Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 1 Chapter 1 Number Theory and...

Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 1

Chapter 1

Number Theory and the Real Number System


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


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


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.


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.


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.


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


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


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.


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.


Example of branching method

Therefore, the prime factorization of

3190 = 2 • 5 • 11 • 29.


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


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


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.


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.


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.


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.


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.


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.


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


Section 2

The Integers


Whole Numbers

The set of whole numbers contains the set of natural numbers and the number 0.

Whole numbers = {0,1,2,3,4,…}


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.


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


Subtraction of Integers

a – b = a + (b)

Evaluate:

a) –7 – 3 = –7 + (–3) = –10

b) –7 – (–3) = –7 + 3 = –4


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


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.


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


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.


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


Section 3

The Rational Numbers


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


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.


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


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”.


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


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.


Example

Convert to an improper fraction.

5

7

10

(10 5 7)

10

50 7

10

57

10

5

7

10


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.


Convert to a mixed number.

The mixed number is

Example

7 236

21

33

26

21

5

236

7

33

5

7.


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.


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


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


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


Addition and Subtraction of Fractions

a

c

b

c

a b

c, c 0;

a

c

b

c

a b

c, c 0


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


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.


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


Section 4

The Irrational Numbers and the Real Number System


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


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...


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


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


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


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


Multiplication of Irrational Numbers

Simplify:

6 54

6 54 654 324 18


Section 5

Real Numbers and their Properties


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 .


Relationships Among Sets

Irrational numbers

Rational numbers

Integers

Whole numbersNatural numbers

Real numbers


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.


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.


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.


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.


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.


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

Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 1 Chapter 1 Number Theory and...

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