addition subtraction-multi-divison(6)

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-1

Presented byProfessional Aided Supplemental Instruction

(PASI)Ivy Tech Community College

Indianapolis

Basic MathematicsAddition

SubtractionMultiplication

Division

Real Numbers

Chapter 1

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-2

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-3

Addition of Real Numbers

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-4

Number LinesEvaluate 3 + (- 4) using a number line1. Always begin with 0.

2. Since the first number is positive, the first arrow starts at 0 and is drawn 3 units to the right.

3. The second arrow starts at 3 and is drawn 4 units to the left , since the second addend is negative.

3 + (– 4) = -1

-5 -4 -3 -2 -1 0 1 2 3 4 5

3

-5 -4 -3 -2 -1 0 1 2 3 4 5

-4

3

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-5

Add Fractions

The LCD is 48. Rewriting the first fraction with the LCD gives the following.

32

167 Add

4811

4821

1616

33

167

4832

32

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-6

Identify OppositesAny two numbers whose sum is 0 are said to be opposites, or additive inverses, of each other.

a + (– a) = 0

The opposite of a is –a.

The opposite of –a is a.

Example:

The opposite of –5 is 5, since –5 + 5 = 0

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-7

Add Using Absolute Values

To add real numbers with the same sign,add their absolute values. The sum has the same sign as the numbers being added.

Example:

–6 + (–9) = –15 4 + 8 = 12

The sum of two positive numbers will always be positive and the sum of two negative numbers will always be negative.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-8

Add Using Absolute Values

To add two signed numbers with different signs, subtract the smaller absolute value from the larger absolute value. The answer has the sign of the number with the larger absolute value.

Example:13 + (–4) = 9 –35 + 15 = -20

The sum of two numbers with different signs may be positive or negative. The sign of the sum will be the same as the sign of the number with the larger absolute value.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-9

Subtraction of Real Numbers

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-10

SubtractingIn general, if a and b represent any two real numbers, then

a – b = a + (-b)

Examples 1 and 2:

1.) 9 – (+4) =9 + (– 4) = 5

2.) 5 – 3 = 5 + (– 3) = 2

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-11

More Examples

Example: 3

1.) 3 – 10 =3 + (– 10) = -7

2.) -6 – 4 = -6 + (– 4) = -10

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-12

Kuta Software - Infinite Algebra 1Adding and Subtracting Positive and Negative Numbers1 (-2) + 3 9 (-14) + (-7)

2 3 - (-8) 10 (-9) + 14

3 (-8) - (-2) 11 5 + (-8)

4 (-27) - 24 12 (-41) + (-40)

5 38 - (-17) 13 (-44) + (-9)

6 (-16) - (-36) 14 (-6) - 24

7 (-16) - 6 + (-5) 15 15 - 13 + 2

8 16 - (-13) - (-5) 16 (-7) - (-2) - 9

1

11

-6

-51

55

20

-27

34

-21

5

-3

-81

-53

-30

4

-14

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-13

Multiplication and Division of Real Numbers

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-14

Sign of the Product Rule

The product of two numbers with like signs is a positive number.

The product of two numbers with unlike signs is a negative number.

Example:

a.) 4(– 5) = – 20

b.) (– 6)(7) = -42

c.) (– 9)(-3) = 27

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-15

Helpful Hint

At this point some students begin confusing problems like -2 – 3 with (-2)(-3) and problems like 2-3 and (-2)(-3). Make sure you understand the difference between these problems.

Subtraction Problems

– 2 – 3 = – 5

2- 3 = – 1

Multiplication Problems

(-2) (– 3) = 6

(2)(-3)= – 6

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-16

Divide Numbers

1. The quotient of two numbers with like signs is a positive number.

2. The quotient of two numbers with unlike signs is a negative number.

Example:2

510 a.)

9- 545

b.)

6 6

36

c.)

The Sign of the Quotient of Two Real Numbers

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-17

Helpful Hint

(+)(+) = + (+)(+) = +

(–)(–) = + (–)(–) = +

(+)(–) = – (+)(–) = –

(–)(+) = – (–)(+) = –

Like signs give positive products and quotients.

Unlike signs give negative products and quotients.

For multiplication and division of two real numbers:

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-18

Remove Negative Signs from Denominators

If a and b represent any real numbers, b 0, then

ba

ba

ba

We generally do not write fractions with a negative sign in the denominator. When a negative sign appears in a denominator, we can move it to the numerator or place it in front of the fraction.

