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College Algebra Fifth Edition

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

2 P Prerequisites

3 Real Numbers and Their Properties

4 Types of Real Numbers

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

6 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 We construct the rational numbers by taking ratios of integers.
Thus, any rational number r can be expressed as: where m and n are integers and n ≠ 0.

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

10 Hence, they are called 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 The set of all real numbers is usually denoted by:
The symbol R

12 When we use the word ‘number’ without qualification, we will mean:
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 Every real number has a decimal representation.
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 ≈ 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 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 ), 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

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 We all know that: 2 + 3 = 3 + 2 5 + 7 = 7 + 5 513 + 87 = 87 + 513
Introduction We all know that: = = = 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 E.g. 3—Using the Properties
Example (a) 2 + (3 + 7) = 2 + (7 + 3) (Commutative Property of Addition) = (2 + 7) (Associative Property of Addition)

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

35 E.g. 3—Using the Properties
Example (c) (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 The number 0 is special for addition.
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 Every real number a has a negative, –a, that satisfies a + (–a) = 0.
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 Don’t assume that –a is a negative number.
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 – (Property 5: –(a + b) = –a – b) –(x + 2) = –x – (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 undoes multiplication.
Every nonzero real number a has an inverse, 1/a, that satisfies a . (1/a). Division undoes 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 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 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: = =

51 E.g. 5—Using LCD to Add Fractions
We find the LCD by forming the product of all the factors that occur in these factorizations, using the highest power of each factor. Thus, the LCD is: = 360

52 E.g. 3—Using LCD to Add Fractions
So, we have: (Use common denominator) (Property 3: Adding fractions with the same denominator)


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