section 1.1 numbers and their properties. objectives a write a set of numbers using roster or set–...
TRANSCRIPT
Section 1.1
Numbers and Their Properties
OBJECTIVES
A Write a set of numbers using roster or set–builder notation.
OBJECTIVES
B Write a rational number as a decimal.
OBJECTIVES
C Classify a number as natural, whole, integer, rational, irrational, or real.
OBJECTIVES
D Find the additive inverse of a number.
OBJECTIVES
E Find the absolute value of a number.
OBJECTIVES
F Given two numbers, use the correct notation to indicate equality or which is larger.
DEFINITION
The set of numbers used for counting.
NATURAL NUMBERS
N = {1, 2, 3, . . . }
DEFINITION
The set of natural numbers and zero.
WHOLE NUMBERS
W = { 0, 1, 2, 3, . . .}
DEFINITION
The set of whole numbers and their opposites(negatives).
INTEGERS
I = {. . . , –2, –1, 0, 1, 2, . . .}
DEFINITION
All numbers that can be written as the ratio of two integers.
RATIONAL NUMBERS
{ and integers }aQ = r |r = , a b , b 0b
DEFINITION
Numbers that cannot be written as ratios of two integers.
IRRATIONAL NUMBERS
H = {x |x is a number that is not rational}
DEFINITION
Numbers that are either rational or irrational:
REAL NUMBERS
R = {x | x a number that is rational or irrational}
DEFINITION
The additive inverse(opposite) of a is –a.
ADDITIVE INVERSE
DEFINITION
The distance between a and 0 on the real-number line
ABSOLUTE VALUE
a if a is positive |11 | = 11
0 if a is zero |0 | = 0
-a if a is negative | -5 | = - (-5) = 5 |a | =
CAUTION
The absolute value is always positive or zero.
–|– 3| = –3, –|4.2 | = –4.2
DEFINITION
If given any two real numbers, only one of three things is true:
TRICHOTOMY LAW
1. a is equal to b, denoted by a = b, or2. a is less than b, denoted by a < b, or3. a is greater than b, denoted by a >b.
Practice Test
Exercise #1
Chapter 1The Real NumbersSection 1.1A
Use roster notation to list the natural numbers between 5 and 9.
The set of natural numbers between 5 and 9 is {6, 7,8}
Note 5 and 9 are not included
Practice Test
Exercise #2
Chapter 1The Real NumbersSection 1.1B
Write as a decimal:
= 8 3 b. 2
3 a. 3
8
= 8 3.000
= 0.375
= 3 2
= 3 2.000
= 0.666...
= 0.6
Practice Test
Exercise #3
Chapter 1The Real NumbersSection 1.1C
Set
Classify the given number by making a check mark () in the appropriate row(s).
Natural numberWhole numberIntegerRational numberIrrational numberReal number
0.5 0 – 6 27 5
Practice Test
Exercise #4
Chapter 1The Real NumbersSection 1.1D
Find the additive inverse of 4
5.
= – 4
5
Practice Test
Exercise #5
Chapter 1The Real NumbersSection 1.1E
= 9 a. | –9 |
Find:
b. | 0.5 | = 0.5
Practice Test
Exercise #6
Chapter 1The Real Numbers1.1F
14
= 4 1
– 0.66 is farther from 0 than – 0.25so – 0.25 is greater than – 0.66
a. – 1
4 _____ – 1
3
Fill in the blank with <, >, or = to make the resulting statement true:
= 4 1.00 = 0.25 so – 1
4 = – 0.25
13
= 3 1 = 3 1.00 = 0.66 so – 1
3 = – 0.66
>
b. 0.4 _____ 2
5
25
= 5 2 = 5 2.0 = 0.4
=
Section 1.2
Operations and Properties of Real Numbers
OBJECTIVES
A Add, subtract, multiply, and divide signed numbers.
OBJECTIVES
B Identify uses of the properties of the real numbers.
TO ADD TWO NUMBERS WITH THE SAME SIGN:
PROCEDURE
Add their absolute values and give the sum the common sign.
TO ADD TWO NUMBERS WITH DIFFERENT SIGNS:
PROCEDURE
1. Find the absolute value.2. Subtract the smaller from the
greater number.3. Use the sign of the number with
the greater absolute value.
DEFINITION
For any real number a:
ADDITIVE IDENTITY
a + 0 = a = 0 + a
DEFINITION
If a and b are real numbers:
SUBTRACTION OF SIGNED NUMBERS
a - b = a + (-b)
DEFINITION
For any real number a:
ADDITIVE INVERSE
a + (-a) = (-a) + a = 0
SIGNIFY MULTIPLICATIONPROCEDURE
raised dot : a • bnext to each other : abparentheses: (a)(b), a(b), or (a)b
MULTIPLYING NUMBERS WITH OPPOSITE SIGNS
PROCEDURE
To multiply a positive number by a negative number, multiply their absolute values and make the product negative.
