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Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 1
Chapter 1
Number Theory and the Real Number System
Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 2
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 3
Section 1
Number Theory
Chapter 5 Section 1 - Slide 4Copyright © 2009 Pearson Education, Inc.
Number Theory
The study of numbers and their properties. The numbers we use to count are called natural
numbers, N , or counting numbers.
={1,2,3,4,5,...}N
Chapter 5 Section 1 - Slide 5Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 6Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 7Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 8Copyright © 2009 Pearson Education, Inc.
Rules of Divisibility
285The number ends in 0 or 5.
5
844
since 44
is divisible by 4
The number formed by the last two digits of the number is divisible by 4.
4
846
since 8 + 4 + 6 = 18 and 18 is divisible by 3
The sum of the digits of the number is divisible by 3.
3
846The number is even.2
ExampleTestDivisible by
Chapter 5 Section 1 - Slide 9Copyright © 2009 Pearson Education, Inc.
Divisibility Rules (continued)
730The number ends in 0.10
846
since 8 + 4 + 6 = 18 and 18 is divisible by 9
The sum of the digits of the number is divisible by 9.
9
3848
since 848 is divisible by 8
The number formed by the last three digits of the number is divisible by 8.
8
846The number is divisible by both 2 and 3.
6
ExampleTestDivisible by
Chapter 5 Section 1 - Slide 10Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 11Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 12Copyright © 2009 Pearson Education, Inc.
Example of branching method
Therefore, the prime factorization of
3190 = 2 • 5 • 11 • 29.
Chapter 5 Section 1 - Slide 13Copyright © 2009 Pearson Education, Inc.
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.
Division Method
Chapter 5 Section 1 - Slide 14Copyright © 2009 Pearson Education, Inc.
Write the prime factorization of 663.
The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is 3 •13 •17
Example of division method
13
3
17
221
663
Chapter 5 Section 1 - Slide 15Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 16Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 17Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 18Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 19Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 20Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 21Copyright © 2009 Pearson Education, Inc.
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
Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 22
Section 2
The Integers
Chapter 5 Section 1 - Slide 23Copyright © 2009 Pearson Education, Inc.
Whole Numbers
The set of whole numbers contains the set of natural numbers and the number 0.
Whole numbers = {0,1,2,3,4,…}
Chapter 5 Section 1 - Slide 24Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 25Copyright © 2009 Pearson Education, Inc.
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 < 7c) 0 4 d) 6 8
0 > 4 6 < 8
Chapter 5 Section 1 - Slide 26Copyright © 2009 Pearson Education, Inc.
Subtraction of Integers
a – b = a + (b)
Evaluate:
a) –7 – 3 = –7 + (–3) = –10
b) –7 – (–3) = –7 + 3 = –4
Chapter 5 Section 1 - Slide 27Copyright © 2009 Pearson Education, Inc.
Properties
Multiplication Property of Zero
Division
For any a, b, and c where b 0, means that c • b = a.
0 0 0a a
a
b= c
Chapter 5 Section 1 - Slide 28Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 29Copyright © 2009 Pearson Education, Inc.
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
Chapter 5 Section 1 - Slide 30Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 31Copyright © 2009 Pearson Education, Inc.
Example
Evaluate:
a) b)
c) d)
Solution:
a) b)
c) d)
72
9
72
9
72
8
72
8
72
98
72
98
72
89
72
89
Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 32
Section 3
The Rational Numbers
Chapter 5 Section 1 - Slide 33Copyright © 2009 Pearson Education, Inc.
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,
3
4,
7
8, 1
2
3, 2, 0,
15
7
Chapter 5 Section 1 - Slide 34Copyright © 2009 Pearson Education, Inc.
Fractions
Fractions are numbers such as:
The numerator is the number above the fraction line.
The denominator is the number below the fraction line.
1
3,
2
9, and
9
53.
Chapter 5 Section 1 - Slide 35Copyright © 2009 Pearson Education, Inc.
Reducing Fractions
In order to reduce a fraction to its lowest terms, we divide both the numerator and denominator by the greatest common divisor.
Example: Reduce to its lowest terms.
Solution:
72
81
72 72 9 8
81 81 9 9
Chapter 5 Section 1 - Slide 36Copyright © 2009 Pearson Education, Inc.
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”.
Chapter 5 Section 1 - Slide 37Copyright © 2009 Pearson Education, Inc.
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
Chapter 5 Section 1 - Slide 38Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 39Copyright © 2009 Pearson Education, Inc.
Example
Convert to an improper fraction.
5
7
10
(10 5 7)
10
50 7
10
57
10
5
7
10
Chapter 5 Section 1 - Slide 40Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 41Copyright © 2009 Pearson Education, Inc.
Convert to a mixed number.
The mixed number is
Example
7 236
21
33
26
21
5
236
7
33
5
7.
