Mathematics Bridge Program ©2002 DeVry University

Algebra

Chapter 1

The Real Number System

1.1. The Number Sets

• Natural Numbers • Whole Numbers • Integers • Rational Numbers • Irrational Numbers • Real Numbers

1.2. Operations With Real Numbers • Absolute Value • Addition • Subtraction • Multiplication • Division • Order of Operations

1.3. Answers to Exercises

1.1 The Number Sets
A set is a collection of objects. The objects in a set are called the elements of the set. A set of numbers is simply a listing, within braces {}. For example, the set of numbers used for counting can be represented as S = {1, 2, 3, 4, 5, . . .}. The braces
{ } indicate that we are representing a set.

Some important sets of numbers that we will study are the following:

Natural numbers: The natural numbers N are the counting numbers. N= {1, 2, 3, 4, . . .}.

Whole numbers: The whole numbers W are the natural numbers combined with the number 0. W = {0, 1, 2, 3, 4, . . .}.

Integers: The integers I are the whole numbers combined with the negative of all natural numbers. I = { . . ., -4, -3, -2, -1, 0, 1, 2, 3, 4, . . .}. Z can also be used to represent the integers.

The integers consist of the following numbers: • Positive integers: {1, 2, 3, 4, . . .} • Negative integers: { . . . -4, -3, -2, -1}. (negative bank balance) • Zero is also an integer but it is neither positive nor negative.

Rational numbers: Q represents the rational numbers. A rational number is any number in the form [pic], where a and b are integers and b [pic]0. For example, [pic]. Note that all integers are also rational numbers, since any integer can also be expressed in the form [pic]

Example 1: Show that 3 is a rational number: [pic]

Exercise 1: Write 5 as an improper fraction. Answer in Section 1.3

A rational number may be expressed as a decimal number by dividing the numerator by the denominator (i.e. the top by the bottom). The result is either a terminating or a non-terminating-and-repeating decimal number.

Example 2: Express the following rational numbers in decimal form: Type 1: [pic] Dividing 3 by 5 gives 0.6, a terminating decimal.

Type 2: [pic] Dividing 2 by 3 actually gives 0.666 . . .(never-ending). This is a repeating decimal and is written with a bar above the first repeating segment behind the decimal point.

Exercise 2: Express the following rational numbers in decimal form a) [pic] b) [pic] Answer in Section 1.3

Irrational numbers:
Irrational numbers are those numbers that cannot be expressed as rational numbers. In their decimal form, irrational numbers neither terminate nor repeat.

Example 3: Listed below are 3 examples of irrational numbers and with their decimal approximations rounded to the nearest ten-thousandths place: [pic]

The irrational numbers are often encountered when finding the square root or higher roots. The square root of any number that is not a perfectly squared number is an irrational number.

Example 4: Are [pic] [pic] irrational numbers? Do they belong to any other number sets? [pic]is an irrational number because 3 is not a perfect square number. [pic]is also a real number.

[pic] is not an irrational number. 8 is a perfect cube, so [pic] is 2. 2 belongs to the natural numbers, the whole numbers, the integers, the rational numbers, and the real numbers at the same time.

Exercise 4: Determine if the following are irrational numbers and explain: a) [pic] b) [pic] Answer in Section 1.3

Real Numbers: R represents the real numbers.
The set R of real numbers is the set of all numbers that are rational or irrational numbers (and by default that includes all natural numbers, whole numbers, and integers). Each real number corresponds to a point on the number line.

Example 5: Given the set [pic], list the numbers in this set that belong to the:

a) Natural numbers b) Whole numbers c) Integers d) Rational numbers e) Irrational numbers f) Real numbers

a. The natural numbers are 11, and 114. b. The whole numbers are 0, 11, and 114. c. The integers are4, – 3, 0, 11, and 114. d. The rational numbers are –4, -3, 0, [pic], 11, and 114. e. The irrational number is[pic]. f. The real numbers are all numbers in the given set.

Exercise 5: Given the set [pic], list the numbers in this set that belong to the:

a) Natural numbers b) Whole numbers c) Integers d) Rational numbers e) Irrational numbers f) Real numbers Answer in Section 1.3

1.2 Operations with Real Numbers

Signed numbers: Positive and negative numbers are sometimes called signed numbers. Real numbers is a more appropriate term.

Real number operations are done exactly like integer operations – except you need to watch out for the rules of fraction operations and decimal operations as well.

Absolute value of numbers describes the number’s distance from zero on the number line.

Example 6: [pic] because 5 is 5 units to the right of 0 on the number line. [pic] because -5 is 5 units to the left of 0 on the number line.

Exercise 6: Evaluate: [pic] and [pic]. Explain the results. Answer in Section 1.3

Example 7: Evaluate: [pic] [pic]

Exercise 7: Evaluate: [pic] Answer in Section 1.3

Adding Real Numbers: If the numbers have the same signs, add the absolute values of the numbers, keep the sign.

Example 8: Add [pic] [pic]

Exercise 8: Evaluate: [pic] Answer in Section 1.3

Adding Real Numbers: If the numbers have different signs, subtract the absolute values of the numbers, keep the sign of the number with the larger absolute value

Example 9: Add: [pic]

[pic]

Exercise 9: Evaluate: [pic]

Answer in Section 1.3

Subtracting Real Numbers: Change “subtract an integer” to “add its opposite”. Then proceed like real number addition.

Example 10: Subtract: [pic] [pic]

Exercise 10: Evaluate: [pic]

Answer in Section 1.3

Multiplying and Dividing In pairs, the product or quotient of numbers with
Real Numbers: different signs give a negative answer. In pairs, the product or quotient of numbers with same signs give a positive answer.

Example 11: Multiply: [pic] [pic]

Exercise 11: Evaluate: [pic]

Answer in Section 1.3

Example 12: Divide: [pic] [pic]

Exercise 12: Evaluate: [pic]

Answer in Section 1.3

Order of Operations:[pic] (Please Excuse My Dear Aunt Sally)
Work an expression from left to right, except if mixing operations resolve:

Parentheses (grouping symbols) first

before Exponents before Multiplying OR Dividing, from left to right before Adding OR Subtracting, from left to right or grouping, as preferred

Example 13: Evaluate [pic]

[pic] Parentheses first: [pic] In the parentheses, multiply or divide from left to right [pic] [pic] In the parentheses, multiply [pic] Multiply before you subtract [pic] Subtract [pic]

Exercise 13: Evaluate [pic] Answer in Section 1.3

1.3 Answers to Exercises
1. [pic] or any of its equivalent improper fractions, like [pic]

2. a) 0.125 b) [pic]

4. a) not an irrational number, [pic] b) Yes, [pic] is irrational: [pic] and [pic] so [pic]is a non-terminating, non-repeating decimal number somewhere between 11 an 12.
5. The set was [pic] c) The natural numbers are [pic]. d) The whole numbers are [pic]. e) The integers are [pic] f) The rational numbers are [pic]. g) The irrational number is[pic]. h) The real numbers are all numbers in the given set.

6. [pic] and [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]