R.2 Real Numbers and Their Properties
Sets of Numbers and the Number Line As mentioned previously, the set of natural numbers is written in set notation as follows.
51, 2, 3, 4,N6 Natural numbers
Including 0 with the set of natural numbers gives the set of whole numbers.
50, 1, 2, 3, 4,N6 Whole numbers
Including the negatives of the natural numbers with the set of whole numbers gives the set of integers.
5N, −3, −2, −1, 0, 1, 2, 3,N6 Integers
Integers can be graphed on a number line. See Figure 5. Every number corresponds to one and only one point on the number line, and each point cor- responds to one and only one number. The number associated with a given point is the coordinate of the point. This correspondence forms a coordinate system.
The result of dividing two integers (with a nonzero divisor) is a rational number, or fraction. A rational number is an element of the set defined as follows.
ep
q `p and q are integers and q 30f Rational numbers
The set of rational numbers includes the natural numbers, the whole numbers, and the integers. For example, the integer -3 is a rational number because it can be written as -13 . Numbers that can be written as repeating or terminating decimals are also rational numbers. For example, 0.6 = 0.66666c represents a rational number that can be expressed as the fraction 23 .
The set of all numbers that correspond to points on a number line is the real numbers, shown in Figure 6. Real numbers can be represented by decimals.
Because every fraction has a decimal form—for example, 14 = 0.25—real numbers include rational numbers.
Some real numbers cannot be represented by quotients of integers. These numbers are irrational numbers. The set of irrational numbers includes 12 and 15. Another irrational number is p, which is approximately equal to 3.14159. Some rational and irrational numbers are graphed in Figure 7.
The sets of numbers discussed so far are summarized as follows.
■ Sets of Numbers and the Number Line
■ Exponents
■ Order of Operations
■ Properties of Real Numbers
■ Order on the Number Line
■ Absolute Value
Figure 5
–5–4 –3 –2 0 1 2 3 4 5
Graph of the Set of Integers –1
Origin
Figure 6
–5–4 –3 –2 –1 0 1 2 3 4 5 Graph of the Set of Real Numbers
–1 0 1 2 3 4
– 23 Ë2 Ë5 p
Figure 7 Graph of the Set
E-23 , 0, 12, 15, p, 4F
12, 15, and p are irrational. Because 12 is approximately equal to 1.41, it is located between 1 and 2, slightly closer to 1.
Sets of Numbers
Set Description
Natural numbers 51, 2, 3, 4,c6 Whole numbers 50, 1, 2, 3, 4,c6
Integers 5c , -3, -2, -1, 0, 1, 2, 3,c6 Rational numbers Epq Pp and q are integers and q≠0F
Irrational numbers 5xx is real but not rational6
Real numbers 5xx corresponds to a point on a number line6
EXAMPLE 1 Identifying Sets of Numbers
Let A = E-8, -6, -124 , -34 , 0, 38 , 12 , 1, 22, 25, 6F. List all the elements of A that belong to each set.
(a) Natural numbers (b) Whole numbers (c) Integers (d) Rational numbers (e) Irrational numbers (f) Real numbers SOLUTION
(a) Natural numbers: 1 and 6 (b) Whole numbers: 0, 1, and 6 (c) Integers: -8, -6, -124 (or -3), 0, 1, and 6
(d) Rational numbers: -8, -6, -124 (or -3), -34 , 0, 38 , 12 , 1, and 6 (e) Irrational numbers: 22 and 25
(f) All elements of A are real numbers. ■✔ Now Try Exercises 11, 13, and 15.
Figure 8 Real Numbers
Rational numbers Irrational numbers
Integers –11, –6, –3, –2, –1
Whole numbers 0 Natural numbers 1, 2, 3, 4, 5, 37, 40
, – , 49 5
8 11
7 2
8 p
p4 15
Figure 8 shows the relationships among the subsets of the real numbers. As shown, the natural numbers are a subset of the whole numbers, which are a sub- set of the integers, which are a subset of the rational numbers. The union of the rational numbers and irrational numbers is the set of real numbers.
Exponents Any collection of numbers or variables joined by the basic operations of addition, subtraction, multiplication, or division (except by 0), or the operations of raising to powers or taking roots, formed according to the rules of algebra, is an algebraic expression.
-2x2 + 3x, 15y
2y- 3 , 2m3 - 64, 13a + b24 Algebraic expressions The expression 23 is an exponential expression, or exponential, where the 3 indicates that three factors of 2 appear in the corresponding product. The number 2 is the base, and the number 3 is the exponent.
Exponent: 3 23=(+)+*2 # 2 # 2 =8 Base: 2 Three factors
of 2
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R.2 Real Numbers and Their Properties
Read an as “a to the nth power” or simply “a to the nth.”
