Slide 5-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION

Copyright © 2005 Pearson Education, Inc. Chapter 5 Number Theory and the Real Number System

Copyright © 2005 Pearson Education, Inc. 5.1 Number Theory

Slide 5-4 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.

Slide 5-5 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-6 Copyright © 2005 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.

Slide 5-7 Copyright © 2005 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.

Slide 5-8 Copyright © 2005 Pearson Education, Inc. Rules of Divisibility 285The number ends in 0 or since 44  4 The number formed by the last two digits of the number is divisible by since = 18 The sum of the digits of the number is divisible by The number is even.2 ExampleTestDivisible by

Slide 5-9 Copyright © 2005 Pearson Education, Inc. Divisibility Rules, continued 730The number ends in since = 18 The sum of the digits of the number is divisible by since 848  8 The number formed by the last three digits of the number is divisible by The number is divisible by both 2 and 3. 6 ExampleTestDivisible by

Slide 5-10 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-11 Copyright © 2005 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, then continue factoring each number until all numbers are prime.

Slide 5-12 Copyright © 2005 Pearson Education, Inc. Example of branching method Therefore, the prime factorization of 3190 =

Slide 5-13 Copyright © 2005 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

Slide 5-14 Copyright © 2005 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 Example of division method

Slide 5-15 Copyright © 2005 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.

Slide 5-16 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-17 Copyright © 2005 Pearson Education, Inc. Example (GCD) Find the GCD of 63 and = = Smallest exponent of each factor: 3 and 7 So, the GCD is 3 7 = 21

Slide 5-18 Copyright © 2005 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.

Slide 5-19 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-20 Copyright © 2005 Pearson Education, Inc. Example (LCM) Find the LCM of 63 and = = Greatest exponent of each factor: 3 2, 5 and 7 So, the GCD is = 315

Slide 5-21 Copyright © 2005 Pearson Education, Inc. Example of GCD and LCM Find the GCD and LCM of 48 and 54. Prime factorizations of each:  48 = =  54 = =  GCD = 2 3 = 6  LCM = = 432

Copyright © 2005 Pearson Education, Inc. 5.2 The Integers

Slide 5-23 Copyright © 2005 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,…}

Slide 5-24 Copyright © 2005 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.

Slide 5-25 Copyright © 2005 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 <  7 c) 0  4d) >  4 6 < 8

Slide 5-26 Copyright © 2005 Pearson Education, Inc. Subtraction of Integers a – b = a + (  b) Evaluate: a) –7 – 3 = –7 + (–3) = –10 b) –7 – (–3) = –7 + 3 = –4

Slide 5-27 Copyright © 2005 Pearson Education, Inc. Properties Multiplication Property of Zero Division For any a, b, and c where b  0, means that c b = a.

Slide 5-28 Copyright © 2005 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.

Slide 5-29 Copyright © 2005 Pearson Education, Inc. Examples Evaluate: a) (3)(  4)b) (  7)(  5) c) 8 7d) (  5)(8) Solution: a) (3)(  4) =  12b) (  7)(  5) = 35 c) 8 7 = 56d) (  5)(8) =  40

Slide 5-30 Copyright © 2005 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.

Slide 5-31 Copyright © 2005 Pearson Education, Inc. Example Evaluate: a) b) c) d)

Copyright © 2005 Pearson Education, Inc. 5.3 The Rational Numbers

Slide 5-33 Copyright © 2005 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.

Slide 5-34 Copyright © 2005 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.

Slide 5-35 Copyright © 2005 Pearson Education, Inc. 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:

Slide 5-36 Copyright © 2005 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”.

Slide 5-37 Copyright © 2005 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.

Slide 5-38 Copyright © 2005 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

Slide 5-39 Copyright © 2005 Pearson Education, Inc. Example Convert to an improper fraction.

Slide 5-40 Copyright © 2005 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.

Slide 5-41 Copyright © 2005 Pearson Education, Inc. Convert to a mixed number. The mixed number is Example

Slide 5-42 Copyright © 2005 Pearson Education, Inc. 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, Examples of repeating decimal numbers … which may be written

Slide 5-43 Copyright © 2005 Pearson Education, Inc. Division of Fractions Multiplication of Fractions

Slide 5-44 Copyright © 2005 Pearson Education, Inc. Example: Multiplying Fractions Evaluate the following. a) b)

Slide 5-45 Copyright © 2005 Pearson Education, Inc. Example: Dividing Fractions Evaluate the following. a) b)

Slide 5-46 Copyright © 2005 Pearson Education, Inc. Addition and Subtraction of Fractions

Slide 5-47 Copyright © 2005 Pearson Education, Inc. Example: Add or Subtract Fractions Add: Subtract:

Slide 5-48 Copyright © 2005 Pearson Education, Inc. Fundamental Law of Rational Numbers If a, b, and c are integers, with b  0, c  0, then

Slide 5-49 Copyright © 2005 Pearson Education, Inc. Example: Evaluate: Solution:

Copyright © 2005 Pearson Education, Inc. 5.4 The Irrational Numbers and the Real Number System

Slide 5-51 Copyright © 2005 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 (a 2 ) added to the square of the length of the other side (b 2 ) equals the square of the length of the hypotenuse (c 2 ). a 2 + b 2 = c 2

Slide 5-52 Copyright © 2005 Pearson Education, Inc. Irrational Numbers An irrational number is a real number whose decimal representation is a nonterminating, nonrepeating decimal number.

Slide 5-53 Copyright © 2005 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

Slide 5-54 Copyright © 2005 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.

