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Copyright © 2005 Pearson Education, Inc.
SEVENTH EDITION and EXPANDED SEVENTH EDITION
Slide 5-1
Chapter 5
Number Theory and the Real
Number System
Copyright © 2005 Pearson Education, Inc.
5.1
Number Theory
Copyright © 2005 Pearson Education, Inc.
Number Theory
„
„
The study of numbers and their properties.
The numbers we use to count are called the
Natural Numbers or Counting Numbers.
= {1,2,3,4,5,...}
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Slide 5-4
Factors
„
„
The natural numbers that are multiplied together
to equal another natural number are called
factors of the product.
Example:
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Copyright © 2005 Pearson Education, Inc.
Slide 5-5
Divisors
„
If a and b are natural numbers and the quotient
of b divided by a has a remainder of 0, then we
say that a is a divisor of b or a divides b.
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Slide 5-6
Prime and Composite Numbers
„
„
„
A prime number is a natural number greater
than 1 that has exactly two factors (or divisors),
itself and 1.
A composite number is a natural number that is
divisible by a number other than itself and 1.
The number 1 is neither prime nor composite, it
is called a unit.
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Slide 5-7
Rules of Divisibility
Divisible
by
Test
2
The number is even.
3
The sum of the digits of the
number is divisible by 3.
4
The number formed by the last
two digits of the number is
divisible by 4.
5
The number ends in 0 or 5.
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Example
846
846
since 8 + 4 + 6 = 18
844
since 44 ÷ 4
285
Slide 5-8
Divisibility Rules, continued
Divisible
by
Test
6
The number is divisible by
both 2 and 3.
8
The number formed by the last
three digits of the number is
divisible by 8.
9
The sum of the digits of the
number is divisible by 9.
10
The number ends in 0.
Copyright © 2005 Pearson Education, Inc.
Example
846
3848
since 848 ÷ 8
846
since 8 + 4 + 6 = 18
730
Slide 5-9
The Fundamental Theorem of Arithmetic
„
Every composite number can be written as a
unique product of prime numbers.
„
This unique product is referred to as the prime
factorization of the number.
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Slide 5-10
Finding Prime Factorizations
„
Branching Method:
‰
‰
Select any two numbers whose product is the
number to be factored.
If the factors are not prime numbers, then
continue factoring each number until all
numbers are prime.
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Slide 5-11
Example of branching method
3190
319
11
10
29
5
2
Therefore, the prime factorization of
3190 = 2 • 5 • 11 • 29
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Slide 5-12
Division Method
1.
2.
3.
4.
Divide the given number by the smallest prime
number by which it is divisible.
Place the quotient under the given number.
Divide the quotient by the smallest prime number by
which it is divisible and again record the quotient.
Repeat this process until the quotient is a prime
number.
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Slide 5-13
Example of division method
„
Write the prime factorization of 663.
3 663
13 221
17
„
The final quotient 17, is a prime number, so we stop.
The prime factorization of 663 is
3 •13 •17
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Slide 5-14
Greatest Common Divisor
„
The greatest common divisor (GCD) of a set of
natural numbers is the largest natural number
that divides (without remainder) every number
in that set.
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Slide 5-15
Finding the GCD
„
„
„
Determine the prime factorization of each
number.
Find each prime factor with smallest
exponent that appears in each of the prime
factorizations.
Determine the product of the factors found
in step 2.
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Slide 5-16
Example (GCD)
„
„
„
Find the GCD of 63 and 105.
63 = 32 • 7
105 = 3 • 5 • 7
Smallest exponent of each factor:
3 and 7
So, the GCD is 3 • 7 = 21
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Slide 5-17
Least Common Multiple
„
The least common multiple (LCM) of a set of
natural numbers is the smallest natural number
that is divisible (without remainder) by each
element of the set.
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Slide 5-18
Finding the LCM
„
„
„
Determine the prime factorization of each
number.
List each prime factor with the greatest
exponent that appears in any of the prime
factorizations.
Determine the product of the factors found in
step 2.
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Slide 5-19
Example (LCM)
„
„
„
Find the LCM of 63 and 105.