Exponential Notation
If n is any positive integer and a is any real number, then the nth power of a is written using exponential notation as follows.
an= a # a # a #N# a
en factors of a
EXAMPLE 2 Evaluating Exponential Expressions
Evaluate each exponential expression, and identify the base and the exponent.
(a) 43 (b) 1-622 (c) -62 (d) 4 # 32 (e) 14 # 322 SOLUTION
(a) 43 = 4 # 4 # 4 = 64 The base is 4 and the exponent is 3.
•
3 factors of 4
(b) 1-622 = 1-621-62 = 36 The base is -6 and the exponent is 2.
(c) -62= -16 # 62= -36
The base is 6 and the exponent is 2.
Notice that parts (b) and (c) are different.
(d) 4 # 32 = 4 # 3 # 3 = 36 The base is 3 and the exponent is 2.
32=3#3, NOT 3#2
(e) 14 # 322 = 122 =144 14#322≠4#32 The base is 4 #3, or 12, and the exponent is 2.
■✔ Now Try Exercises 17, 19, 21, and 23.
Order of Operations When an expression involves more than one opera- tion symbol, such as 5# 2 + 3, we use the following order of operations.
Order of Operations
If grouping symbols such as parentheses, square brackets, absolute value bars, or fraction bars are present, begin as follows.
Step 1 Work separately above and below each fraction bar.
Step 2 Use the rules below within each set of parentheses or square brackets. Start with the innermost set and work outward.
If no grouping symbols are present, follow these steps.
Step 1 Simplify all powers and roots. Work from left to right.
Step 2 Do any multiplications or divisions in order. Work from left to right.
Step 3 Do any negations, additions, or subtractions in order. Work from left to right.
EXAMPLE 3 Using Order of Operations Evaluate each expression.
(a) 6 ,3 + 23 # 5 (b) 18 + 62 , 7 # 3- 6 (c) 4+ 32
6- 5 # 3 (d) -21-1-38223-+51-13522 SOLUTION
(a) 6 ,3 + 23 # 5
= 6, 3+ 8 # 5 Evaluate the exponential.
= 2+ 8 # 5 Divide.
= 2+ 40 Multiply.
= 42 Add.
Multiply or divide in order from left to right.
(b) 18+ 62, 7 # 3 -6
= 14, 7 # 3 -6 Work inside the parentheses.
= 2 # 3- 6 Divide.
= 6- 6 Multiply.
= 0 Subtract.
Be careful to divide before multiplying
here.
(c) Work separately above and below the fraction bar, and then divide as a last step.
4+ 32 6- 5 # 3
= 4+ 9
6 -15 Evaluate the exponential and multiply.
= 13
-9 Add and subtract.
= -13
9 -ab = -ab
(d) -1-323 +1-52 21-82 - 5132 = -1-272+ 1-52
21-82- 5132 Evaluate the exponential.
= 27 +1-52
-16- 15 Multiply.
= 22
-31 Add and subtract.
= -22
31 -ab= -ab
■✔ Now Try Exercises 25, 27, and 33.
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R.2 Real Numbers and Their Properties
EXAMPLE 4 Using Order of Operations
Evaluate each expression for x= -2, y = 5, and z = -3.
(a) -4x2 - 7y+ 4z (b) 21x - 522 + 4y
z +4 (c)
x 2 - y
5 3z
9 + 8y 5 SOLUTION
(a) -4x2 - 7y+ 4z
= -41-222- 7152+ 41-32 Substitute: x= -2, y=5, and z= -3.
= -4142 -7152 + 41-32 Evaluate the exponential.
= -16 -35 -12 Multiply.
= -63 Subtract.
Use parentheses around substituted values to
avoid errors.
(b) 21x - 522+ 4y z + 4
= 21-2- 522+ 4152
-3 +4 Substitute: x= -2, y=5, and z= -3.
= 21-722 + 4152
-3 +4 Work inside the parentheses.
= 21492+ 4152
-3 + 4 Evaluate the exponential.
= 98 +20
1 Multiply in the numerator.
Add in the denominator.
= 118 Add; a1 =a.
(c) This is a complex fraction. Work separately above and below the main frac- tion bar, and then divide as a last step.
x 2 - y
5 3z
9 + 8y 5
=
-2 2 - 5
5 31-32
9 + 8152 5
Substitute: x= -2, y=5, and z= -3.
= -1- 1
-1+ 8 Simplify the fractions.
= -2
7 Subtract and add; -ba= -ab .
■✔ Now Try Exercises 35, 43, and 45.
CAUTION With the commutative properties, the order changes, but with the associative properties, the grouping changes.
Commutative Properties Associative Properties 1x +42+ 9=14+x2+9 1x +42+ 9=x +14+92 7# 15 # 22=15 # 22 # 7 7# 15 # 22=17 # 52 # 2
Properties of Real Numbers Let a, b, and c represent real numbers.
Property Description
Closure Properties
a+ b is a real number.
ab is a real number.
