Transcript chapterP_Sec2

College Algebra
Fifth Edition
James Stewart  Lothar Redlin

Saleem Watson
P Prerequisites
P.2
Real Numbers
and Their Properties
Types of Real Numbers
Introduction
Let’s review the types
of numbers that make up
the real number system.
Natural Numbers
We start with the natural
numbers:
1, 2, 3, 4, …
Integers
The integers consist of the natural
numbers together with their negatives
and 0:
. . . , –3, –2, –1, 0, 1, 2, 3, 4, . . .
Rational Numbers
We construct the rational numbers
by taking ratios of integers.
• Thus, any rational number r
can be expressed as:
m
r 
n
where m and n are integers and n ≠ 0.
Rational Numbers
Examples are:
1
2

3
7
46 
46
1
17
0.17  100
• Recall that division by 0 is always ruled out.
• So, expressions like 3/0 and 0/0 are undefined.
Irrational Numbers
There are also real numbers, such as 2,
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:
3
5
3
2

3

2
Set of All Real Numbers
The set of all real numbers is
usually denoted by:
• The symbol R
Real Numbers
When we use the word ‘number’
without qualification, we will mean:
• “Real number”
Real Numbers
Figure 1 is a diagram of the types
of real numbers that we work with
in this book.
Repeating Decimals
Every real number has a decimal
representation.
If the number is rational, then its
corresponding decimal is repeating.
Repeating Decimals
For example,
1
2
 0.5000...  0.50
2
3
 0.66666...  0.6
157
495
9
7
 0.3171717...  0.317
 1.285714285714...  1.285714
• The bar indicates that the sequence of digits
repeats forever.
Non-Repeating Decimals
If the number is irrational, the decimal
representation is non-repeating:
2  1.414213562373095...
  3.141592653589793...
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 ≈ 3.14159265
where the symbol ≈ is read
“is approximately equal to.”
• The more decimal places we retain,
the better our approximation.
E.g. 1—Classifying Real Numbers
Determine whether
a) 999
d)
25
b) –6/5
e)
3
c) –6/3
is a natural number, an integer, a rational
number, or an irrational number.
E.g. 1—Classifying Real Numbers
a) 999 is a positive whole number, so it is a
natural number.
b) –6/5 is a ratio of two integers, so it is a
rational number.
c) –6/3 equals –2, so it is an integer.
E.g. 1—Classifying Real Numbers
d)
25 equals 5, so it is a natural number.
e)
3 is a nonrepeating decimal
(approximately 1.7320508075689),
so it is an irrational number.
Operations on Real Numbers
Operations on Real Numbers
Real numbers can be combined using the
familiar operations:
•
•
•
•
Addition
Subtraction
Multiplication
Division
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:
Order of Operations on Real Numbers
1. 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.
2. Perform all multiplication and division
•
Working from left to right
3. Perform all addition and subtraction
•
Working from left to right
E.g. 2—Evaluating an Arithmetic Expression
Find the value of the expression
 8  10

3
 4   2 5  9
 2 3

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.
 8  10

 18

3
 4   2 5  9   3 
 4   2 5  9 
 6

 2 3

 3 3  4   2 5  9 
 3 7  2 14
 21  28
 7
Properties of Real Numbers
Introduction
We all know that:
2+3=3+2
5+7=7+5
513 + 87 = 87 + 513
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.
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.
Properties of Real Numbers
From our experience with numbers, we
know that these properties are also valid.
Distributive Property
The Distributive Property
applies:
• Whenever we multiply a number
by a sum.
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.
E.g. 3—Using the Properties
Example (a)
2 + (3 + 7)
= 2 + (7 + 3)
(Commutative Property of Addition)
= (2 + 7) + 3
(Associative Property of Addition)
E.g. 3—Using the Properties
Example (b)
2(x + 3)
=2.x+2.3
(Distributive Property)
= 2x + 6
(Simplify)
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.
Addition and Subtraction
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.
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)
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
Properties of Negatives
To combine real numbers involving
negatives, we use these properties.
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.
E.g. 4—Using Properties of Negatives
Let x, y, and z be real numbers.
a) –(3 + 2) = –3 – 2
(Property 5: –(a + b) = –a – b)
b) –(x + 2) = –x – 2
(Property 5: –(a + b) = –a – b)
c) –(x + y – z) = –x – y – (–z)
= –x – y + z
(Property 5)
(Property 2:
–(– a) = a)
Multiplication and Division
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.
Division
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.
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).
Division
To combine real numbers using division,
we use these properties.
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.
LCD
This denominator is the Least
Common Denominator (LCD).
• It is described in the next example.
E.g. 5—Using LCD to Add Fractions
Evaluate: 5
7

36 120
• Factoring each denominator
into prime factors gives:
36 = 22 . 32
120 = 23 . 3 . 5
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:
23 . 32 . 5 = 360
E.g. 3—Using LCD to Add Fractions
So, we have:
5
7

36 120
5  10
73


36  10 120  3
50
21


360 360
71

360
(Use common denominator)
(Property 3: Adding fractions
with the same denominator)