−12\) on a number line. Therefore, \(−2 > −12\), which reads “negative two is greater than negative twelve.”

\(-2>-12\)

In this text, we will often point out the equivalent notation used to express mathematical quantities electronically using the standard symbols available on a keyboard. We begin with the equivalent textual notation for inequalities:

\[\begin{align*} &\geq &&">=" \\ &\leq &&"<=" \\ &\neq &&"!=" \end{align*}\]

Many calculators, computer algebra systems, and programming languages use this notation.

The opposite of any real number \(a\) is \(−a\). Opposite real numbers are the same distance from the origin on a number line, but their graphs lie on opposite sides of the origin and the numbers have opposite signs.

For example, we say that the opposite of \(10\) is \(−10\).

Next, consider the opposite of a negative number. Given the integer \(−7\), the integer the same distance from the origin and with the opposite sign is \(+7\), or just \(7\).

Therefore, we say that the opposite of \(−7\) is \(−(−7) = 7\). This idea leads to what is often referred to as the double-negative property . For any real number \(a\),

\(-(-a)=a\)

Example \(\PageIndex{3}\)

What is the opposite of \(-\frac{3}{4}\)?

Here we apply the double-negative property.

\(-(-\frac{3}{4})=\frac{3}{4}\)

Example \(\PageIndex{4}\)

Simplify \(-(-(4))\)

Start with the innermost parentheses by finding the opposite of \(+4\).

\[\begin{align*} -(-(4)) &= -(\color{Cerulean}{-(4)} \color{Black}{)} \\ &= -(\color{Cerulean}{-4} \color{Black}{)} \\ &=4 \end{align*}\]

Example \(\PageIndex{5}\)

Simplify: \(-(-(-2))\).

Apply the double-negative property starting with the innermost parentheses.

\[\begin{align*} -(-(-2)) &= -(\color{Cerulean}{-(-2)} \color{Black}{)} \\ &= -(\color{Cerulean}{2} \color{Black}{)} \\ &=-2 \end{align*}\]

If there is an even number of consecutive negative signs, then the result is positive. If there is an odd number of consecutive negative signs, then the result is negative.

Exercise \(\PageIndex{1}\)

Simplify: \(-(-(-(5)))\).

\[\begin{align*} -(-(-(5))) &= -(\color{Cerulean}{-(-(5))} \color{Black}{)}\\ &= -(\color{Cerulean}{-(-5)} \color{Black}{)} \\ &= -(\color{Cerulean}{5} \color{Black}{)} \\&= -5 \end{align*} \]

Video Solution:

(click to see video)

Absolute Value

The absolute value of a real number \(a\), denoted \(|a|\), is defined as the distance between zero (the origin) and the graph of that real number on the number line. Since it is a distance, it is always positive. For example,

\(|-4|=4 \quad \text{and} \quad |4|=4\)

Both \(4\) and \(−4\) are four units from the origin, as illustrated below:

Example \(\PageIndex{6}\)

a. \(|-12|\)

Both \(−12\) and \(12\) are twelve units from the origin on a number line. Therefore,

\(|-12|=12 \quad \text{and} \quad |12|=12\)

a.\(12\) b.\(12\)

Also, it is worth noting that

The absolute value can be expressed textually using the notation abs\((a)\). We often encounter negative absolute values, such as \(−|3|\) or \(−\) abs\((3)\). Notice that the negative sign is in front of the absolute value symbol. In this case, work the absolute value first and then find the opposite of the result.

Try not to confuse this with the double-negative property, which states that \(−(−7)=+7\).

Example \(\PageIndex{7}\)

Simplfy: \(-|-(-7)|\).

First, find the opposite of \(−7\) inside the absolute value. Then find the opposite of the result.

\[\begin{align*} -|\color{Cerulean}{-(-7)} \color{Black}{|} &= -|\color{Cerulean}{7} \color{Black}{|} \\ &=-7 \end{align*}\]

At this point, we can determine what real numbers have a particular absolute value. For example,

Think of a real number whose distance to the origin is \(5\) units. There are two solutions: the distance to the right of the origin and the distance to the left of the origin, namely, \(\{\pm 5\}\). The symbol \( (\pm) \) is read “plus or minus” and indicates that there are two answers, one positive and one negative.

