Download Chapter 1-3

Chapter 1
Basic Concepts
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 1-1
1
Chapter Sections
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
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 1-2
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§ 1.3
Properties of and
Operations with
Real Numbers
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Chapter 1-3
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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 Inverse
For any real number a, its additive inverse is –a

Double Negative Property
For any real number, a –(-a) = a
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Chapter 1-4
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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
4 units
-5
-4
-3
-2
-1
4 units
0
1
2
3
4
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Chapter 1-5
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Number Lines
Add 4 + (– 2) using a number line
1. Always begin with 0.
2. Since the first number is positive, the first
arrow starts at 0 and is drawn 4 units to the
4
right.
-5
-4
-3
-2
-1
0
1
2
3
4
5
3. The second arrow starts at 4 and is drawn 2
units to the left , since the second number is
negative.
4 -2
4 + (– 2) = 2
-5
-4
-3
-2
-1
0
1
2
3
4
5
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Chapter 1-6
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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.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 1-7
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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.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 1-8
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Least Common Denominator

The least common denominator (LCD) of
a set of denominators is the smallest
number that each denominator divides
into without remainder.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 1-9
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Least Common Denominator
Example:
Add - 4  5
9 27
The LCD is 27. Rewriting the first fraction with the
LCD gives the following.
-4  3  5 
9 3 27
-12  5  - 7
27 27
27
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Chapter 1-10
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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
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Chapter 1-11
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Subtracting a Negative Number
If a and b represent two real numbers, then
a – (-b) = a + b
Example:
a.) -4 – (-11) = -4 +11 = 7
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Chapter 1-12
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More Examples
Example:
a.) – 42 – 35 = -77














3  5  3   25 27  2
5



b.) 9
5 9 5 45 45 45
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Chapter 1-13
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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
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Chapter 1-14
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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
Multiplication
– 3 – 5 = –8
– 3(–5) = 15
– 2 – 4 = –6
2(–4) = –8
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Chapter 1-15
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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
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Chapter 1-16
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Multiplication vs. Division
For multiplication and division of two real numbers:
(+)(+) = +
(+) ÷ (+) = +
(–)(–) = +
(–) ÷ (–) = +
(+)(–) = –
(+) ÷ (–) = –
(–)(+) = –
(–) ÷ (+) = –
Like signs give
positive products and
quotients.
Unlike signs give
negative products and
quotients.
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Chapter 1-17
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Signs of a Fraction
If a and b represent any real numbers, b 0, then
a  a   a
b
b
b
We generally do not write fractions with a negative
sign in the denominator.
The fraction 5 would be written as  5 or  5 .
9
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Chapter 1-18
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Dividing with Zero
If a represents any real number except 0, then
0  a = a0 = 0
Division by 0 is undefined.
a
 ?
0
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Chapter 1-19
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