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

2. 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

3. Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 3 Section 1 Number Theory

4. Chapter 5 Section 1 - Slide 4 Copyright © 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.

5. Chapter 5 Section 1 - Slide 5 Copyright © 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.

6. Chapter 5 Section 1 - Slide 6 Copyright © 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.

7. Chapter 5 Section 1 - Slide 7 Copyright © 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.

8. Chapter 5 Section 1 - Slide 8 Copyright © 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

9. Chapter 5 Section 1 - Slide 9 Copyright © 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

10. Chapter 5 Section 1 - Slide 10 Copyright © 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.

11. Chapter 5 Section 1 - Slide 11 Copyright © 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.

12. Chapter 5 Section 1 - Slide 12 Copyright © 2009 Pearson Education, Inc. Example of branching method Therefore, the prime factorization of 3190 = 2 5 11 29.

13. Chapter 5 Section 1 - Slide 13 Copyright © 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

14. Chapter 5 Section 1 - Slide 14 Copyright © 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

15. Chapter 5 Section 1 - Slide 15 Copyright © 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.

16. Chapter 5 Section 1 - Slide 16 Copyright © 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.

17. Chapter 5 Section 1 - Slide 17 Copyright © 2009 Pearson Education, Inc. Example (GCD) Find the GCD of 63 and 105. 63 = 3 2 7 105 = 3 5 7 Smallest exponent of each factor: 3 and 7 So, the GCD is 3 7 = 21.

18. Chapter 5 Section 1 - Slide 18 Copyright © 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.

19. Chapter 5 Section 1 - Slide 19 Copyright © 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.

20. Chapter 5 Section 1 - Slide 20 Copyright © 2009 Pearson Education, Inc. Example (LCM) Find the LCM of 63 and 105. 63 = 3 2 7 105 = 3 5 7 Greatest exponent of each factor: 3 2, 5 and 7 So, the LCM is 3 2 5 7 = 315.

21. Chapter 5 Section 1 - Slide 21 Copyright © 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 = 2 4 3 54 = 2 3 3 3 = 2 3 3 GCD = 2 3 = 6 LCM = 2 4 3 3 = 432

22. Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 22 Section 2 The Integers

23. Chapter 5 Section 1 - Slide 23 Copyright © 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,…}

24. Chapter 5 Section 1 - Slide 24 Copyright © 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.

25. Chapter 5 Section 1 - Slide 25 Copyright © 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 <  7 c) 0  4d) 6 8 0 >  4 6 < 8

26. Chapter 5 Section 1 - Slide 26 Copyright © 2009 Pearson Education, Inc. Subtraction of Integers a – b = a + (  b) Evaluate: a) –7 – 3 = –7 + (–3) = –10 b) –7 – (–3) = –7 + 3 = –4

27. Chapter 5 Section 1 - Slide 27 Copyright © 2009 Pearson Education, Inc. Properties Multiplication Property of Zero Division For any a, b, and c where b  0, means that c b = a.

28. Chapter 5 Section 1 - Slide 28 Copyright © 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.

29. Chapter 5 Section 1 - Slide 29 Copyright © 2009 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

30. Chapter 5 Section 1 - Slide 30 Copyright © 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.

31. Chapter 5 Section 1 - Slide 31 Copyright © 2009 Pearson Education, Inc. Example Evaluate: a) b) c) d) Solution: a) b) c) d)

32. Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 32 Section 3 The Rational Numbers

33. Chapter 5 Section 1 - Slide 33 Copyright © 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:

34. Chapter 5 Section 1 - Slide 34 Copyright © 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.

35. Chapter 5 Section 1 - Slide 35 Copyright © 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:

36. Chapter 5 Section 1 - Slide 36 Copyright © 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”.

37. Chapter 5 Section 1 - Slide 37 Copyright © 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.

38. Chapter 5 Section 1 - Slide 38 Copyright © 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.

39. Chapter 5 Section 1 - Slide 39 Copyright © 2009 Pearson Education, Inc. Example Convert to an improper fraction.

40. Chapter 5 Section 1 - Slide 40 Copyright © 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.

41. Chapter 5 Section 1 - Slide 41 Copyright © 2009 Pearson Education, Inc. Convert to a mixed number. The mixed number is Example

42. Chapter 5 Section 1 - Slide 42 Copyright © 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

43. Chapter 5 Section 1 - Slide 43 Copyright © 2009 Pearson Education, Inc. Division of Fractions Multiplication of Fractions

44. Chapter 5 Section 1 - Slide 44 Copyright © 2009 Pearson Education, Inc. Example: Multiplying Fractions Evaluate the following. a) b)

45. Chapter 5 Section 1 - Slide 45 Copyright © 2009 Pearson Education, Inc. Example: Dividing Fractions Evaluate the following. a) b)

46. Chapter 5 Section 1 - Slide 46 Copyright © 2009 Pearson Education, Inc. Addition and Subtraction of Fractions

47. Chapter 5 Section 1 - Slide 47 Copyright © 2009 Pearson Education, Inc. Example: Add or Subtract Fractions Add: Subtract:

48. Chapter 5 Section 1 - Slide 48 Copyright © 2009 Pearson Education, Inc. Fundamental Law of Rational Numbers If a, b, and c are integers, with b  0, c  0, then

49. Chapter 5 Section 1 - Slide 49 Copyright © 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:

50. Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 50 Section 4 The Irrational Numbers and the Real Number System

51. Chapter 5 Section 1 - Slide 51 Copyright © 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 (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

52. Chapter 5 Section 1 - Slide 52 Copyright © 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:

53. Chapter 5 Section 1 - Slide 53 Copyright © 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

54. Chapter 5 Section 1 - Slide 54 Copyright © 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,

55. Chapter 5 Section 1 - Slide 55 Copyright © 2009 Pearson Education, Inc. Product Rule for Radicals Simplify: a) b)

56. Chapter 5 Section 1 - Slide 56 Copyright © 2009 Pearson Education, Inc. Example: Adding or Subtracting Irrational Numbers Simplify:

57. Chapter 5 Section 1 - Slide 57 Copyright © 2009 Pearson Education, Inc. Multiplication of Irrational Numbers Simplify:

58. Copyright © 2009 Pearson Education, Inc. Chapter 5 Section 1 - Slide 58 Section 5 Real Numbers and their Properties

59. Chapter 5 Section 1 - Slide 59 Copyright © 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.

60. Chapter 5 Section 1 - Slide 60 Copyright © 2009 Pearson Education, Inc. Relationships Among Sets Irrational numbers Rational numbers Integers Whole numbers Natural numbers Real numbers

61. Chapter 5 Section 1 - Slide 61 Copyright © 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.

62. Chapter 5 Section 1 - Slide 62 Copyright © 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.

63. Chapter 5 Section 1 - Slide 63 Copyright © 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.

64. Chapter 5 Section 1 - Slide 64 Copyright © 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.

65. Chapter 5 Section 1 - Slide 65 Copyright © 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.

66. Chapter 5 Section 1 - Slide 66 Copyright © 2009 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 + 6 12 = 6r + 72