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Chapter 1
Number Theory and the Real
Number System
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 1
WHAT YOU WILL LEARN
• An introduction to number theory
• Prime numbers
• Integers, rational numbers, irrational
numbers, and real numbers
• Properties of real numbers
• Rules of exponents and scientific
notation
• Arithmetic and geometric sequences
• The Fibonacci sequence
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 2
Section 1
Number Theory
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 3
Number Theory


The study of numbers and their properties.
The numbers we use to count are called natural
numbers, N , or counting numbers.
N = {1,2,3,4,5,...}
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Chapter 5 Section 1 - Slide 4
Factors



The natural numbers that are multiplied together
to equal another natural number are called
factors of the product.
A natural number may have many factors.
Example:
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 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.
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 7
Rules of Divisibility
Divisible
Test
Example
by
2
The number is even.
846
846
3
The sum of the digits of
since 8 + 4 + 6 = 18
the number is divisible
and 18 is divisible by 3
by 3.
844
4
The number formed by
since 44
the last two digits of the
is divisible by 4
number is divisible by 4.
5
The number ends in 0 or
285
5.
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Chapter 5 Section 1 - Slide 8
Divisibility Rules (continued)
Divisible
Test
by
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
9
The8.sum of the digits of
the number is divisible
by 9.
10
The number ends in 0.
Copyright © 2009 Pearson Education, Inc.
Example
846
3848
since 848 is divisible
by 8
846
since 8 + 4 + 6 = 18
and 18 is divisible by 9
730
Chapter 5 Section 1 - Slide 9
The Fundamental Theorem of Arithmetic

Every composite number can be expressed as
a unique product of prime numbers.

This unique product is referred to as the prime
factorization of the number.
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Chapter 5 Section 1 - Slide 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,
continue factoring each number until all
numbers are prime.
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Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 12
Division Method
1. Divide the given number by the smallest prime
number by which it is divisible.
2. Place the quotient under the given number.
3. Divide the quotient by the smallest prime
number by which it is divisible and again
record the quotient.
4. Repeat this process until the quotient is a
prime number.
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Chapter 5 Section 1 - Slide 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
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 15
Finding the GCD of Two or More
Numbers



Determine the prime factorization of each
number.
List 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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Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 18
Finding the LCM of Two or More
Numbers



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.
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 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 LCM is 32 • 5 • 7 = 315.
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 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
GCD = 2 • 3 = 6
LCM = 24 • 33 = 432
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Chapter 5 Section 1 - Slide 21
Section 2
The Integers
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Chapter 5 Section 1 - Slide 22
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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Chapter 5 Section 1 - Slide 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.
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Chapter 5 Section 1 - Slide 24
Writing an Inequality

Insert either > or < in the box between the
paired numbers to make the statement correct.
a) 3
1
3 < 1
c) 0
4
0 > 4
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b) 9
7
9 < 7
d) 6
8
6 < 8
Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 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 © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 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.
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 30
Example

Evaluate:
72
a)
b)
72
9
d)
72
8
9
72
8
c)

Solution:
a) 72  8
9
c)
72
9
8
Copyright © 2009 Pearson Education, Inc.
b)
72
 8
9
d)
72
 9
8
Chapter 5 Section 1 - Slide 31
Section 3
The Rational Numbers
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Chapter 5 Section 1 - Slide 32
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.
The following are examples of rational
numbers:
1 3
7
2
15
,
,  , 1 , 2, 0,
3 4
8
3
7
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Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 34
Reducing Fractions

In order to reduce a fraction to its lowest terms,
we divide both the numerator and denominator
by the greatest common divisor.

72
Example: Reduce
to its lowest terms.
81

Solution: 72  72  9  8
81 81  9 9
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Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 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.
12
An example of an improper fraction is
.
5
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 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 number.
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 38
Example

7
Convert 5
to an improper fraction.
10
7 (10  5  7) 50  7 57
5



10
10
10
10
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 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.
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 40
Example

236
Convert
to a mixed number.
7
33
7 236
21
26
21
5

5
The mixed number is 33 .
7
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Chapter 5 Section 1 - Slide 41
Terminating or Repeating Decimal
Numbers



Every rational number when expressed as a
decimal number will be either a terminating or
a repeating decimal number.
Examples of terminating decimal numbers
are 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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Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 43
Example: Multiplying Fractions
Evaluate the following.

a)
 3   1
 1 4    2 2 
2 7

3 16
b)
2 7
27


3 16 3 16
14
7


48 24
 3   1 7 5
 1 4    2 2   4  2
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35
3

4
8
8
Chapter 5 Section 1 - Slide 44
Example: Dividing Fractions
Evaluate the following.
a) 2 6
b) 5 4


3 7
8 5

2 6 2 7
  
3 7 3 6
2  7 14 7



3  6 18 9
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5 4 5 5
 

8 5
8 4
5  5 25


84
32
Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 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
Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 48
Example:




7
1
 .
12 10
Evaluate:
Find LCM of the denominators. LCM of 12 and 10 is 60.
Using the Fundamental Law of Rational Numbers, express each
fraction as an equivalent fraction with a denominator of 60.
Solution:
7
1  7 5  1 6

     
12 10  12 5   10 6 
35 6


60 60
29

60
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Chapter 5 Section 1 - Slide 49
Section 4
The Irrational Numbers and the
Real Number System
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Chapter 5 Section 1 - Slide 50
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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Chapter 5 Section 1 - Slide 51
Irrational Numbers


An irrational number is a real number whose
decimal representation is a nonterminating,
nonrepeating decimal number.
Examples of irrational numbers:
5.12639573...
6.1011011101111...
0.525225222...
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Chapter 5 Section 1 - Slide 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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Chapter 5 Section 1 - Slide 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.
For example,
16 = 4 since 4  4 = 16
49 = 7 since 7  7 = 49
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Chapter 5 Section 1 - Slide 54
Product Rule for Radicals
a  b  a  b,

a  0, b  0
Simplify:
a)
40
40  4 10  4  10  2  10  2 10
b)
125
125  25  5  25  5  5  5  5 5
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Chapter 5 Section 1 - Slide 55
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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Chapter 5 Section 1 - Slide 56
Multiplication of Irrational Numbers

Simplify:
6  54
6  54  6  54  324  18
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Chapter 5 Section 1 - Slide 57
Section 5
Real Numbers and their
Properties
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Chapter 5 Section 1 - Slide 58
Real Numbers


The set of real numbers is formed by the union
of the rational and irrational numbers.
The symbol for the set of real numbers is .
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Chapter 5 Section 1 - Slide 59
Relationships Among Sets
Real numbers
Rational numbers
Integers
Whole numbers
Natural numbers
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Irrational
numbers
Chapter 5 Section 1 - Slide 60
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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Chapter 5 Section 1 - Slide 61
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.
Chapter 5 Section 1 - Slide 62
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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Chapter 5 Section 1 - Slide 63
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.
Chapter 5 Section 1 - Slide 64
Example

(3 + 5) + 6 = 3 + (5 + 6) is true.

(4  6)  2 = 4  (6  2) is true.

Note: The associative property does not hold
true for subtraction or division.
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Chapter 5 Section 1 - Slide 65
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
Copyright © 2009 Pearson Education, Inc.
Chapter 5 Section 1 - Slide 66