DEFINITION
Same signs: Positive(+)Different signs: Negative(–)
SIGNS OF MULTIPLICATION PRODUCTS
DEFINITION
For any real number a:
IDENTITY FOR MULTIPLICATION
a • 1 = 1 • a = a
DEFINITIONMULTIPLICATION OF FRACTIONS
•••
a c a c = (b, d 0)b d b d
DEFINITION
If a and b are real numbers and b is not zero:
DIVISION OF REAL NUMBERS
ab
= q means that a = b • q
DEFINITION
For any real number a and nonzero real number b, there are two cases of signs:
SIGNS OF A FRACTION
-ab
= a-b
= - ab
or -a-b
= ab
DEFINITION
For a ≠ 0:
ZERO IN DIVISION
0a = 0 and a
0 = undefined
CAUTION
0k
is okay but n0
is a no-no!
DEFINITION
Every nonzero real number a has a reciprocal such that:
MULTIPLICATIVE INVERSE (RECIPROCAL)
a •
1a = 1
DEFINITIONDIVISION OF FRACTIONS
a c a d = (b, c and d 0)cb d b
•÷
Practice Test
Exercise #7
Chapter 1The Real NumbersSection 1.2A
= –4
a. –9 + 5
Find.
–0.8 + b. –0.7
= –1.5
Practice Test
Exercise #8
Chapter 1The Real NumbersSection 1.2A
a. –16 – 7
Find.
– 0.6 – b. –0.4
= –16 + –7
= –23
= –0.6 + 0.4
= –0.2
Practice Test
Exercise #9
Chapter 1The Real NumbersSection 1.2A
Find.
a. – 1
8 – 3
4
Least common denominator = 8.
1 3= – + –8 4
1 6= – + –8 8
Now add numerators.
–1 + –6= 8
= –7
8
= – 7
8
3 5= – + 4 6
Least common denominator = 12.
= – 9
12 + 10
12
3 5 – – –4
b.6
Find.
= –9 +10
12
= 1
12
Now add numerators.
Practice Test
Exercise #10
Chapter 1The Real NumbersSection 1.2A
6a. –9
Find.
–4 b. –1.2
= – 6 9
= –54
= + 4 12
= 4.8
Practice Test
Exercise #11
Chapter 1The Real NumbersSection 1.2A
= – 1
9
1
1
a. – 1
2 2
9
Find.
= – 4
3
= – 3
2 8
9
1
31
4
Find.
b. – 3
2 ÷ 9
8
Practice Test
Exercise #12
Chapter 1The Real NumbersSection 1.2B
7 + 3 + 6 = 3 + 7a. + 6
Name the property illustrated in the statement.
Commutative Property of Addition
2 + 9 + 4 = 2 + 9b. + 4
Associative Property of Addition
Practice Test
Exercise #13
Chapter 1The Real NumbersSection 1.2B
Name the property illustrated in the statement.
a. 3 1
3 = 1
0.3 + –0.3b. = 0
Inverse Property of Multiplication.
Inverse Property of Addition.
Section 1.3
Properties of Exponents
OBJECTIVES
A Evaluate expressions containing natural numbers as exponents.
OBJECTIVES
B Write an expression containing negative exponents as a fraction.
OBJECTIVES
C Multiply and divide expressions containing exponents.
OBJECTIVES
D Raise a power to a power.
OBJECTIVES
E Raise a quotient to a power.
OBJECTIVESF Convert between
ordinary decimal notation and scientific notation, and use scientific notation in computations.
DEFINITION
If a is a real number and n is a natural number:
EXPONENT AND BASE
an = a • a • a • • • a
n factors
Practice Test
Exercise #14
Chapter 1The Real NumbersSection 1.3A
a.
Evaluate.
b.
–3 4
= +9 +9
= 81
= – 3 3 3 3
= – 81
= –3 –3 –3 –3
–34
Practice Test
Exercise #15
Chapter 1The Real NumbersSection 1.3B
a. 7–2
Write without negative exponents.
b. x –8
= 1
72 = 1
7 7 = 1
49
= 1
x8
Practice Test
Exercise #16
Chapter 1The Real NumbersSection 1.3C
Perform the indicated operation and simplify.
a. (3x 4y)(–4x –8y8)
b. 48x 4
16x –8
Perform the indicated operation and simplify.