\(|-5|=5\ \quad \text{and} \quad |5|=5\)

Now consider the following:

Here we wish to find a value for which the distance to the origin is negative. Since negative distance is not defined, this equation has no solution. If an equation has no solution, we say the solution is the empty set: \(\varnothing\).

Key Takeaways

• Any real number can be associated with a point on a line.
• Create a number line by first identifying the origin and marking off a scale appropriate for the given problem.
• Negative numbers lie to the left of the origin and positive numbers lie to the right.
• Smaller numbers always lie to the left of larger numbers on the number line.
• The opposite of a positive number is negative and the opposite of a negative number is positive.
• The absolute value of any real number is always positive because it is defined to be the distance from zero (the origin) on a number line.
• The absolute value of zero is zero.

Exercise \(\PageIndex{2}\)

Use set notation to list the described elements.

• The hours on a clock.
• The days of the week.
• The first ten whole numbers.
• The first ten natural numbers.
• The first five positive even integers.
• The first five positive odd integers.

1. \(\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}\)

3. \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\)

5. \(\{2, 4, 6, 8, 10\}\)

Exercise \(\PageIndex{3}\)

Determine whether the following real numbers are integers, rational, or irrational.

• \(−3\)
• \(−5\)
• \(0.3 \overline{6} \)
• \(0. \overline{3} \)
• \(1.001000100001 \dots\)
• \(1.00 \overline{1} \)
• \(e=2.71828 \dots \)
• \(\sqrt{7}=2.645751 \dots \)
• \(−7\)
• \(8,675,309\)

1: Integer, Rational

3: Rational

5: Rational

7: Irrational

9: Irrational

11: Integer, Rational

13: Rational

15: Integer, Rational

Exercise \(\PageIndex{4}\)

True or false.

• All integers are rational numbers.
• All integers are whole numbers.
• All rational numbers are whole numbers.
• Some irrational numbers are rational.
• All terminating decimal numbers are rational.
• All irrational numbers are real.

Exercise \(\PageIndex{5}\)

Choose an appropriate scale and graph the following sets of real numbers on a number line.

• \(\{−3, 0, 3\}\)
• \(\{−2, 2, 4, 6, 8, 10\}\)
• \(\{−2, −1/3, 2/3, 5/3\}\)
• \(\{−5/2, −1/2, 0, 1/2 , 2\}\)
• \(\{−5/7, 0, 2/7 , 1\}\)
• \(\{ –5, –2, –1, 0\}\)
• \(\{ −3, −2, 0, 2, 5\}\)
• \(\{−2.5, −1.5, 0, 1, 2.5\}\)
• \(\{0, 0.3, 0.6, 0.9, 1.2\}\)
• \(\{−10, 30, 50\}\)
• \(\{−6, 0, 3, 9, 12\}\)
• \(\{−15, −9, 0, 9, 15\}\)

1. \(\{−3, 0, 3\}\)

3. \(\{−2, −1/3, 2/3, 5/3\}\)

5. \(\{−5/7, 0, 2/7 , 1\}\)

7. \(\{ −3, −2, 0, 2, 5\}\)

9. \(\{0, 0.3, 0.6, 0.9, 1.2\}\)

11. \(\{−6, 0, 3, 9, 12\}\)

Exercise \(\PageIndex{6}\)

Fill in the blank with \(<, =\), or \(>\).

• \(−7\) ___ \(0\)
• \(30\) ___ \(2\)
• \(10\) ___ \(−10\)
• \(−150\) ___ \(−75\)
• \(−0.5\) ___ \(−1.5\)
• \(0\) ___ \(0\)
• \(-500\) ___ \(200\)
• \(−1\) ___ \(−200\)
• \(−10\) ___ \(−10\)
• \(−40\) ___ \(−41\)

1. \(<\)

3. \(>\)

5. \(>\)

7. \(<\)

Exercise \(\PageIndex{7}\)

• \(−5>−10\)
• \(4 \leq 4\)
• \(−12 \geq 0\)
• \(−10=−10\)
• \(−1000<−20\)

Exercise \(\PageIndex{8}\)

List the numbers.

