Copyright SEVENTH EDITION and EXPANDED SEVENTH EDITION · 2009-01-05 · Copyright © 2005 Pearson...

Chapter 5

Number Theory and the Real Number System

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,...}

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.

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 ÷ 4

The number formed by the last two digits of the number is divisible by 4.

846 since 8 + 4 + 6 = 18

The sum of the digits of the number is divisible by 3.

846The number is even.2

ExampleTestDivisible by

Divisibility Rules, continued

730The number ends in 0.10

846 since 8 + 4 + 6 = 18

The sum of the digits of the number is divisible by 9.

3848since 848 ÷ 8

The number formed by the last three digits of the number is divisible by 8.

846The number is divisible by both 2 and 3.

ExampleTestDivisible by

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.

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.

Example of branching method

Therefore, the prime factorization of

3190 = 2 • 5 • 11 • 29

319 10

11 29 5 2

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

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

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.

Example (GCD)

Find the GCD of 63 and 105.63 = 32 • 7

105 = 3 • 5 • 7 Smallest exponent of each factor:

3 and 7So, 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

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 7So, the GCD 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 • 354 = 2 • 3 • 3 • 3 = 2 • 33

GCD = 2 • 3 = 6

LCM = 24 • 33 = 432

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

DivisionFor any a, b, and c where b ≠ 0, means that c • b = a.

⋅ = ⋅ =0 0 0a a

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) = 35c) 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)

8 −= −

−−

9 = −−72

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.

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 2 9, , and .3 9 53

Reducing Fractions

In order to reduce a fraction, we divide both the numerator and denominator by the greatest common divisor.Example: Reduce to its lowest terms.

Solution:

÷72 72 9 881 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

Example

• + += = =

7 (10 5 7) 50 7 57510 10 10 10

Convert to an improper fraction.7510

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

337 236

533 .7

Terminating or Repeating Decimal Numbers

Every rational number when expressed as a decimal number will be either a terminating or repeatingdecimal number.Examples of terminating decimal numbers 0.7, 2.85, 0.000045Examples 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

•• = = ≠ ≠

•, 0, 0.a c a c ac b d

b d b d bd

÷ = • = ≠ ≠ ≠, 0, 0, c 0.a c a d ad b db d b c bc

Example: Multiplying Fractions

Evaluate the following.

⋅2 73 16

⎛ ⎞ ⎛ ⎞⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

3 11 24 2

⋅⋅ = = =

⋅2 7 2 7 14 73 16 3 16 48 24

⎛ ⎞ ⎛ ⎞⋅ = ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

3 1 7 51 24 2 4 2

35 348 8

Example: Dividing Fractions

Evaluate the following.a)

÷2 63 7

÷ = ⋅

⋅= = =

2 6 2 73 7 3 6

2 7 14 73 6 18 9

5 48 5

− −÷ = ⋅

− ⋅ −= =

5 4 5 58 5 8 4

5 5 258 4 32

Addition and Subtraction of Fractions

++ = ≠

−− = ≠

, c 0.

a b a bc c c

Example: Add or Subtract Fractions

Add: Subtract:+4 39 9 −

11 316 16

++ = =

4 3 4 3 79 9 9 9

−− = =

11 3 11 3 816 16 16 16

Fundamental Law of Rational Numbers

⋅= ⋅ = =

⋅.a a c a c ac

b b c b c bc

If a, b, and c are integers, with b ≠0, c ≠ 0, then

Example:

Evaluate:

Solution:

−7 9 .

⎛ ⎞ ⎛ ⎞− = ⋅ − ⋅⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

7 1 7 112 10 12 10

35 660 60

5 65 6

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.

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.

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.

Product Rule for Radicals

⋅ = ⋅ ≥ ≥, 0, 0.a b a b a bSimplify:

40= ⋅ = ⋅ =40 4 10 2 10 2 10

= ⋅ = ⋅ =125 5 25 5 5 5 5

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.

