1 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-1

Basic Concepts

Chapter 1

2 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-2

1.1 – Study Skills for Success in Mathematics, and Use of a Calculator

1.2 – Sets and Other Basic Concepts

1.3 – Properties of and Operations with Real Numbers

1.4 – Order of Operations

1.5 – Exponents

1.6 – Scientific Notation

Chapter Sections

3 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-3

Properties of and

Operations with Real Numbers

§ 1.3

4 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-4

Additive Inverse

Two numbers that are the same distance from 0 on the number line but in opposite directions are called additive inverses, or opposites, of each other.

Additive InverseFor any real number a, its additive inverse is –

a Double Negative Property

For any real number, a –(-a) = a

5 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-5

Absolute Value

The absolute value of a number is its distance from the number 0 on the real number line. The absolute value of every number will be either 0 or positive

4 4 and 4 4

-5 -4 -3 -2 -1 0 1 2 3 4 5

4 units 4 units

6 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-6

Number LinesAdd 4 + (– 2) using a number line1. Always begin with 0.

2. Since the first number is positive, the first arrow starts at 0 and is drawn 4 units to the right.

3. The second arrow starts at 4 and is drawn 2 units to the left , since the second number is negative.

4 + (– 2) = 2

-5 -4 -3 -2 -1 0 1 2 3 4 5

4

-5 -4 -3 -2 -1 0 1 2 3 4 5

4 -2

7 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-7

Add Real Numbers

To add real numbers with the same sign,add their absolute values. The sum has the same sign as the numbers being added.Example:

–4 + (–7) = 11

The sum of two positive numbers will always be positive and the sum of two negative numbers will always be negative.

8 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-8

Adding with Different Signs

To add real numbers with the different signs, subtract the smaller absolute value from the larger absolute value. The sum has the sign of the number with the larger absolute value. Example:5 + (–9) = -4

The sum of two numbers with different signs may be positive or negative. The sign of the sum will be the same as the sign of the number with the larger absolute value.

9 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-9

Least Common Denominator

The least common denominator (LCD) of a set of denominators is the smallest number that each denominator divides into without remainder.

10 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-10

Least Common Denominator

The LCD is 27. Rewriting the first fraction with the LCD gives the following.

275

94- Add

277

275

2712-

275

33

94-

-

Example:

11 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-11

Subtraction of Real Numbers

If a and b represent two real numbers, then

a – b = a + (– b)In other words, to subtract b from a, add the additive inverse of b to a.Example:

a.) 3 – (8) =3 + (– 8) = -5

b.) – 6 – 4 = – 6 + (– 4) = – 10

12 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-12

Subtracting a Negative Number

If a and b represent two real numbers, then

a – (-b) = a + bExample:

a.) -4 – (-11) = -4 +11 = 7

13 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-13

More Examples

452

4527

4525

53

95

53

95

b.)

a.) – 42 – 35 = -77

Example:

14 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-14

Multiply Two Real Numbers

1. To multiply two numbers with like signs, multiply their absolute values. The product is positive.

2. To multiply two numbers with unlike signs, one positive and the other negative, multiply their absolute values. The product is negative.Example:

a.) (4.2)(–1.6) = –6.72

b.) (-18)(-1/2) = 9

15 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-15

Caution!

It is very easy to mix up subtraction and multiplication problems.

– 3 – 5 is not the same as –3(–5).

2 – 4 is not the same as 2(–4)

Subtraction

– 3 – 5 = –8

– 2 – 4 = –6

Multiplication

– 3(–5) = 15

2(–4) = –8

16 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-16

Divide Real Numbers

1. To divide two numbers with like signs, either both positive or both negative, divide their absolute values. The quotient is positive.

2. To divide two numbers with unlike signs, one positive and the other negative, divide their absolute values. The quotient is negative.

Example:

a.) -24 (4) = –6

b.) –6.45 (–0.4) = 16.125

17 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-17

Multiplication vs. Division

(+)(+) = + (+) ÷ (+) = +

(–)(–) = + (–) ÷ (–) = +

(+)(–) = – (+) ÷ (–) = –

(–)(+) = – (–) ÷ (+) = –

Like signs give positive products and quotients.

Unlike signs give negative products and quotients.

For multiplication and division of two real numbers:

18 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-18

Signs of a Fraction

If a and b represent any real numbers, b 0, then

ba

ba

ba

We generally do not write fractions with a negative sign in the denominator.

The fraction would be written as or . 9

5

9

5

9

5

19 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-19

Dividing with Zero

If a represents any real number except 0, then

0 a = = 0a0

Division by 0 is undefined.

0

a ?