• List three integers less than \(−5\).
• List three integers greater than \(−10\).
• List three rational numbers less than zero.
• List three rational numbers greater than zero.
• List three integers between \(−20\) and \(−5\).
• List three rational numbers between \(0\) and \(1\).

1. \(−10, −7, −6\) (answers may vary)

3. \(−1, −2/3, −1/3\) (answers may vary)

5. \(−15, −10, −7\) (answers may vary)

Exercise \(\PageIndex{9}\)

Translate each statement into an English sentence.

• \(10<20\)
• \(−50 \leq −10\)
• \(−4 \neq 0\)
• \(30 \geq −1\)
• \(e \approx 2.718\)

1. Ten is less than twenty.

3. Negative four is not equal to zero.

5. Zero is equal to zero.

Exercise \(\PageIndex{10}\)

Translate the following into a mathematical statement.

• Negative seven is less than zero.
• Twenty-four is not equal to ten.
• Zero is greater than or equal to negative one.
• Four is greater than or equal to negative twenty-one.
• Negative two is equal to negative two.
• Negative two thousand is less than negative one thousand.

1. \(−7<0\)

3. \(0 \geq −1\)

5. \(−2=−2\)

Exercise \(\PageIndex{11}\)

• \(−(−9)\)
• \(−(−35)\)
• \(−(10)\)
• \(−(3)\)
• \(−(5)\)
• \(−(34)\)
• \(−(−1)\)
• \(−(−(−1))\)
• \(−(−(1))\)
• \(−(−(−3))\)
• \(−(−(−(−11)))\)

3. \(−10\)

5. \(−5\)

Exercise \(\PageIndex{12}\)

Answer the following questions.

• What is the opposite of \(-12\)
• What is the opposite of \(\pi \) ?
• What is the opposite \(−0.01\)?
• Is the opposite of \(−12\) smaller or larger than \(−11\)?
• Is the opposite of \(7\) smaller or larger than \(−6\)?

2. \(-\pi \)

Exercise \(\PageIndex{13}\)

• \(−7\) ___ \(−(−8)\)
• \(6\) ___ \(−(6)\)
• \(13\) ___ \(−(−12)\)
• \(−(−5)\) ___ \(−(−2)\)
• \(−100\) ___ \(−(−(−50))\)
• \(44\) ___ \(−(−44)\)

5. \(<\)

Exercise \(\PageIndex{14}\)

• \(|−20|\)
• \(|−33|\)
• \(|−0.75|\)
• \(|−\frac{3}{5}|\)
• \(−|12|\)
• \(−|−20|\)
• \(−|20|\)
• \(−|−8|\)
• \(−|7|\)
• \(−|−316|\)
• \(−(−|\frac{8}{9}|)\)
• \(|−(−2)|\)
• \(−|−(−3)|\)
• \(−(−|5|)\)
• \(−(−|−45|)\)
• \(−|−(−21)|\)
• abs\((−7)\)
• \(−\)abs\((5)\)
• \(−\)abs\((−19)\)
• \(−(−\)abs\((9))\)
• \(−\)abs\((−(−12))\)

5. \(\frac{3}{5}\)

9. \(−12\)

11. \(−20\)

13. \(−7\)

15. \(\frac{8}{9}\)

17. \(−3\)

23. \(−5\)

Exercise \(\PageIndex{15}\)

Determine the unknown.

• \(| ? |=9\)
• \(| ? |=15\)
• \(| ? |=0\)
• \(| ? |=1\)
• \(| ? |=−8\)
• \(| ? |=−20\)
• \(|?|−10=−2\)
• \(|?|+5=14\)

1. \(\pm 9\)

5. \(\varnothing\), No solution

7. \(\pm 8\)

Exercise \(\PageIndex{16}\)

• \(|−2|\) ____ \(0\)
• \(|−7|\) ____ \(|−10|\)
• \(−10\) ____ \(−|−2|\)
• \(|−6|\) ____ \(|−(−6)|\)
• \(−|3|\) ____ \(|−(−5)|\)
• \(0\) ____ \(−|−(−4)|\)

1. \(>\)

3. \(<\)

Exercise \(\PageIndex{17}\)

Discussion Board Topics.

