Chapter 1 Real Numbers and their Properties

8/7/2019 Chapter 1 Real Numbers and their Properties

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Chapter 1 Real Numbers and theirProperties

1.1 Basic Definitions

Exercise A. Encircle the correct answer on the right.

1. a whole number 1.5 2

3

1 0

2. NOT a naturalnumber

2 100

0 50

3. a whole number

between4

12 and

3

15

44

32

2

13 3.35

4. a fraction between

314 and

524

2

14

4

34

5

14

3

24

5. a real number25.12

0

8

π 2

Exercise B. Enumerate the following.

1. first four whole numbers ______ ______ ______ ______

2. first four natural numbers ______ ______ ______ ______

1

NATURAL NUMBERS (counting numbers) = 1, 2, 3, 4,…

FRACTIONS = written asb

a

number)natural(anyrdenominato

number)whole(anynumerator

WHOLE NUMBERS = 0, 1, 2, 3, 4,…

REAL NUMBERS = whole numbers, fractions, decimals and mixed numbers



3. fractions between 1 and 2 ______ ______ ______ ______

4. decimals between 5 and 6 ______ ______ ______ ______

5. whole numbers between 5 and

50 which are divisible by 5 ______ ______ ______ ______

Exercise C. Magic Rectangles

1. Write the numbers 1-8 in each square so that the sum in each column is 9.

2. Write the numbers 1-15 in each square so that the sum in each column is 24.

1.2 Introduction to Signed Numbers

The Number Line

2



Trichotomy Property

For any numbers a and b, exactly one of the followingsentences is true.

bababa >=<

Absolute Value

x (read as: the absolute value of x) represents the distanceof x from 0, which is a positive quantity. The absolute value of anumber is never negative.

−= n u mn e g a t i v eai sxi f x

0o rn u m bp o s i t i v eai sxi f x ,x

Exercise A. Write <, >, or = in each blank.

1. 1010 −_____

2. 63 −_____

3. 350 −_____

4. 1212 −− _____

Number Absolute Value of the Number5 55 =

-3 33 =−

6

5−

6

5

6

5=−

3

Negative Numbers Positive Numbers



5. 88 −−− _____

6. 151 −− _____

7. 3658 −−+ _____

8. 37458 −− _____

9. 225152 −++ _____

10. 822

−_____

Exercise B. Answer the following.

1. ______ =−+− 810

2. ______ =−+ 6824

3. ______ =−++− 81412

4. ______ ))(( =−+−− 4554

5. ______ ))(())(( =−−−− 2256

6. ______ =−+− 32918

7. ______ ))(( =−+−− 141258

8. ( )( ) ______ ))(())(( =−−− 5228

9. ( ) ______ =++−− 3125

10. ______ =−−+− 1536

Exercise C. Write I if the number is irrational and R if rational.

1. 4 ______

2. π ______

3.4

9______

4. 520 + ______

5. 030. ______

6. 13 + ______

7.9

2______

8.2

5______

9.3

1______

10. 4100 + _______

1.3 Adding Signed Numbers

Exercise A. Add the following.

1. _______ )()( =−+−+ 111428

2. _______ )( =−++− 81412

3. _______ )()()( =−+−+− 453513

4. _______ )()( =−+−+ 111428

4



5. _______ )()( =−++− 212355

6. _______ ).().(.. =−+−+ 030250280

7. _______ ).(.).( =−++− 0202812534

8. _______ ).(.)( =−++− 03122384125

9. _______ ).()().( =−+−+− 71294850

10. _______ ).(.)( =−++− 12252314200

11. _______ =

−+

−+

8

1

8

5

8

3

12. _______ =

−++

−

5

11

5

1

5

22

13. _______ =

−+

−+

2

1

5

2

4

3

14. _______ =

−++

−

6

51

3

12

4

13

15. _______ )( =

−++−

6

1

3

26

Exercise B. Solve the following problems. Show all necessary solutions.

Box and label your final answer.

1.The temperature in a certain area is

C°−9

during the day. At night itdrops °4 . What is the temperature at night?

2. In a quiz show you get a point for each correct answer and lose a point

for each incorrect answer. Rica gets 23 correct answers and 17 wrong

answers. What is her score?

5



3. Camille went on four-month diet. She lost 1.2kg in the first month,

1.1kg in the second, 2.1kg in the third, and 0.5kg in the fourth. Howmuch weight did she lose altogether?

4. An airplane 1,245m above sea level drops an object into the ocean. It

settles at a point 465m below sea level. How far did the object fall?

