**Unformatted text preview: **Basic Math OER
Material Contents
Chapter
1 2 3 4 5 6 Title
REAL NUMBERS
1.1 Classification of Real Numbers 1 1.1 Classification of Real Numbers 9 1.1 Classification of Real Numbers 23 INTEGER & RATIONAL EXPONENTS AND
RADICALS
2.1 Integer Exponents 47 2.2 Rational Exponents 60 2.3 Radicals 67 POLYNOMIALS
3.1 Add and Subtract Polynomials 94 3.2 Multiply Polynomials 101 FACTORING POLYNOMIALS
Greatest Common Factor and Factor by Grouping 107 Factor Trinomials 113 Factor Special Products 121 RATIONAL EXPRESSIONS
5.1 Multiply and Divide Rational Expressions 127 5.2 Add and Subtract Rational Expressions 137 5.3 Simplify Complex Rational Expressions 151 SYSTEMS OF MEASUREMENT
6.1 Systems of Measurement 7 8 9 10 Page No. 162 LINEAR EQUATIONS & INEQUALITIES
7.1 Linear Equations 177 7.2 Linear Inequalities 203 QUADRATIC EQUATIONS
8.1 Solve Quadratic Equations Using the Quadratic Formula 212 8.2 Solve Applications Modeled by Quadratic Equations 221 COORDINATE GEOMETRY
9.1 Rectangular Coordinate System 227 9.2 Straight Lines 236 9.3 Circle 243 9.4 Testing Equations for Symmetry 248 TRIGONOMETRY
10.1 Angles and Circle 252 10.2 Right Triangle Trigonometry 261 10.3 Trigonometric Identities 267 Basic Mathematics 1 CHAPTER 1 REAL NUMBERS
Chapter Outline
1.1 Classification of Real Numbers
1.2 Properties of Real Numbers
1.3 Fractions, Ratios and Percent
1.4 Systems of Measurement
Learning outcome covered:
a. Describe the set of real numbers, all its subsets and their relationship.
b. Identify and use the arithmetic properties of subsets of integers,
rational, irrational, and real numbers, including closure properties for
the four basic arithmetic operations where applicable.
c. Manipulate fractions, ratios, decimals, and percentages. Learning Objectives
By the end of this chapter, the students will be able to:
à Identify rational numbers and irrational numbers
à Classify different types of real numbers
à Use the commutative and associative properties
à Simplify expressions using the distributive property
à Recognize and use the identity and inverse properties of addition and multiplication
à Use the properties of zero
à Simplify expressions using the properties of identities, inverses, and zero
à Add and Subtract Fractions
à Convert percents to fractions and decimals
à Convert decimals and fractions to percents
Introduction
Even though counting is first taught at a young age, mastering mathematics, which is the study of
numbers, requires constant attention. If it has been a while since you have studied math, it can be
helpful to review basic topics. In this chapter, we will focus on numbers as well as four arithmetic
operations—addition, subtraction, multiplication, and division. We will also discuss some vocabulary
that we will use throughout this book. Download for free at Basic Mathematics 2 1.1 Classification of Real Numbers
1.1.1 Identify Counting Numbers and Whole Numbers
Algebra uses numbers and symbols to represent words and ideas. Let’s look at the numbers first.
The most basic numbers used in algebra are those we use to count objects: 1, 2, 3, 4, 5, . . . and so
on. These are called the counting numbers. Counting numbers are also called natural numbers.
Counting Numbers
The counting numbers start with 1 and continue.
1, 2, 3, 4, 5, · · ·
Counting numbers and whole numbers can be visualized on a number line as shown in
Figure 1.
larger
smaller
0 1 2 3 4 5 6 Figure 1: The numbers on the number line increase from left to right, and decrease from right to left. The discovery of the number zero was a big step in the history of mathematics. Including zero with
the counting numbers gives a new set of numbers called the whole numbers.
Whole Numbers
The whole numbers are the counting numbers and zero.
0, 1, 2, 3, 4, 5, · · ·
Example 1: Which of the following are (a) counting numbers? (b) whole numbers?