• Research and discuss the history of the number zero.
• Research and discuss the various numbering systems throughout history.
• Research and discuss the definition and history of \(\pi\).
• Research the history of irrational numbers. Who is credited with proving that the square root of \(2\) is irrational and what happened to him?
• Research and discuss the history of absolute value.
• Discuss the “just make it positive” definition of absolute value

1-1 Properties of Real Numbers

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Properties of Real Numbers - Word Docs & PowerPoints

1-1 Assignment - Properties of Real Numbers 1-1 Bellwork - Properties of real numbers 1-1 Exit Quiz - Properties of Real Numbers 1-1 Guided Notes SE - Properties of Real Numbers 1-1 Guided Notes TE - Properties of Real Numbers 1-1 Lesson Plan - Properties of Real Numbers 1-1 Online Activity - Properties of Real Numbers 1-1 Slide Show - Properties of Real Numbers

Properties of Real Numbers - PDFs

1-1 Assignment SE - Properties of Real Numbers 1-1 Assignment - Properties of Real Numbers 1-1 Bellwork SE - Properties of real numbers 1-1 Bellwork - Properties of real numbers 1-1 Exit Quiz SE - Properties of Real Numbers 1-1 Exit Quiz - Properties of Real Numbers 1-1 Guided Notes TE - Properties of Real Numbers 1-1 Guided Notes SE - Properties of Real Numbers 1-1 Lesson Plan - Properties of Real Numbers 1-1 Online Activity - Properties of Real Numbers 1-1 Slide Show - Properties of Real Numbers

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Unit 1 Algebra Basics Homework 1 The Real Numbers

Unit 1 Algebra Basics Homework 1 The Real Numbers - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Unit 1 real number system homework, Lesson 1 classification and real numbers, Just the maths, Order of operations, Unit 1 the real number system, Prentice hall mathematics courses 1 3, Two step equations date period, Coordinate geometry mathematics 1.

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1. Unit 1 Real Number System Homework

2. lesson 1 (classification and real numbers), 3. ''just the maths'', 4. order of operations -, 5. unit 1: the real number system, 6. prentice hall mathematics courses 1-3, 7. two-step equations date period, 8. coordinate geometry mathematics 1 -.

Algebra Basics (Algebra 1 Curriculum - Unit 1) | All Things Algebra®

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Description

This Algebra Basics Unit Bundle contains guided notes, homework assignments, three quizzes, a study guide, and a test that address the following topics:

• The Real Number System

• Properties (includes a performance task!)

• Order of Operations

• Evaluating Expressions

• Translating Expressions

• Combining Like Terms

• Distributing

• Simplifying Expressions

• Two Step Equations

• Two-Step Inequalities

ADDITIONAL COMPONENTS INCLUDED:

(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice.  Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.

(3) Google Slides Version of the PDF: The second page of the Video links document contains a link to a Google Slides version of the PDF. Each page is set to the background in Google Slides. There are no text boxes;  this is the PDF in Google Slides.  I am unable to do text boxes at this time but hope this saves you a step if you wish to use it in Slides instead! 

This resource is included in the following bundle(s):

Algebra 1 First Semester Notes Bundle

Algebra 1 Curriculum Algebra 1 Curriculum (with Activities)

More Algebra 1 Units:

Unit 2 – Multi-Step Equations & Inequalities

Unit 3 – Relations & Functions

Unit 4 – Linear Equations

Direct and Inverse Variation (Mini-Unit)

Unit 5 – Systems of Equations & Inequalities

Unit 6 – Exponents and Exponential Functions

Unit 7 – Polynomials & Factoring

Unit 8 – Quadratic Equations Unit 9 – Linear, Quadratic, and Exponential Functions

Unit 10 – Radical Expressions & Equations

Unit 11 – Rational Expressions & Equations

Unit 12 – Statistics

LICENSING TERMS: This purchase includes a license for one teacher only for personal use in their classroom. Licenses are non-transferable , meaning they can not be passed from one teacher to another. No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses. If you are a coach, principal, or district interested in transferable licenses to accommodate yearly staff changes, please contact me for a quote at [email protected].

COPYRIGHT TERMS: This resource may not be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students. © All Things Algebra (Gina Wilson), 2012-present

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