Slide 5-55 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-56 Copyright © 2005 Pearson Education, Inc. Product Rule for Radicals Simplify:  a)  b)

Slide 5-57 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-58 Copyright © 2005 Pearson Education, Inc. Example: Adding or Subtracting Irrational Numbers Simplify:

Slide 5-59 Copyright © 2005 Pearson Education, Inc. Multiplication of Irrational Numbers Simplify:

Slide 5-60 Copyright © 2005 Pearson Education, Inc. Division of Irrational Numbers

Slide 5-61 Copyright © 2005 Pearson Education, Inc. Example: Division Divide: Solution: Divide: Solution:

Slide 5-62 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-63 Copyright © 2005 Pearson Education, Inc. Example: Rationalize Rationalize the denominator of Solution:

Copyright © 2005 Pearson Education, Inc. 5.5 Real Numbers and their Properties

Slide 5-65 Copyright © 2005 Pearson Education, Inc. Real Numbers The set of real numbers is formed by the union of the rational and irrational numbers.

Slide 5-66 Copyright © 2005 Pearson Education, Inc. Relationships Among Sets Irrational numbers Rational numbers Integers Whole numbers Natural numbers Real numbers

Slide 5-67 Copyright © 2005 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.

Slide 5-68 Copyright © 2005 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.

Slide 5-69 Copyright © 2005 Pearson Education, Inc. Example = is a true statement. 5  9 = 9  5 is a true statement. Note: The commutative property does not hold true for subtraction or division.

Slide 5-70 Copyright © 2005 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.

Slide 5-71 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-72 Copyright © 2005 Pearson Education, Inc. 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 = 6r + 72

Copyright © 2005 Pearson Education, Inc. 5.6 Rules of Exponents and Scientific Notation

Slide 5-74 Copyright © 2005 Pearson Education, Inc. Exponents When a number is written with an exponent, there are two parts to the expression: base exponent The exponent tells how many times the base should be multiplied together.

Slide 5-75 Copyright © 2005 Pearson Education, Inc. Product Rule Simplify: = = 3 13 Simplify: = = 6 9

Slide 5-76 Copyright © 2005 Pearson Education, Inc. Quotient Rule Simplify: Simplify:

Slide 5-77 Copyright © 2005 Pearson Education, Inc. Zero Exponent Simplify: (3y) 0 (3y) 0 = 1 Simplify: 3y 0 3y 0 = 3 (y 0 ) = 3(1) = 3

Slide 5-78 Copyright © 2005 Pearson Education, Inc. Negative Exponent Simplify: 6  4

Slide 5-79 Copyright © 2005 Pearson Education, Inc. Power Rule Simplify: (3 2 ) 3 (3 2 ) 3 = 3 23 = 3 6 Simplify: (2 3 ) 5 (2 3 ) 5 = 2 35 = 2 15

Slide 5-80 Copyright © 2005 Pearson Education, Inc. Scientific Notation Many scientific problems deal with very large or very small numbers. 93,000,000,000,000 is a very large number is a very small number.

Slide 5-81 Copyright © 2005 Pearson Education, Inc. Scientific notation is a shorthand method used to write these numbers. 9.3 x and 4.82 x are two examples of the scientific numbers. Scientific Notation continued

Slide 5-82 Copyright © 2005 Pearson Education, Inc. 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 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.

Slide 5-83 Copyright © 2005 Pearson Education, Inc. 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

Slide 5-84 Copyright © 2005 Pearson Education, Inc. Example Write each number in scientific notation. a)1,265,000,  10 9 b)  10  10

Slide 5-85 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-86 Copyright © 2005 Pearson Education, Inc. Example Write each number in decimal notation. a)4.67  ,000 b)1.45 

Copyright © 2005 Pearson Education, Inc. 5.7 Arithmetic and Geometric Sequences

Slide 5-88 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.

Slide 5-89 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-90 Copyright © 2005 Pearson Education, Inc. General Term of an Arithmetic Sequence Find the 5 th term of the arithmetic sequence whose first term is 4 and whose common difference is  8. a 5 = 4 + (5  1)(  8) = 4 + (4)(  8) = 4 + (  32) =  28

Slide 5-91 Copyright © 2005 Pearson Education, Inc. 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

Slide 5-92 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-93 Copyright © 2005 Pearson Education, Inc. General Term of a Geometric Sequence Find the 6th term for the geometric sequence with the first term = 3 and the common ratio = 4.

Slide 5-94 Copyright © 2005 Pearson Education, Inc. Sum of the First n Terms in an Geometric Sequence

Slide 5-95 Copyright © 2005 Pearson Education, Inc. Example Find the sum of the first 4 terms of the geometric sequence for r = 2 and the first term = 3.

Copyright © 2005 Pearson Education, Inc. 5.8 Fibonacci Sequence

Slide 5-97 Copyright © 2005 Pearson Education, Inc. 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.

Slide 5-98 Copyright © 2005 Pearson Education, Inc. 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 2 nd and 3 rd terms give us the 4 th term (3) and so on.

Slide 5-99 Copyright © 2005 Pearson Education, Inc. In Nature In the middle of the 19 th 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.

Slide Copyright © 2005 Pearson Education, Inc. Divine Proportions Golden Number : The value obtained when the ratio of any term to the term preceding it in the Fibonacci sequence.

Slide Copyright © 2005 Pearson Education, Inc. Golden or Divine Proportion

Slide Copyright © 2005 Pearson Education, Inc. Golden Rectangle