1
, 3, 5.2, 15, 105
4
Solution: 0, (a) The counting numbers start at 1, so 0 is not a counting number. The numbers 3, 15, and 105
are all counting numbers.
(b) Whole numbers are counting numbers and 0. The numbers 0, 3, 15, and 105 are whole numbers.
1
The numbers
and 5.2 are neither counting numbers nor whole numbers. We will discuss
4
these numbers later. Try It # 1: Which of the following are (a) counting numbers? (b) whole numbers?
0, 2
, 2, 9, 11.8, 241, 376
3 Download for free at Basic Mathematics 3 Introduction to Integers
Do you live in a place that has very cold winters? Have you ever experienced a temperature
below zero? If so, you are already familiar with negative numbers. A negative number
is a number that is less than 0. Very cold temperatures are measured in degrees below
zero and can be described by negative numbers. For example, −1◦ F (read as “negative one
degree Fahrenheit”) is 1 degree below 0.
Both positive and negative numbers can be represented on a number line. Recall that the
number line started at 0 and showed the counting numbers increasing to the right. The
counting numbers (1, 2, 3, . . . ) on the number line are all positive. We could write a plus
sign, +, before a positive number such as +2 or +3, but it is customary to omit the plus sign
and write only the number. If there is no sign, the number is assumed to be positive
Now we need to extend the number line to include negative numbers. We mark several units
to the left of zero, keeping the intervals the same width as those on the positive side. We
label the marks with negative numbers, starting with −1 at the first mark to the left of 0,
−2 at the next mark, and so on. See Figure 2.
-4 -3 -2 -1 Negative numbers 0
Zero 1 2 3 4 Positive numbers Figure 2: The arrows at either end of the line indicate that the number line extends forever in each
direction. There is no greatest positive number and there is no smallest negative number.
Order Positive and Negative Numbers
We can use the number line to compare and order positive and negative numbers. Going
from left to right, numbers increase in value. Going from right to left, numbers decrease in
value. See Figure 3.
increasing -4 -3 -2 -1 0 1 2 3 4 decreasing
Figure 3: Just as we did with positive numbers, we can use inequality symbols to show the ordering
of positive and negative numbers. Remember that we use the notation a < b (read a is less
than b ) when a is to the left of b on the number line. We write a > b (read a is greater than
b ) when a is to the right of b on the number line. Download for free at Basic Mathematics 4 Example 2: Order each of the following pairs of numbers using < or >:
a. 14 6 −1 b. 9 c. −1 −4 d. 2 − 20 Solution:
a. 14 Compare 14 and 6. 14 > 6 14 is to the right of 6 on the number line.
b. Compare −1 and 9. −1 −1 is to the left of 9 on the number line.
Compare −1 and −4. c. 9 −1 < 9
14 − 1 −1 is to the right of −4 on the number line.
d. 6 −4 −1 > −4 Compare 2 and −20. 2 − 20
2 > −20 2 is to the right of -20 on the number line. Try It # 2: Order each of the following pairs of numbers using < or >:
a. 15 7 b. −2 5 c. 5 − 17 d. −3 −7 1.1.2 Rational and Irrational Numberss
Rational Numbers
Rational Numbers
A rational number is a number that can be written in the form p
, where p and q are
q integers and q 6= 0.
4
7
All fractions, both positive and negative, are rational numbers. A few examples are , − ,
5
8
13
20
and − .
4
3
Each numerator and each denominator is an integer.
We need to look at all the numbers we have used so far and verify that they are rational. The
definition of rational numbers tells us that all fractions are rational. We will now look at the
counting numbers, whole numbers, integers, and decimals to make sure they are rational.
Are integers rational numbers? To decide if an integer is a rational number, we try to write it
as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator
one.
3
−8
0
3=
−8 =
0=
1
1
1 Download for free at Basic Mathematics 5 Since any integer can be written as the ratio of two integers, all integers are rational numbers.
Remember that all the counting numbers and all the whole numbers are also integers, and
so they, too, are rational.
What about decimals? Are they rational? Let’s look at a few to see if we can write each
of them as the ratio of two integers. We’ve already seen that integers are rational numbers.
The integer −8 could be written as the decimal −8.0. So, clearly, some decimals are rational.
Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means
73
3
. So 7.3 is the ratio of the integers 73 and
7 10
, we can write it as an improper fraction,
10
10. It is a rational number.
In general, any decimal that ends after a number of digits (such as 7.3 or −1.2684) is a
rational number. We can use the place value of the last digit as the denominator when
writing the decimal as a fraction.
Every rational number can be written both as a ratio of integers and as a decimal that either
stops or repeats. The table below shows the numbers we looked at expressed as a ratio of
integers and as a decimal.
Rational Numbers
Fractions Integers Number 4
, − 87 , 13
, −20
5
4
3 −2, −1, 0, 1, 2, 3 Ratio of Integer 4 −7 13
, , 4 , − 20
5 8
3 −2 −1 0 1 2 3
, 1 , 1, 1, 1, 1
1 Decimal number 0.8, −0.875, 3.25, −6.6, −2.0, −1.0, 0.0, 1.0, 2.0, 3.0 Irrational Numbers
Are there any decimals that do not stop or repeat? Yes. The number π (the Greek letter
π, pronounced ‘pie’), which is very important in describing circles, has a decimal form that
does not stop or repeat.
π = 3.141592654 · · ·
Similarly, the decimal representations of square roots of numbers that are not perfect squares
never stop and never repeat. For example,
√
5 = 2.236067978 · · ·
A decimal that does not stop and does not repeat cannot be written as the ratio of integers.
We call this kind of number an irrational number. Download for free at Basic Mathematics 6 Irrational Number
An irrational number is a number that cannot be written as the ratio of two integers. Its
decimal form does not stop and does not repeat.
Let’s summarize a method we can use to determine whether a number is rational or irrational.
If the decimal form of a number
< stops or repeats, the number is rational.
< does not stop and does not repeat, the number is irrational.
Example 3: Identify each of the following as rational or irrational:
a. 0.583 b. 0.475 c. 3.605551275... Solution:
(a) 0.583
The bar above the 3 indicates that it repeats. Therefore, 0.583 is a repeating decimal,
and is therefore a rational number.
(b) 0.475
This decimal stops after the 5 , so it is a rational number.
(c) 3.605551275...
The ellipsis (...) means that this number does not stop. There is no repeating pattern
of digits. Since the number doesn’t stop and doesn’t repeat, it is irrational. Try It # 3: Identify each of the following as rational or irrational:
a. 0.29 b. 0.816 c. 2.515115111... Let’s think about square roots now. Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not
perfect squares never stop and never repeat, so these square roots are irrational.
Example 4: Identify each of the following as rational or irrational:
√
√
a.
36
b.
44
Solution:
(a) The number 36 is a perfect square, since 62 = 36. So √ 36 = 6. Therefore 36 is rational. (b) Remember that 62 = 36 and 72 = 49, so 44 is not a perfect square.
√
This means 44 is irrational. Download for free at Basic Mathematics 7 Try It # 4: Identify each of the following as rational or irrational:
√
√
71
b.
17
a.
Classify Real Numbers
We have seen that all counting numbers are whole numbers, all whole numbers are integers,
and all integers are rational numbers. Irrational numbers are a separate category of their
own. When we put together the rational numbers and the irrational numbers, we get the set
of real numbers.
Figure 4 illustrates how the number sets are related.
Real Numbers
Rational Numbers
Integers
Whole Numbers Counting Numbers Irrational Numbers Figure 4: Real Numbers
Real numbers are numbers that are either rational or irrational.
Example 5: Determine whether each of the numbers in the following list is a (a) whole
number, (b) integer, (c) rational number, (d) irrational number, and (e) real number.
-7, √
√
14
, 8, 5, 5.9, − 64
5 Solution:
(a) The whole numbers are 0, 1, 2, 3,. . . The number 8 is the only whole number given.
(b) The integers are the whole numbers, their opposites, and 0. From the given numbers,
√
−7 and 8 are integers. Also, notice that 64 is the square of 8 so − 64 = −8. So the
√
integers are −7, 8, − 64.
√
(c) Since all integers are rational, the numbers −7, 8, and − 64 are also rational. Rational
Download for free at Basic Mathematics 8 14
and 5.9 are
numbers also include fractions and decimals that terminate or repeat, so
5
rational.
