Post on 16-Aug-2015
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