63 = 32 • 7
105 = 3 • 5 • 7
Greatest exponent of each factor:
32, 5 and 7
So, the GCD is 32 • 5 • 7 = 315
Copyright © 2005 Pearson Education, Inc.
Slide 5-20
Example of GCD and LCM
„
„
Find the GCD and LCM of 48 and 54.
Prime factorizations of each:
‰
‰
48 = 2 • 2 • 2 • 2 • 3 = 24 • 3
54 = 2 • 3 • 3 • 3 = 2 • 33
ƒ GCD = 2 • 3 = 6
ƒ LCM = 24 • 33 = 432
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Slide 5-21
5.2
The Integers
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Whole Numbers
„
„
The set of whole numbers contains the set of
natural numbers and the number 0.
Whole numbers = {0,1,2,3,4,…}
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Slide 5-23
Integers
„
„
„
The set of integers consists of 0, the natural
numbers, and the negative natural numbers.
Integers = {…-4,-3,-2,-1,0,1,2,3,4,…}
On a number line, the positive numbers extend
to the right from zero; the negative numbers
extend to the left from zero.
Copyright © 2005 Pearson Education, Inc.
Slide 5-24
Writing an Inequality
„
„
„
Insert either > or < in the box between the
paired numbers to make the statement correct.
a) −3
−1
b) −9
−7
−3 < −1
−9 < −7
c) 0
−4
d) 6
8
0 > −4
6 < 8
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Slide 5-25
Subtraction of Integers
a – b = a + (−b)
Evaluate:
a) –7 – 3 = –7 + (–3) = –10
b) –7 – (–3) = –7 + 3 = –4
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Slide 5-26
Properties
„
Multiplication Property of Zero
a⋅0 = 0⋅a = 0
„
Division
a
For any a, b, and c where b ≠ 0, = c means
b
that c • b = a.
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Slide 5-27
Rules for Multiplication
„
The product of two numbers with like signs
(positive × positive or negative × negative) is a
positive number.
„
The product of two numbers with unlike signs
(positive × negative or negative × positive) is a
negative number.
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Slide 5-28
Examples
„
„
„
„
„
„
Evaluate:
a) (3)(−4)
b) (−7)(−5)
c) 8 • 7
d) (−5)(8)
Solution:
a) (3)(−4) = −12
b) (−7)(−5) = 35
c) 8 • 7 = 56
d) (−5)(8) = −40
Copyright © 2005 Pearson Education, Inc.
Slide 5-29
Rules for Division
„
The quotient of two numbers with like signs
(positive ÷ positive or negative ÷ negative) is a
positive number.
„
The quotient of two numbers with unlike signs
(positive ÷ negative or negative ÷ positive) is a
negative number.
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Slide 5-30
Example
„
„
Evaluate:
a) 72 = 8
b)
−72
= −8
9
d)
72
= −9
−8
9
„
c)
−72
=9
−8
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Slide 5-31
5.3
The Rational Numbers
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The Rational Numbers
„
The set of rational numbers, denoted by Q,
is the set of all numbers of the form p/q,
where p and q are integers and q ≠ 0.
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Slide 5-33
Fractions
„
„
„
Fractions are numbers such as:
1 2
9
, , and
.
3 9
53
The numerator is the number above the fraction
line.
The denominator is the number below the
fraction line.
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Slide 5-34
Reducing Fractions
„
In order to reduce a fraction, we divide both the
numerator and denominator by the greatest
common divisor.
72
Example: Reduce
to its lowest terms.
„
Solution: 72 = 72 ÷ 9 = 8
„
81
81
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81 ÷ 9
9
Slide 5-35
Mixed Numbers
„
„
A mixed number consists of an integer and a
fraction. For example, 3 ½ is a mixed number.
3 ½ is read “three and one half” and means
“3 + ½”.
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Slide 5-36
Improper Fractions
„
„
Rational numbers greater than 1 or less than -1
that are not integers may be written as mixed
numbers, or as improper fractions.
An improper fraction is a fraction whose
numerator is greater than its denominator.
An example of an improper fraction is 12/5.