√
(d) The number 5 is not a perfect square, so 5 is irrational.
(e) All of the numbers listed are real.
We’ll summarize the results in a table. Number Whole Integer Rational -7 X X X X X X X X 14
5
8
√ X X Irrational 5 X 5.9
√
− 64 X Real X X X X X Try It # 5: Determine whether each of the numbers in the following list is a (a) whole
number, (b) integer, (c) rational number, (d) irrational number, and (e) real number.
√
√
√
9
-3, − 2, 0.3, , 8, 5, 4, 49
5 1.1 Section Exercises
Rational Numbers
In the following exercises, determine which of the given numbers are rational and which are
irrational.
1. 0.75, 0.223, 1.39174 · · · 2. 0.36, 0.94729 · · · , 2.528 3. 0.45, 1.919293 · · · , 3.59 4. 0.13, 0.42982 · · · , 1.875
In the following exercises, identify whether each number is rational or irrational.
√
√
√
√
√
√
5. a. 25
b. 30
6. a. 44
b. 49
7. a. 164
b. 169
8. √
a. 225 √
b. 216
Download for free at Basic Mathematics 9 Classifying Real Numbers
In the following exercises, determine whether each number is whole, integer, rational, irrational, and real.
4 √
11
12 √
10. −9, −3 , − 9, 0.409, , 7
9. −8, 0, 1.95286...., , 36, 9
5
9
6
√
1
4 √
8
11
11. − 100, − , −1, 0.07, 3
12. −9, −3 , − 9, 0.409, , 7
3
4
9
6 1.2 Properties of Real Numbers
In the next few sections, we will take a look at the properties of real numbers. Many of
these properties will describe things you already know, but it will help to give names to the
properties and define them formally.
Commutative and Associative Properties
Think about adding two numbers, such as 5 and 3.
5+3
3+5
8
8
The results are the same. 5 + 3 = 3 + 5
Notice, the order in which we add does not matter. The same is true when multiplying 5
and 3.
5·3
3·5
15
15
Again, the results are the same! 5 · 3 = 3 · 5. The order in which we multiply does not
matter.
These examples illustrate the commutative properties of addition and multiplication.
Commutative Properties
Commutative Property of Addition: if a and b are real numbers, then
a+b=b+a
Commutative Property of Multiplication: if a and b are real numbers, then
a·b=b·a
Example 6: Use the commutative properties to rewrite the following expressions:
a. −1 + 3 = b. 4·9= Solution:
a. −1+3=
−1 + 3 = 3 + (−1) Use the commutative property of addition to change the order.
b. 4 · 9 =
Download for free at Basic Mathematics 4·9=9·4 10 Use the commutative property of multiplication to change the order. Try It # 6: Use the commutative properties to rewrite the following expressions:
a. −4 + 7 = b. 4·9= What about subtraction? Does order matter when we subtract numbers? Does 7 − 3 give
the same result as 3 − 7?
73
3−7
4
−4
4 6= −4
The results are not the same. 7 − 3 6= 3 − 7
Since changing the order of the subtraction did not give the same result, we can say that
subtraction is not commutative. Let’s see what happens when we divide two numbers. Is
division commutative?
12 ÷ 4
4 ÷ 12
1
3
3
1
3 6=
3
The results are not the same. So 12 ÷ 4 6= 4 ÷ 12.
Since changing the order of the division did not give the same result, division is not commutative. Addition and multiplication are commutative. Subtraction and division are not
commutative.
Associative Properties
Associative Property of Addition: if a, b, and c are real numbers, then
(a + b) + c = a + (b + c)
Associative Property of Multiplication: if a, b, and c are real numbers, then
(a · b) · c = a · (b · c)
Example 7: Use the commutative properties to rewrite the following expressions:
2
a. (3 + 0.6) + 0.4 =
b.
−4 ·
· 15 =
5
Solution:
a. (3 + 0.6) + 0.4 =
(3 + 0.6) + 0.4 = 3 + (0.6 + 0.4) Change the grouping. Notice that 0.6 + 0.4 is 1, so the addition will be easier if we group as shown on the right.
b.
