SEVENTH EDITION and EXPANDED SEVENTH EDITION

1. SEVENTH EDITION and EXPANDED SEVENTH EDITION

2. Chapter 5 Number Theory and the Real Number System

3. 5.1 Number Theory

4. Number Theory • The study of numbers and their properties. • The numbers we use to count are called the Natural Numbers or Counting Numbers.

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.

6. 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. 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. Rules of Divisibility Divisible by Test Example 2 The number is even. 846 3 The sum of the digits of the number is divisible by 3. 846 since 8 + 4 + 6 = 18 4 The number formed by the last two digits of the number is divisible by 4. 844 since 44  4 5 The number ends in 0 or 5. 285

9. Divisible by Test Example 6 The number is divisible by both 2 and 3. 846 8 The number formed by the last three digits of the number is divisible by 8. 3848 since 848  8 9 The sum of the digits of the number is divisible by 9. 846 since 8 + 4 + 6 = 18 10 The number ends in 0. 730 Divisibility Rules, continued

10. 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.

11. 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.

12. Example of branching method Therefore, the prime factorization of 3190 = 2 • 5 • 11 • 29

13. Division Method 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.

14. 3 663 13 221 17 Example of 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

15. 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. 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.

17. 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

18. 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. 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.

20. 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 GCD is 32 • 5 • 7 = 315

21. 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

22. 5.2 The Integers

23. 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. 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. 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 4 d) 6 8 0 > 4 6 < 8

26. Subtraction of Integers a – b = a + (b) Evaluate: a) –7 – 3 = –7 + (–3) = –10 b) –7 – (–3) = –7 + 3 = –4

27. Multiplication Property of Zero Division For any a, b, and c where b 0, means that c• b = a. Properties

28. Rules for Multiplication • The product of two numbers with likesigns (positive  positive or negative  negative) is a positivenumber. • The product of two numbers with unlikesigns (positive  negative or negative  positive) is a negative number.

29. 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

30. Rules for Division • The quotient of two numbers with likesigns (positive  positive or negative  negative) is a positivenumber. • The quotient of two numbers with unlikesigns (positive  negative or negative  positive) is a negative number.

31. Example • Evaluate: • a) b) • c) d)

32. 5.3 The Rational Numbers

33. 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.

34. 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. 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:

36. 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. 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.

38. 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

39. Example • Convert to an improper fraction.

40. 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. Example • Convert to a mixed number. • The mixed number is

42. 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, 0.000045 • Examples of repeating decimal numbers 0.44444… which may be written

43. Multiplication of Fractions • Division of Fractions

44. Evaluate the following. a) b) Example: Multiplying Fractions

45. Evaluate the following. a) b) Example: Dividing Fractions

46. Addition and Subtraction of Fractions

47. Add: Subtract: Example: Add or Subtract Fractions

48. Fundamental Law of Rational Numbers • If a, b, and c are integers, with b 0, c  0, then

49. Example: • Evaluate: • Solution:

50. 5.4 The Irrational Numbers and the Real Number System