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Slide 5-37
Converting a Positive Mixed Number to
an Improper Fraction
„
„
Multiply the denominator of the fraction in the mixed
number by the integer preceding it.
Add the product obtained in step 1 to the numerator of
the fraction in the mixed number. This sum is the
numerator of the improper fraction we are seeking.
The denominator of the improper fraction we are
seeking is the same as the denominator of the fraction
in the mixed
Copyright © 2005 Pearson Education, Inc.
Slide 5-38
Example
„
Convert
7
5
10
to an improper fraction.
7 (10 • 5 + 7) 50 + 7 57
5
=
=
=
10
10
10
10
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Slide 5-39
Converting a Positive Improper
Fraction to a Mixed Number
„
„
Divide the numerator by the denominator. Identify the
quotient and the remainder.
The quotient obtained in step 1 is the integer part of
the mixed number. The remainder is the numerator of
the fraction in the mixed number. The denominator in
the fraction of the mixed number will be the same as
the denominator in the original fraction.
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Slide 5-40
Example
„
Convert
236
7
to a mixed number.
33
7 236
21
26
21
5
„
The mixed number is
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5
33 .
7
Slide 5-41
Terminating or Repeating Decimal
Numbers
„
„
„
Every rational number when expressed as a decimal
number will be either a terminating or repeating
decimal number.
Examples of terminating decimal numbers 0.7, 2.85,
0.000045
Examples of repeating decimal numbers 0.44444…
which may be written 0.4,
and 0.2323232323... which may be written 0.23.
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Slide 5-42
Multiplication of Fractions
a c a • c ac
• =
=
, b ≠ 0, d ≠ 0.
b d b • d bd
„
Division of Fractions
a c a d ad
÷ = • =
, b ≠ 0, d ≠ 0, c ≠ 0.
b d b c bc
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Slide 5-43
Example: Multiplying Fractions
„
Evaluate the following.
„
a) 2 ⋅ 7
3 16
2 7
2 ⋅ 7 14
7
⋅
=
=
=
3 16 3 ⋅ 16 48 24
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„
b)
⎛ 3 ⎞ ⎛ 1⎞
⎜1 4 ⎟ ⋅ ⎜ 2 2 ⎟
⎝ ⎠ ⎝
⎠
⎛ 3 ⎞ ⎛ 1⎞ 7 5
⎜1 4 ⎟ ⋅ ⎜ 2 2 ⎟ = 4 ⋅ 2
⎝ ⎠ ⎝
⎠
35
3
=
=4
8
8
Slide 5-44
Example: Dividing Fractions
„
„
Evaluate the following.
a) 2 6
÷
3 7
2 6 2 7
÷ = ⋅
3 7 3 6
2 ⋅ 7 14 7
=
=
=
3 ⋅ 6 18 9
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„
b) −5 4
÷
8 5
−5 4 −5 5
÷ =
⋅
8 5 8 4
−5 ⋅ 5 −25
=
=
8⋅4
32
Slide 5-45
Addition and Subtraction of Fractions
a b a+b
, c ≠ 0.
+ =
c c
c
a b a−b
− =
, c ≠ 0.
c c
c
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Slide 5-46
Example: Add or Subtract Fractions
„
Add: 4 + 3
9 9
„
„
4 3 4+3 7
+ =
=
9 9
9
9
„
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Subtract:
11 3
−
16 16
11 3 11 − 3 8
−
=
=
16 16
16
16
1
=
2
Slide 5-47
Fundamental Law of Rational Numbers
„
If a, b, and c are integers, with b ≠
0, c ≠ 0, then
a a c a ⋅ c ac
= ⋅ =
=
.
b b c b ⋅ c bc
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Slide 5-48
Example:
„
Evaluate:
„
Solution:
7
9
− .