5. Michael took a joy ride in an elevator of ABC building. He took the

elevator at the 8th floor, went up to the 53rd floor, went down 17 floors,

and then went down again 15 floors. He then got out of the elevator to

see a Science exhibit. In what floor is the Science exhibit located?

1.4 Subtracting Signed Numbers

Exercise A. Subtract the following.

1. ______ =−− 112315

2. ______ )( =−−− 25158

3. ______ )( =−−− 321724

4. ______ )()( =−−−− 431121

5. ______ )()( =−−−− 19816

6. ______ ... =−− 24012152319

7. ______ )( =−−− 472140

6



8. ______ )()( =−−−− 12915200

9. ______ )()( =−−−− 1412446

10. ______ )()()( =−−−−− 1241454

11. ______ =−−12

1

12

8

12

5

12. ______ =−−6

7

6

12

6

15

13. ______ =

−−

−−

2

1

5

12

8

5

14. ______ =

−−

−−

−

2

12

3

2

6

59

15. ______ )( =

−−−7

32

5

323

Exercise B. Solve for M in each equation. Write your answer in each

blank.

1. 1254 −=− M _____

2. 88 =−− M)( _____

3. 1556 −=−− M)( _____

4. 3614 −=− M _____

5. 623 −=−− M)( _____

6. 1520 −=−M _____

7. 216 −=−− )(M _____

8. 11789 −=−− )(M _____

9. 1515 =−− )(M _____

10. 2412 =−− )(M _____

Exercise C. Solve the following problems. Show all necessary solutions.

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1. Plato, the Classical Greek philosopher, was born in 428 B.C. and died on 348B.C. How old was he when he died?

2. You are in debt of P325.50. How much money will you need to make yourtotal assets P850.00?

3. An airplane is 5,439m above sea level. At the same time, a submarine isdirectly under the plane at 1,235m below sea level. How far above thesubmarine is the plane?

1.5 Multiplying and Dividing Signed Numbers

Exercise A. Solve for N for each equation. Write your answer in eachblank.

1. ( ) ( ) 100N25 −=− ______

2. ( ) ( ) ( ) 60N34 −=−− ______

3. ( ) ( ) ( ) 12636N −=− ______

4. ( ) ( ) ( ) 451N9 −=−− ______

5. ( ) ( ) 2828N −= ______

6. ( ) ( ) 4.14N2.1 =− ______

7. ( ) ( ) 5.125.0N −=− ______

8. ( ) ( ) ( ) 7.2N2.15.2 =− ______

9. ( ) ( ) ( ) 7.1825.13.20N −= ______

10. ( ) ( ) ( ) 55.12N105.125 =− ______

8



Exercise B. Answer the following.

1. ( ) ( ) ( ) ________ 2813 =−−

2. ( ) ( ) ( ) ________ 10102.13 =−

3. ( ) ( ) ( ) ________ 6512 =−−−

4. ( ) ( ) ( ) ________ 545.8 =−−−

5. ( ) ( ) ( ) ________ 2274 =−

6.________

5

9

27

10

5

1=

−

−

7. ________ 4

31

7

21

2

13 =

−

−

8. ( ) ________ 243

22

8

11 =−

−

9. ( ) ( ) ________ 28

356 =−

−−

10. ________ 9

51

16

15

9

32 =

−

Exercise C. Solve for S for each equation. Write your answer in each

blank.

1. 29S87 −=÷ _______

2. ( ) 3.5S5.26 =÷− _______

3. ( ) 72S4.14 −=÷− _______

4.35

2S

49

5−=÷ _______

5. 3

21S

4

12 =÷

− _______

6. ( ) 24S −=−÷ _______

7. 1246S −=÷ _______

8. ( ) 253.12S −=−÷ _______

9. 205

3S −=

−÷ _______

10.4

3

7

21S =

−÷ _______

Exercise D. Answer the following.

1. ( ) ( ) ________ 24140 =−÷−÷

2. ( ) ( ) ( ) _________ 121000 =−÷−÷−

3. ( ) ( ) ________ 2.03.02.73 =−÷÷−

4. ( ) ( ) ( ) _________ 1525 =−÷−÷−

5. ( ) _________ 621584 =−÷÷

6. _________ 2

9

15

4

3

2=

÷

−÷

7. ______ 3

5

15

12

15

3=

÷

−÷

−

8. ( ) _________ 248

3

2

1=−÷

÷

−

9. ( ) ( ) _________ 1.05255.0 =÷−÷

10. ( ) ( ) ( ) _________ 2.03372 =−÷−÷−

9



Exercise E. Perform the indicated operations. Write your answer in the

box.

