College Algebra Fifth Edition James Stewart  Lothar Redlin  Saleem Watson

1. College Algebra Fifth Edition James StewartLothar RedlinSaleem Watson

2. Prerequisites P

3. Real Numbersand Their Properties P.2

4. Types of Real Numbers

5. Introduction • Let’s review the types of numbers that make up the real number system.

6. Natural Numbers • We start with the natural numbers: 1, 2, 3, 4, …

7. Integers • The integers consist of the natural numbers together with their negatives and 0: . . . , –3, –2, –1, 0, 1, 2, 3, 4, . . .

8. Rational Numbers • We construct the rational numbersby taking ratios of integers. • Thus, any rational number r can be expressed as: where m and n are integers and n≠ 0.

9. Rational Numbers • Examples are: • Recall that division by 0 is always ruled out. • So, expressions like 3/0 and 0/0 are undefined.

10. Irrational Numbers • There are also real numbers, such as , that can’t be expressed as a ratio of integers. • Hence, they are called irrational numbers. • It can be shown, with varying degrees of difficulty, that these numbers are also irrational:

11. Set of All Real Numbers • The set of all real numbers is usually denoted by: • The symbol R

12. Real Numbers • When we use the word ‘number’ without qualification, we will mean: • “Real number”

13. Real Numbers • Figure 1 is a diagram of the types of real numbers that we work with in this book.

14. Repeating Decimals • Every real number has a decimal representation. • If the number is rational, then its corresponding decimal is repeating.

15. Repeating Decimals • For example, • The bar indicates that the sequence of digits repeats forever.

16. Non-Repeating Decimals • If the number is irrational, the decimal representation is non-repeating:

17. Approximation • If we stop the decimal expansion of any number at a certain place, we get an approximation to the number. • For instance, we can write π≈ 3.14159265 where the symbol ≈ is read “is approximately equal to.” • The more decimal places we retain, the better our approximation.

18. E.g. 1—Classifying Real Numbers • Determine whether • 999 d) • –6/5 e) • –6/3 • is a natural number, an integer, a rational number, or an irrational number.

19. E.g. 1—Classifying Real Numbers • 999 is a positive whole number, so it is a natural number. • –6/5 is a ratio of two integers, so it is a rational number. • –6/3 equals –2, so it is an integer.

20. E.g. 1—Classifying Real Numbers • d) equals 5, so it is a natural number. • e) is a nonrepeating decimal(approximately 1.7320508075689),so it is an irrational number.

21. Operations on Real Numbers

22. Operations on Real Numbers • Real numbers can be combined using the familiar operations: • Addition • Subtraction • Multiplication • Division

23. Order of Operations on Real Numbers • When evaluating arithmetic expressions that contain several of these operations, we use the following convention to determine the order in which operations are performed:

24. Order of Operations on Real Numbers • Perform operations inside parenthesis first, beginning with the innermost pair. • In dividing two expressions, the numerator and denominator of the quotient are treated as if they are within parentheses. • Perform all multiplication and division • Working from left to right • Perform all addition and subtraction • Working from left to right

25. E.g. 2—Evaluating an Arithmetic Expression • Find the value of the expression

26. E.g. 2—Evaluating an Arithmetic Expression • First we evaluate the numerator and denominator of the quotient. • Recall, these are treated as if they are inside parentheses.

27. Properties of Real Numbers

28. Introduction • We all know that: 2 + 3 = 3 + 2 5 + 7 = 7 + 5 513 + 87 = 87 + 513 • and so on. • In algebra, we express all these (infinitely many) facts by writing:a +b =b +a where a and b stand for any two numbers.

29. Commutative Property • In other words, “a +b =b +a” is a concise way of saying that:“when we add two numbers, the order of addition doesn’t matter.” • This is called the Commutative Property for Addition.

30. Properties of Real Numbers • From our experience with numbers, we know that these properties are also valid.

31. Distributive Property • The Distributive Property applies: • Whenever we multiply a number by a sum.

32. Distributive Property • Figure 2 explains why this property works for the case in which all the numbers are positive integers. • However, it is true for any real numbers a, b, and c.

33. Example (a) E.g. 3—Using the Properties • 2 + (3 + 7) = 2 + (7 + 3) (Commutative Property of Addition)= (2 + 7) + 3 (Associative Property of Addition)

34. Example (b) E.g. 3—Using the Properties • 2(x + 3) = 2 .x + 2 . 3 (Distributive Property)= 2x + 6 (Simplify)

35. Example (c) E.g. 3—Using the Properties • (a + b)(x + y) • = (a + b)x + (a + b)y(Distributive Property) • = (ax + bx) + (ay + by)(Distributive Property) • = ax + bx + ay + by(Associative Property of Addition) • In the last step, we removed the parentheses. • According to the Associative Property, the order of addition doesn’t matter.

36. Addition and Subtraction

37. Additive Identity • The number 0 is special for addition. • It is called the additive identity. • This is because a + 0 = a for a real number a.

38. Subtraction • Every real number a has a negative, –a, that satisfies a + (–a) = 0. • Subtraction undoes addition. • To subtract a number from another, we simply add the negative of that number. • By definition, a – b = a + (–b)

39. Note on “–a” • Don’t assume that –a is a negative number. • Whether –a is a negative or positive number depends on the value of a. • For example, if a = 5, then –a = –5. • A negative number • However, if a = –5, then –a = –(–5) = 5. • A positive number

40. Properties of Negatives • To combine real numbers involving negatives, we use these properties.

41. Property 5 & 6 of Negatives • Property 5 is often used with more than two terms: • –(a + b + c) = –a – b – c • Property 6 states the intuitive fact that: • a –b and b –a are negatives of each other.

42. E.g. 4—Using Properties of Negatives • Let x, y, and z be real numbers. • –(3 + 2) = –3 – 2 (Property 5: –(a + b) = –a – b) • –(x + 2) = –x – 2 (Property 5: –(a + b) = –a – b) • –(x + y – z) = –x – y – (–z) (Property 5) • = –x – y + z (Property 2: –(– a) = a)

43. Multiplication and Division

44. Multiplicative Identity • The number 1 is special for multiplication. • It is called the multiplicative identity. • This is because a .1 = a for any real number a.

45. Division • Every nonzero real number a has an inverse, 1/a, that satisfies a. (1/a). • Divisionundoes multiplication. • To divide by a number, we multiply by the inverse of that number. • If b ≠ 0, then, by definition, a÷ b = a. 1/b • We write a. (1/b) as simply a/b.

46. Division • We refer to a/b as: The quotient of a and b or as the fraction a over b. • a is the numerator. • b is the denominator (or divisor).

47. Division • To combine real numbers using division, we use these properties.

48. Property 3 & 4 • When adding fractions with different denominators, we don’t usually use Property 4. • Instead, we rewrite the fractions so that they have the smallest common denominator (often smaller than the product of the denominators). • Then, we use Property 3.

49. LCD • This denominator is the Least Common Denominator (LCD). • It is described in the next example.

50. E.g. 5—Using LCD to Add Fractions • Evaluate: • Factoring each denominator into prime factors gives: 36 = 22. 32 120 = 23. 3 . 5