12 10
7
1 ⎛ 7 5⎞ ⎛ 1 6⎞
−
= ⎜ ⋅ ⎟−⎜ ⋅ ⎟
12 10 ⎝ 12 5 ⎠ ⎝ 10 6 ⎠
35 6
=
−
60 60
29
=
60
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Slide 5-49
5.4
The Irrational Numbers and the
Real Number System
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Pythagorean Theorem
„
„
Pythagoras, a Greek mathematician, is credited
with proving that in any right triangle, the square
of the length of one side (a2) added to the
square of the length of the other side (b2)
equals the square of the length of the
hypotenuse (c2) .
a2 + b2 = c2
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Slide 5-51
Irrational Numbers
„
An irrational number is a real number whose
decimal representation is a nonterminating,
nonrepeating decimal number.
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Slide 5-52
Radicals
„
2,
17,
53
are all irrational numbers.
The symbol
is called the radical sign. The
number or expression inside the radical sign
is called the radicand.
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Slide 5-53
Principal Square Root
„
The principal (or positive) square root of a
number n, written n is the positive number
that when multiplied by itself, gives n.
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Slide 5-54
Perfect Square
„
„
Any number that is the square of a natural
number is said to be a perfect square.
The numbers 1, 4, 9, 16, 25, 36, and 49 are the
first few perfect squares.
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Slide 5-55
Product Rule for Radicals
a ⋅ b = a ⋅ b,
„
a ≥ 0, b ≥ 0.
Simplify:
‰
a)
40
40 = 4 ⋅ 10 = 2 ⋅ 10 = 2 10
‰
b)
125
125 = 5 ⋅ 25 = 5 ⋅ 5 = 5 5
Copyright © 2005 Pearson Education, Inc.
Slide 5-56
Addition and Subtraction of Irrational
Numbers
„
„
To add or subtract two or more square roots
with the same radicand, add or subtract their
coefficients.
The answer is the sum or difference of the
coefficients multiplied by the common radical.
Copyright © 2005 Pearson Education, Inc.
Slide 5-57
Example: Adding or Subtracting
Irrational Numbers
„
Simplify: 4 7 + 3 7
„
Simplify: 8 5 − 125
4 7 +3 7
8 5 − 125
= (4 + 3) 7
= 8 5 − 25 ⋅ 5
=7 7
= 8 5 −5 5
= (8 − 5) 5
=3 5
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Slide 5-58
Multiplication of Irrational Numbers
„
Simplify:
6 ⋅ 54
6 ⋅ 54 = 6 ⋅ 54 = 324 = 18
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Slide 5-59
Division of Irrational Numbers
„
a
a
=
,
b
b
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a ≥ 0, b ≥ 0.
Slide 5-60
Example: Division
„
Divide:
16
„
Divide:
144
4
„
Solution:
16
16
=
= 4 =2
4
4
2
„
Solution:
144
144
=
= 72
2
2
= 36 ⋅ 2 = 36 ⋅ 2
=6 2
Copyright © 2005 Pearson Education, Inc.
Slide 5-61
Rationalizing the Denominator
„
„
A denominator is rationalized when it contains
no radical expressions.
To rationalize the denominator, multiply BOTH
the numerator and the denominator by a
number that will result in the radicand in the
denominator becoming a perfect square. Then
simplify the result.
Copyright © 2005 Pearson Education, Inc.
Slide 5-62
Example: Rationalize
„
Rationalize the denominator of
8
12
„
Solution:
.
8
8
2
=
=
12
3
12
=
2
3
=
2
3
⋅
3
3
6
=
3
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Slide 5-63
5.5
Real Numbers and their
Properties
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Real Numbers
„
The set of real numbers is formed by the union
of the rational and irrational numbers.
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Slide 5-65
Relationships Among Sets
Real numbers
Rational numbers
Integers
Whole numbers
Natural numbers
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Irrational
numbers
Slide 5-66
Properties of the Real Number System
„
Closure
‰
If an operation is performed on any two elements
of a set and the result is an element of the set,
we say that the set is closed under that given
operation.
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Slide 5-67
Commutative Property
„
Addition
a+b=b+a
for any real numbers
a and b.
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„
Multiplication
a.b = b.a
for any real numbers
a and b.
Slide 5-68
Example
„
„
„
8 + 12 = 12 + 8 is a true statement.
5 × 9 = 9 × 5 is a true statement.
Note: The commutative property does not hold
true for subtraction or division.