1. 4.

2. 5.

3.1. 6 Powers and Roots of Signed Numbers

10

( )( )

=−−−32

723

( ) ( )

( ) ( )=

−−

−÷−

75

428

( ) ( )

( ) ( )=

−−−

−−

75

432

( ) ( ) ( )( ) ( )

=

−−

−−

108

8435

( ) ( ) ( )( )

=

−−

−

27

15112

823=

base

exponent

Third power of 2

EVEN POWER OF THE BASE – a base that has an exponent thatis exactly divisible by 2.

Ex. 862253 )(,, −

ODD POWER OF THE BASE – a base that has an exponent thatis not exactly divisible by 2.

Ex. 973253 )(,, −

An even power of a negative number is positive.Ex. 42

2=− )(

An odd power of a negative number is negative.Ex. 82

3−=− )(

Power of ZeroIf a is any positive real number, then 00 =a



Exercise A. Find each indicated root.

1. 36−

2. 225

3.25

4

4.25

36−

5.121

1−

6. 61.−

7. 112 .−

8.63

211

.

.−

9. 414 .

10.112

1

.−

11. 36100 +

12. ( ) ( )436 −+−

13. 3600196 −

14. 144211 −.

15. 4400 +−−

Exercise B. Find each indicated root.

1. 3 8−

2. 364

3. 33 27729 +

4. 3 84 −−

5. ( )4164 −−

6. 33 15120 −+

7. ( ) ( )6483 −

8. 31000169 −

9. 35 2732 +−

10. 253433 −

Exercise C. Solve the following problems. Show all necessary solutions.Box and label your final answer.

11



1. The area of a square lot is 2361m . Find the length of each side.

2. The volume of a box is .33375cm Find the length of each side.

1.7 Properties of Real Numbers

12

The Closure Property

A set S is closed under the operation ♦ if whenever a and b are in S,

a ♦ b is in S.

Commutative Properties of Addition and Multiplication

If a and b represent real numbers, then

abba

abba

•=•+=+

Associative Properties of Addition and Multiplication

If a, b, and c represent real numbers, then

( ) ( )( ) ( )cbacba

cbacba

••=••++=++

Additive and Multiplicative Identity Property

Let a represent a real number.

tion)multiplica for element identity the is (One aa11a

addition) for element identity the is (Zero aa00a

=•=•=+=+

Inverse Properties for Addition and Multiplication

Let a represent a real number.( ) ( )

0)(a 1,aa

1

a

1a

0aaaa

≠=•=•

=+−=−+

Distributive Property of Multiplication Over Addition (DPMA)

For any real numbers a, b, and c,( ) ( ) ( )

( ) ( ) ( )acabacb and

cabacba

•+•=•+

•+•=+•

Distributive Property of Multiplication Over Subtraction (DPMS)( ) ( ) ( )

( ) ( ) ( )acabacb and

cabacba

•−•=•−

•−•=−•



Exercise A. Tell whether the statement is true or false.

_______1. ( ) ( )b8a8ba ++=++

______2. ( ) 828a2a8 +=+

_______3. ( ) 7y27y2 +=+_______4. ( )b7a 22b14a −=−_______5. ( ) 5b5aba5 −=+

_______6.a

0

0

a=

_______7. ab0ab =•

_______8. ( ) ( )b8a8ba ++=++_______9. ( ) 0aa =−_______10. ( ) 550 −=−

Exercise B. Name the property illustrated in going from one step to thenext.

1.

2.

3.

13

Properties of EqualityReflexive Property: aa =

Symmetric Property: If ba = , then ab = .Transitive Property: If ba = and cb = , then ca = .

Substitution Property: If ba = , then a may be replaced by b.

( ) ( )

( )

( ) _____ __________ 5c-11b

_____ __________ 8c)(3c8b)(3b

_____ __________ 8c)-(8b3c3b

_____ __________ 2c)-4(2b3c)(3b2c2b4cb3

+=

−++=++=++=−++

( )

_____ __________ 93a

_____ __________ 3-12)3a4a3

+=

+=−+ (3

( )

_____ __________ 5b5a

_____ __________ 5b3a)(2a

_____ __________ 3a5b2a

_____ __________ 03a5b2a2a03a5b2a

+=

++=++=

+++=+++



4.

14

( ) ( ) ( )

( )

_____ __________

_____ __________ 2440

_____ __________ 2446b 6b-

_____ __________ 246b46b-4b623b-2

28=

++=

+++=+++=+++

Chapter 1 Real Numbers and their Properties

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