Chapter 5 Section 1 - Slide 42Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 43Copyright © 2009 Pearson Education, Inc.
Division of Fractions
Multiplication of Fractions
a
b
c
d
a c
b d
ac
bd, b 0, d 0
a
b
c
d
a
b
d
c
ad
bc, b 0, d 0, c 0
Chapter 5 Section 1 - Slide 44Copyright © 2009 Pearson Education, Inc.
Example: Multiplying Fractions
Evaluate the following.
a)
b)
2
3
7
16
2
3
7
16
27316
14
48
7
24
1
3
4
2
1
2
13
4
2
1
2
7
45
2
35
84
3
8
Chapter 5 Section 1 - Slide 45Copyright © 2009 Pearson Education, Inc.
Example: Dividing Fractions
Evaluate the following.
a)
b)
2
3
6
7
2
3
6
7
2
37
6
2736
14
18
7
9
5
8
4
5
5
8
4
5
5
85
4
5584
25
32
Chapter 5 Section 1 - Slide 46Copyright © 2009 Pearson Education, Inc.
Addition and Subtraction of Fractions
a
c
b
c
a b
c, c 0;
a
c
b
c
a b
c, c 0
Chapter 5 Section 1 - Slide 47Copyright © 2009 Pearson Education, Inc.
Example: Add or Subtract Fractions
Add:
Subtract:
4
9
3
9
4
9
3
9
4 3
9
7
9
11
16
3
16
11
16
3
16
11 3
16
8
16
1
2
Chapter 5 Section 1 - Slide 48Copyright © 2009 Pearson Education, Inc.
Fundamental Law of Rational Numbers
If a, b, and c are integers, with b 0, c 0, then
a
b
a
bc
c
acbc
ac
bc.
Chapter 5 Section 1 - Slide 49Copyright © 2009 Pearson Education, Inc.
Example:
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
12
1
10.
7
12
1
10
7
125
5
1
106
6
35
60
6
60
29
60
Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 50
Section 4
The Irrational Numbers and the Real Number System
Chapter 5 Section 1 - Slide 51Copyright © 2009 Pearson Education, Inc.
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
Chapter 5 Section 1 - Slide 52Copyright © 2009 Pearson Education, Inc.
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...
Chapter 5 Section 1 - Slide 53Copyright © 2009 Pearson Education, Inc.
are all irrational numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.
Radicals
2, 17, 53
Chapter 5 Section 1 - Slide 54Copyright © 2009 Pearson Education, Inc.
Principal Square Root
The principal (or positive) square root of a number n, written is the positive number that when multiplied by itself, gives n.
For example,
16 = 4 since 44 =16
49 = 7 since 77 = 49
n
Chapter 5 Section 1 - Slide 55Copyright © 2009 Pearson Education, Inc.
Product Rule for Radicals
Simplify:a)
b)
ab a b, a 0, b 0
40 410 4 10 2 10 2 10
125 255 25 5 5 5 5 5
40
125
Chapter 5 Section 1 - Slide 56Copyright © 2009 Pearson Education, Inc.
Example: Adding or Subtracting Irrational Numbers
Simplify: Simplify: 4 7 3 7
4 7 3 7
(4 3) 7
7 7
8 5 125
8 5 125
8 5 25 5
8 5 5 5
(8 5) 5
3 5
Chapter 5 Section 1 - Slide 57Copyright © 2009 Pearson Education, Inc.
Multiplication of Irrational Numbers
Simplify:
6 54
6 54 654 324 18
Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 58
Section 5
Real Numbers and their Properties
Chapter 5 Section 1 - Slide 59Copyright © 2009 Pearson Education, Inc.
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 .
Chapter 5 Section 1 - Slide 60Copyright © 2009 Pearson Education, Inc.
Relationships Among Sets
Irrational numbers
Rational numbers
Integers
Whole numbersNatural numbers
Real numbers
Chapter 5 Section 1 - Slide 61Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 62Copyright © 2009 Pearson Education, Inc.
Commutative Property
Addition
a + b = b + a
for any real numbers a and b.
Multiplication
a • b = b • a
for any real numbers a and b.
Chapter 5 Section 1 - Slide 63Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 64Copyright © 2009 Pearson Education, Inc.
Associative Property
Addition
(a + b) + c = a + (b + c),
for any real numbers a, b, and c.
Multiplication
(a • b) • c = a • (b • c),
for any real numbers a, b, and c.
Chapter 5 Section 1 - Slide 65Copyright © 2009 Pearson Education, Inc.
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.
Chapter 5 Section 1 - Slide 66Copyright © 2009 Pearson Education, Inc.
Distributive Property
Distributive property of multiplication over addition
a • (b + c) = a • b + a • cfor any real numbers a, b, and c.
Example: 6 • (r + 12) = 6 • r + 6 • 12
= 6r + 72