4 –8 1 8= 3 –4 x x y y
4 + –8 1 + 8= –12 x y
= –12x –4y 9
94
1= –12 yx
= –12y 9
x 4 = – 12y 9
x 4
a. (3x 4y)(–4x –8y8)
= 48
16 x 4
x –8
Perform the indicated operation and simplify.
b. 48x 4
16x –8
= 3 x 4 – –8
= 3x 4 + 8
= 3x12
Practice Test
Exercise #17
Chapter 1The Real NumbersSection 1.3D, E
Simplify.
a. (–2x8y –2)3
3 8 3 –2 3= (–2) ( ) ( ) x y
8 3 –2 3= –2 –2 –2 x y
= –8x 24 y –6
= –8x 24
y6 = – 8x 24
y6
–35
–3. y
b x
=
(x5)–3
(y–3)–3
= x5(–3)
y (–3)(–3)
= x –15
y 9 = 1
x15 1
y 9 = 1
x15y 9
Simplify.
Practice Test
Exercise #18
Chapter 1The Real NumbersSection 1.3F
The exponent of 10, (–3), means move the decimal point 3 places to the left.
Write in standard notation.
6.5 x 10–3
= .006.5
= 0.0065
Practice Test
Exercise #19
Chapter 1The Real NumbersSection 1.3F
The exponent of 10, (5), means move the decimal point 5 places to the right.
Write as a whole number.
8.5 x 105
= 8.50000
= 850,000
Practice Test
Exercise #20
Chapter 1The Real NumbersSection 1.3F
Perform the calculation and write your answer in scientific notation.
5 – 77.1 10 4 10
5 –7= 7.1 4 10 10
= 28.4 105 + (–7)
–2= 28.4 10
NOTE28.4 = 2.84 101
1 –2= 2.84 10 10
= 2.84 101 + (–2)
= 2.84 10–1
Perform the calculation and write your answer in scientific notation.
–2= 28.4 10
5 – 77.1 10 4 10
Section 1.4
Algebraic Expressions and The Order of Operations
OBJECTIVES
A Evaluate numerical expressions with grouping symbols.
OBJECTIVES
B Evaluate expressions using the correct order of operations.
OBJECTIVES
C Evaluate algebraic expressions.
OBJECTIVES
D Use the distributive property to simplify expressions.
OBJECTIVES
E Simplify expressions by combining like terms.
OBJECTIVES
F Simplify expressions by removing grouping symbols and combining like terms.
ORDER OF OPERATIONSPROCEDURE
1. Do the operations in the ().2. Evaluate exponential
expressions.3. Perform multiplications and
divisions from left to right.4. Perform additions and
subtractions from left to right.
PE
(MD)
(AS)
Identity for Multiplication
a = 1 • a
PROCEDURE
For any real number a:
Additive Inverse
– a = –1 • a
PROCEDURE
For any real number a:
Additive Inverse of a Sum
–(a + b) = –a – b
PROCEDURE
Additive Inverse of a Difference
–(a – b) = –a + b
PROCEDURE
DEFINITION
Constant terms or terms with exactly the same variable factors are called similar or like terms.
LIKE TERMS
Practice Test
Exercise #21
Chapter 1The Real NumbersSection 1.4A
a. [ –7(4 + 3)] + 9
Evaluate.
b.
5 (131 – 32)9
= –7(7) + 9
= –49 + 9
= –40
=
5(99)9
= 55
11
1
Practice Test
Exercise #22
Chapter 1The Real NumbersSection 1.4B
– 43 + 6 – 12
2 + 15 ÷ 3
Evaluate.
= – 64 + 6 – 12
2 + 15 ÷ 3
= – 64 + – 6
2 + 15 ÷ 3
= – 64 – 3 + 5
= –62
Practice Test
Exercise #23
Chapter 1The Real NumbersSection 1.4C
12
(b1 + b2)h gives the area of a
trapezoid. Find the area of thetrapezoid if b1 = 8, b2 = 3, and h = 6.
Evaluate.
a.
= 1
28 + 3 6
= 1
211 6
= 33
b. 7 – x 2 + 20 ÷ y – xy;if x = –2 and y = 4.
= 7 – (–2)2 + (20 ÷ 4) – (–2 4)
= 7 – (–2)2 + 5 – (–8)
= 7 – 4 + 5 + 8
= 16
Evaluate.
Practice Test
Exercise #24
Chapter 1The Real NumbersSection 1.4D, E
a. –5 x + 7
Simplify.
b. 7x – 3x + 1 + 2x + 2
= –5x – 35
= 7x – 3x – 1 + 2x + 2
= 6x + 1
Practice Test
Exercise #25
Chapter 1The Real NumbersSection 1.4F
[(5x 2 – 3) + (3x + 7)] – [(x – 3) + (2x 2 – 2)]
Simplify.
= 5x 2 – 3 + 3x + 7 – x – 3 + 2x 2 – 2
= 5x 2 + 3x + 4 – 2x 2 + x – 5
= 5x 2 + 3x + 4 – 2x 2 – x + 5
= 3x 2 + 2x + 9