The fraction would be written as or . 7

5

7

5

7

5

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-19

Real Number Operations

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-20

Evaluate Divisions Involving Zero

If a represents any real number except 0, then

0 a = = 0a0

Division by 0 is undefined. ?a 0

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-21

Worksheet by Kuta Software LLC Kuta Software - Infinite Pre-Algebra 1 6 × −4 13 4 × 22 3 × −4 14 −6 × 43 5 × −4 15 −3 × 44 −5 × 6 16 −2 × −15 −8 ÷ −2 17 11 × 126 35 ÷ -5 18 9 ÷ −37 10 ÷ 5 19 16 ÷ 28 −49 ÷ 7 20 8 × −129 9 × 10 × 6 21 −6 × −10 ×

−810 7 × 9 × 7 22 6 × 6 × −211 −5 × −4 ×

−1023 9 × 9 × −5

12 8 × 3 × 8 24 7 × 5 × −5

−24

−12

−20

−30

4

−7

2

−7

540

441

−200

192

8

−24

−12

2

132

−3

8

−96

− 480 − 72

− 405

− 175

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-22

Exponents, Parentheses and Order of Operations

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-23

Learn the Meaning of Exponents

In the expression 42, the 4 is called the base, and the 2 is called the exponent.

exponent42

base

43 is read “4 to the third power” and means 4·4·4 = 43

3 factors of 4

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-24

–x2 vs. (-x)2

An exponent refers only to the number or variable that directly precedes it unless parentheses are used to indicate otherwise.

– x2 = -(x)(x)

(– x)2 = (–x)(–x) = x2

Example: – 32 = – (3)(3) = – 9

(– 3)2 = (–3)(–3) = 9

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-25

Learn the Order of Operations

To evaluate mathematical expressions, use the following order:

1. First, evaluate the information within parentheses ( ), brackets , or braces .These are grouping symbols, for they group information together. A fraction bar, —, also serves as a grouping symbol. If the expression contains nested grouping symbols (one pair of grouping symbols within another pair), evaluate the information in the innermost groping symbols first.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-26

Learn the Order of Operations

2. Next, evaluate all exponents.

3. Next, evaluate all the multiplications and divisions in the order in which they occur, working from left to right.

4. Finally, evaluate all additions or subtractions in the order in which they occur, working from left to right.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-27

Order of Operations

Evaluate:

6 + 3 • 52 – 4 = Exponent

Multiply

Add

6 + 3 • 25 – 4 =

6 + 75 - 4=

81 – 4 =

77

Steps Taken

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-28

Order of Operations

Evaluate:

-7 + 2 [-6 + (36 / 32 )] = Exponent

Divide

Add

-7 + 2 [-6 + (36 / 9 )] =

-7 + 2 [-6 + 4] =

-7 + 2 [-2] =

-11

Steps Taken

-7 - 4 =

Multiply

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-29

Properties of the Real

Number System

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-30

Commutative PropertyCommutative Property of Addition

If a and b represent any two real numbers, thena + b = b + a 4 + 3 = 3 + 4

Commutative Property of MultiplicationIf a and b represent any real numbers, then

a · b = b · a 6 · 3 = 3 · 6

Commutative (commute) changes the order.

*Note that the commutative property does not hold for subtraction and division

7 = 7

18 = 18

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-31

Associative PropertyAssociative Property of Addition

If a, b, and c represent three real numbers, then(a + b) + c = a + (b + c) (3 + 4) + 5 = 3 + (4 + 5)

Associative Property of MultiplicationIf a, b, and c represent any three real numbers, then

(a · b) · c = a ·(b · c) (6 · 2) · 4 = 6 · (2 · 4)

Associative (associate) changes the grouping.

*Note that the associative property does not hold for subtraction and division

7 + 5 = 3 + 9 12 = 12

12 · 4 = 6 · 8 48 = 48

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-32

Distributive Property

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

a(b + c) = ab + ac

Distributive involves two operations (usually multiplication and division).

2(3 + 4) = 2(3) + 2(4)

2(7) = 6 + 8

14 = 14

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-33

Identity Properties

If a represents any real number, then

a + 0 = a and 0 + a = a

a · 1 = a and 1 · a = 1

Identity Property of Addition

Identity Property of Multiplication

4 + 0 = 4 0 + 4 = 4

13 · 1 = 13 1 · 13 = 13

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-34

Inverse Properties

If a represents any real number, then

a + (-a)= 0 and (-a) + a = 0

Inverse Property of Addition

Inverse Property of Multiplication

a · = 1 and · a = 1 (a 0)a

1

a

1

7 + (-7) = 0 (-7) + 7 = 0

12 · = 1 · 12 = 112

1

12

1

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-35

Kuta Software -Infinite Pre-Algebra Order of Operations Evaluate each expression. 1 (30 - 3) ÷ 3 13 9+6÷ (8-2)

2 (21 - 5) ÷ 8 14 4(4÷2+4)

3 1 + 72 15 6+ (5+8) × 4

4 5×4-8 16 6×6- (7+5)

5 8+6 × 9 17 (9 × 2) ÷ (2 + 1)

6 3 + 17 × 5 18 2 - (4 + 3 - 6)

7 7 + 12 × 11 19 7 × 7 - (8 - 2)

8 15 + 40 ÷ 20 20 9 - 7 - 6 ÷ 6

9 20 + 16 - 15 21 (4 - 1 + 8 ÷ 8) × 5

10 19 - 15 - 3 22 (10 × 2) ÷ (1 + 1)

11 9 × (3+3) ÷6 23 7 × 9 - 7 - 3 × 5

12 (9 + 18 - 3) ÷8 24 8 - 1 - (18 - 2) ÷ 8

9

2

50

12

62

88

139

17

21

1

9

3

10

24

48

24

6

1

43

1

20

10

41

5