Example: Adding or Subtracting Irrational Numbers

Simplify: Simplify:+4 7 3 7

4 7 3 7

(4 3) 7

−8 5 125

= − ⋅

8 5 125

8 5 25 5

8 5 5 5

(8 5) 5

Multiplication of Irrational Numbers

Simplify: ⋅6 54

⋅ = ⋅ = =6 54 6 54 324 18

Division of Irrational Numbers

= ≥ ≥, 0, 0.a a a bbb

Example: Division

Divide:

Solution:

Divide:

Solution:

= = =16 16 4 2

= ⋅ = ⋅

144 144 7222

36 2 36 2

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.

Example: Rationalize

Rationalize the denominator of

Solution:

= = ⋅

8 8 212 3122 2 33 3 36

Real Numbers and their Properties

Real Numbers

The set of real numbers is formed by the union of the rational and irrational numbers.

Relationships Among Sets

Irrational numbers

Rational numbers

Integers

Whole numbers

Natural numbers

Real numbers

Properties of the Real Number System

ClosureIf 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.

Multiplicationa.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 commutative property does not hold true for subtraction or division.

Distributive Property

Distributive property of multiplication over additiona.(b + c) = a.b + a.cfor any real numbers a, b, and c.

Example: 6(r + 12) = 6.r + 6.12= 6r + 72

Rules of Exponents and Scientific Notation

Exponents

When a number is written with an exponent, there are two parts to the expression: baseexponent

The exponent tells how many times the base should be multiplied together.

= ⋅ ⋅ ⋅ ⋅54 4 4 4 4 4

Product Rule

Simplify: 34 • 39

34 • 39 = 34 + 9 = 313

Simplify: 64 • 65

64 • 65 = 64 + 5 = 69

+= m n m na a a

Quotient Rule

Simplify: Simplify:

−= ≠, 0m

a a aa

−= =5

5 2 32

7 7 77

−= =15

15 8 78

9 9 99

= ≠0 1, 0 a aZero Exponent

Simplify: (3y)0

(3y)0 = 1

Simplify: 3y0

3y0 = 3 (y0)= 3(1) = 3

− = ≠1 , 0 mma a

Negative Exponent

Simplify: 6−4

− = =44

1 1612966

Power Rule

Simplify: (32)3

(32)3 = 32•3 = 36

Simplify: (23)5

(23)5 = 23•5 = 215

=( )m n m na a

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.

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.

Scientific Notation continued

To Write a Number in Scientific Notation:

1. 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.

2. 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.

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.)

To Write a Number in Scientific Notation: continued

Example

Write each number in scientific notation.a) 1,265,000,000.

1.265 × 109

b) 0.0000000004324.32 × 10−10

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.

Example

Write each number in decimal notation.a) 4.67 × 105

467,000

b) 1.45 × 10-7

0.000000145

Arithmetic and Geometric Sequences

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.

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.

General Term of an Arithmetic Sequence

= + −1 ( 1)na a n d

Find the 5th term of the arithmetic sequence whose first term is 4 and whose common difference is −8.

a5 = 4 + (5 − 1)(−8)= 4 + (4)(−8)= 4 + (−32)= −28

= + −1 ( 1)na a n d

Sum of the First n Terms in an Arithmetic Sequence

Find the sum of the first 50 terms in the arithmetic sequence: 2, 4, 6, 8,…100

+= 1( )

nn a a

50(2 100)2

50(102)2

n a as

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.

General Term of a Geometric Sequence−= 1

na a r

Find the 6th term for the geometric sequence with the first term = 3 and the common ratio = 4.

(3)4(3)10243072

Sum of the First n Terms in an Geometric Sequence

−= ≠

−1(1 )

Example

−−

=−−

−= =

3(1 2 )1 2

3(1 16)1

3( 15)1

45 451

Find the sum of the first 4 terms of the geometric sequence for r = 2 and the firstterm = 3.

Fibonacci Sequence

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 Hindu-Arabic number system into Europe.

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.

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.

Divine Proportions

Golden Number :

The value obtained when the ratio of any term to the term preceding it in the Fibonacci sequence.

+= = ≈

5 1 1.6182

AB ACAC CB

Golden or Divine Proportion

+ += = =

length a b awidth a b

Golden Rectangle

Copyright SEVENTH EDITION and EXPANDED SEVENTH EDITION · 2009-01-05 · Copyright © 2005 Pearson...

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