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Slide 5-69
Associative Property
„
Addition
(a + b) + c = a + (b + c),
for any real numbers
a, b, and c.
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„
Multiplication
(a.b) .c = a. (b.c),
for any real numbers
a, b, and c.
Slide 5-70
Example
„
(3 + 5) + 6 = 3 + (5 + 6) is true.
„
(4 × 6) × 2 = 4 × (6 × 2) is true.
„
Note: The commutative property does not hold
true for subtraction or division.
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Slide 5-71
Distributive Property
„
Distributive property of multiplication over
addition
a.(b + c) = a.b + a.c
for any real numbers a, b, and c.
„
Example: 6(r + 12) = 6.r + 6.12
= 6r + 72
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Slide 5-72
5.6
Rules of Exponents and
Scientific Notation
Copyright © 2005 Pearson Education, Inc.
Exponents
„
When a number is written with an exponent,
there are two parts to the expression: base
The exponent tells how many times the base
should be multiplied together.
exponent
„
4 = 4⋅4⋅4⋅4⋅4
5
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Slide 5-74
Product Rule
a a = a
m
„
„
n
m+n
Simplify: 34 • 39
34 • 39 = 34 + 9 = 313
„
Simplify: 64 • 65
„
6 4 • 6 5 = 64 + 5 = 69
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Slide 5-75
Quotient Rule
am
m −n
a
=
, a≠0
n
a
5
„
„
Simplify: 7
2
7
75
5−2
3
=
=
7
7
72
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915
98
„
Simplify:
„
915
15 − 8
7
=
=
9
9
98
Slide 5-76
Zero Exponent
a = 1, a ≠ 0
0
„
Simplify: (3y)0
(3y)0 = 1
„
Simplify: 3y0
3y0 = 3 (y0)
= 3(1) = 3
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Slide 5-77
Negative Exponent
„
a
„
„
−m
1
= m, a≠0
a
Simplify: 6−4
6
−4
1
1
= 4 =
1296
6
Copyright © 2005 Pearson Education, Inc.
Slide 5-78
Power Rule
(a ) = a
m n
„
Simplify: (32)3
„
(32)3 = 32•3 = 36
„
Simplify: (23)5
„
(23)5 = 23•5 = 215
Copyright © 2005 Pearson Education, Inc.
m n
Slide 5-79
Scientific Notation
„
„
„
Many scientific problems deal with very large or
very small numbers.
93,000,000,000,000 is a very large number.
0.000000000482 is a very small number.
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Slide 5-80
Scientific Notation continued
„
Scientific notation is a shorthand method used
to write these numbers.
„
9.3 x 1013 and 4.82 x 10-10 are two examples of
the scientific numbers.
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Slide 5-81
To Write a Number in Scientific Notation:
1.
2.
Move the decimal point in the original number to the
right or left until you obtain a number greater than or
equal to 1 and less than 10.
Count the number of places you have moved the
decimal point to obtain the number in step 1.
If the original decimal point was moved to the left, the
count is to be considered positive. If the decimal point
was moved to the right, the count is to be considered
negative.
Copyright © 2005 Pearson Education, Inc.
Slide 5-82
To Write a Number in Scientific Notation:
continued
3. Multiply the number obtained in step 1 by 10
raised to the count found in step 2. (The count
found in step 2 is the exponent on the base
10.)
Copyright © 2005 Pearson Education, Inc.
Slide 5-83
Example
„
Write each number in scientific notation.
a) 1,265,000,000.
1.265 × 109
b) 0.000000000432
4.32 × 10−10
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Slide 5-84
To Change a Number in Scientific
Notation to Decimal Notation
„
„
Observe the exponent on the 10.
If the exponent is positive, move the decimal point in
the number to the right the same number of places as
the exponent. Adding zeros to the number might be
necessary.
If the exponent is negative, move the decimal point in
the number to the left the same number of places as
the exponent. Adding zeros might be necessary.
Copyright © 2005 Pearson Education, Inc.
Slide 5-85
Example
„
Write each number in decimal notation.
a) 4.67 × 105
467,000
b) 1.45 × 10-7
0.000000145
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Slide 5-86
5.7
Arithmetic and Geometric
Sequences
Copyright © 2005 Pearson Education, Inc.