2
−4 ·
· 15 =
5
Download for free at Basic Mathematics 11
2
2
· 15 = −4 ·
· 15
−4 ·
5
5
Notice that Change the grouping. 2
· 15 is 6. The multiplication will be easier if we group as shown on the right.
5 Try It # 7: Use the associative properties to rewrite the following:
a. (1 + 0.7) + 0.3 = b. (−9 · 8) · 3
=
4 Distributive Property
Distributive Property
if a, b, and c are real numbers, then
a(b + c) = ab + ac
In algebra, we use the Distributive Property to remove parentheses as we simplify expressions.
For example, if we are asked to simplify the expression 3(x + 4), the order of operations says
to work in the parentheses first. But we cannot add x and 4, since they are not like terms.
So we use the Distributive Property, as shown in the Example.
Example 8: Simplify: 3(x + 4).
Solution:
3(x + 4) = 3 · x + 3 · 4 Distribute. = 3x + 12 Multiply. Try It # 8: Simplify: 4(x + 2).
Some students find it helpful to draw in arrows to remind them how to use the Distributive
Property. Then the first step in the Example would look like this:
3 (x + 4)
3·x+3·4
Example 9: Simplify: 6(5y + 1).
Solution:
6 (5y + 1)
6 · 5y + 6 · 1 6(5y + 1) = 6 · 5y + 6 · 1
= 30y + 6 Distribute.
Multiply.
Download for free at Basic Mathematics 12 Try It # 9: Simplify: 9(3y + 8).
The distributive property can be used to simplify expressions that look slightly different from
a(b + c). Here are two other forms.
Distributive Property
if a, b, and c are real numbers, then
a(b − c) = ab − ac
Other forms
(b + c)a = ba + ca
a(b + c) = ab + ac
Example 10: Simplify: 2(x − 3).
Solution:
2 (x − 3)
2·x−2·3
2(x − 3) = 2 · x − 2 · 3 Distribute. = 2x − 6 Multiply. Try It # 10: Simplify: 7(x − 6).
In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.
Example 11: Simplify: m(n − 4).
Solution:
m(n − 4)
m·n−m·4
m(n − 4) = m · n − m · 4 Distribute. = mn − 4m Multiply. Try It # 11: Simplify: r(s − 2).
The next example will use the ‘backwards’ form of the Distributive Property, (b+c)a = ba+ca.
Example 12: Simplify: (x + 8)p. Download for free at Basic Mathematics 13 Solution:
(x + 8) p
px + 8p Distribute. Try It # 12: Simplify: (x + 2)p.
When you distribute a negative number, you need to be extra careful to get the signs correct.
Example 13: Simplify: −2(4y + 1).
Solution:
−2 (4y + 1) −2(4y + 1) = −2 · 4y + (−2) · 1 Distribute. = −8y − 2 Simplify. Try It # 13: Simplify: −3(6m + 5).
Example 14: Simplify: −11(4 − 3a).
Solution:
−11(4 − 3a) = −11 · 4 − (−11) · 3a Distribute. = −44 − (−33a) Multiply. = −44 + 33a
Simplify.
You could also write the result as 33a − 44. Do you know why?
Try It # 14: Simplify: −5(2 − 3a).
In the next example, we will show how to use the Distributive Property to find the opposite
of an expression. Remember, −a = −1 · a.
Example 15: Simplify: −(y + 5).
Solution:
−(y + 5) = −1 · (y + 5) Multiplying by −1 results in the opposite. = −1 · y + (−1) · 5 Distribute. = −y + (−5) Multiply. = −y − 5 Simplify. Try It # 15: Simplify: −(z − 11).
Download for free at Basic Mathematics 14 Sometimes we need to use the Distributive Property as part of the order of operations.
Start by looking at the parentheses. If the expression inside the parentheses cannot be
simplified, the next step would be multiply using the distributive property, which removes
the parentheses. The next two examples will illustrate this.
Example 16: Simplify: 8 − 2(x + 3).
Solution:
8 − 2(x + 3) = 8 − 2 · x − 2 · 3 Distribute. = 8 − 2x − 6 Multiply. =...

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