Sequences
„
„
A sequence is a list of numbers that are related
to each other by a rule.
The terms are the numbers that form the
sequence.
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Slide 5-88
Arithmetic Sequence
„
„
An arithmetic sequence is a sequence in which each
term after the first term differs from the preceding term
by a constant amount.
The common difference, d, is the amount by which each
pair of successive terms differs. To find the difference,
simply subtract any term from the term that directly
follows it.
Copyright © 2005 Pearson Education, Inc.
Slide 5-89
General Term of an Arithmetic Sequence
an = a1 + (n − 1)d
„
Find the 5th term of the arithmetic sequence
whose first term is 4 and whose common
difference is −8.
an = a1 + (n − 1)d
a5 = 4 + (5 − 1)(−8)
= 4 + (4)(−8)
= 4 + (−32)
= −28
Copyright © 2005 Pearson Education, Inc.
Slide 5-90
Sum of the First n Terms in an
Arithmetic Sequence
n(a1 + an )
sn =
2
„
Find the sum of the first
50 terms in the arithmetic
sequence: 2, 4, 6,
8,…100
Copyright © 2005 Pearson Education, Inc.
n(a1 + an )
sn =
2
50(2 + 100)
s50 =
2
50(102)
s50 =
2
s50 = 2550
Slide 5-91
Geometric Sequences
„
„
„
A geometric sequence is one in which the ratio
of any term to the term that directly precedes it
is a constant.
This constant is called the common ratio, r.
r can be found by taking any term except the
first and dividing it by the preceding term.
Copyright © 2005 Pearson Education, Inc.
Slide 5-92
General Term of a Geometric Sequence
an = a1r
„
n −1
Find the 6th term for the geometric sequence
with the first term = 3 and the common ratio = 4.
a6 = (3)4
6 −1
a6 = (3)4
5
a6 = (3)1024
a6 = 3072
Copyright © 2005 Pearson Education, Inc.
Slide 5-93
Sum of the First n Terms in an
Geometric Sequence
a1 (1 − r )
sn =
,
1− r
n
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r ≠1
Slide 5-94
Example
„
Find the sum of the first 4 terms of the
geometric sequence for r = 2 and the first
term = 3.
s4 =
s4 =
s4 =
s4 =
Copyright © 2005 Pearson Education, Inc.
3(1 − 24 )
1− 2
3(1 − 16)
−1
3( −15)
−1
−45
= 45
−1
Slide 5-95
5.8
Fibonacci Sequence
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The Fibonacci Sequence
„
„
„
This sequence is named after Leonardo of Pisa,
also known as Fibonacci.
He was one of the most distinguished
mathematicians of the Middle Ages.
He is also credited with introducing the HinduArabic number system into Europe.
Copyright © 2005 Pearson Education, Inc.
Slide 5-97
Fibonacci Sequence
„
„
„
1, 1, 2, 3, 5, 8, 13, 21, …
In the Fibonacci sequence, the first two terms
are 1. The sum of these two terms gives us the
third term (2).
The sum of the 2nd and 3rd terms give us the 4th
term (3) and so on.
Copyright © 2005 Pearson Education, Inc.
Slide 5-98
In Nature
„
„
„
In the middle of the 19th century, mathematicians found
strong similarities between this sequence and many
natural phenomena.
The numbers appear in many seed arrangements of
plants and petal counts of many flowers.
Fibonacci numbers are also observed in the structure of
pinecones and pineapples.
Copyright © 2005 Pearson Education, Inc.
Slide 5-99
Divine Proportions
5 +1
2
„
Golden Number :
„
The value obtained when the ratio of any term
to the term preceding it in the Fibonacci
sequence.
Copyright © 2005 Pearson Education, Inc.
Slide 5-100
Golden or Divine Proportion
AB AC
5 +1
=
=
≈ 1.618
AC CB
2
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Slide 5-101
Golden Rectangle
length a + b a
5 +1
=
= =
width
a
b
2
Copyright © 2005 Pearson Education, Inc.
Slide 5-102