The sum or product of two real numbers is a real number.
Commutative Properties a + b=b + a
ab=ba
The sum or product of two real numbers is the same regardless of their order.
Associative Properties
1a +b2 + c= a+ 1b+ c2
1ab2c= a1bc2
The sum or product of three real numbers is the same no matter which two are added or multi- plied first.
Identity Properties
There exists a unique real number 0 such that
a+ 0= a and 0 +a =a.
There exists a unique real number 1 such that
a # 1= a and 1 # a =a.
The sum of a real number and 0 is that real number, and the prod- uct of a real number and 1 is that real number.
Inverse Properties
There exists a unique real number -a such that
a+ 1−a2 = 0 and −a +a =0.
If a≠0, there exists a unique real number 1a such that
a # 1a =1 and 1a # a =1.
The sum of any real number and its negative is 0, and the product of any nonzero real number and its reciprocal is 1.
Distributive Properties a1b +c2 =ab +ac a1b −c2 =ab −ac
The product of a real number and the sum (or difference) of two real numbers equals the sum (or difference) of the products of the first number and each of the other numbers.
Multiplication Property of Zero
0 # a=a # 0= 0 The product of a real number
and 0 is 0.
Properties of Real Numbers Recall the following basic properties.
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R.2 Real Numbers and Their Properties
EXAMPLE 5 Simplifying Expressions
Use the commutative and associative properties to simplify each expression.
(a) 6+ 19+ x2 (b) 5
8116y2 (c) -10pa6 5b SOLUTION
(a) 6+ 19+ x2
=16 +92 + x Associative property =15 +x Add.
(b) 5 8116y2 = a5
8 # 16by Associative property = 10y Multiply.
(c) -10pa6 5b = 6
51-10p2 Commutative property = c6
51-102 dp Associative property
= -12p Multiply. ■✔ Now Try Exercises 63 and 65.
Figure 9 helps to explain the distributive property. The area of the entire region shown can be found in two ways, as follows.
415 + 32 = 4182 = 32 or 4152 + 4132 = 20+ 12= 32 The result is the same. This means that
415 + 32 = 4152 + 4132.
Figure 9
5 3
4
Geometric Model of the Distributive Property
EXAMPLE 6 Using the Distributive Property
Rewrite each expression using the distributive property and simplify, if possible.
(a) 31x+ y2 (b) -1m -4n2 (c) 7p + 21 (d) 1 3a4
5 m- 3
2 n- 27b SOLUTION
(a) 31x +y2 = 3x+ 3y
Distributive property
(b) -1m - 4n2 = -11m -4n2
= -11m2+ 1-121-4n2 = -m +4n
Be careful with the negative signs.
(c) 7p+ 21 =7p +7 #3
=7 # p + 7 # 3
=71p +32
Distributive property in reverse
(d) 1 3a4
5 m - 3
2 n -27b = 1
3a4
5 mb + 1 3a-3
2 nb + 1 31-272 = 4
15 m - 1 2 n- 9
■✔ Now Try Exercises 67, 69, and 71.
Absolute Value The undirected distance on a number line from a number to 0 is the absolute value of that number. The absolute value of the number a is written 0a0. For example, the distance on a number line from 5 to 0 is 5, as is the distance from -5 to 0. See Figure 11. Therefore, both of the following are true.
050 = 5 and 0 -50 = 5
Figure 11 Distance
is 5. Distance
is 5.
–5 0 5
1 20
–117 23 7
–5 9 p
–5–4 –3 –2 –1 0 1 2 3 4 5
Figure 10 -15 is to the left of -117 on the number line, so -156 -117 , and 120 is to the right of p, indicating that 1207p.
NOTE Because distance cannot be negative, the absolute value of a number is always positive or 0.
The algebraic definition of absolute value follows.
Absolute Value
Let a represent a real number.
0a0 = e a
−a
if a# 0 if a* 0
That is, the absolute value of a positive number or 0 equals that number, while the absolute value of a negative number equals its negative (or opposite).
EXAMPLE 7 Evaluating Absolute Values Evaluate each expression.
(a) ` -5
8` (b) -080 (c) -0-20 (d) 02x0, for x= p SOLUTION
(a) ` -5 8` = 5
8 (b) -080 = -182 = -8
(c) -0-20 = -122= -2 (d) 02p0 =2p
■✔ Now Try Exercises 83 and 87.
Absolute value is useful in applications where only the size (or magnitude), not the sign, of the difference between two numbers is important.
See Figure 10. Statements involving these symbols, as well as the symbols less than or equal to, …, and greater than or equal to, Ú, are inequalities. The inequality a6b6 c says that b is between a and c because a6b and b6c.
Order on the Number Line If the real number a is to the left of the real number b on a number line, then
a is less than b, written a6 b.
If a is to the right of b, then
a is greater than b, written a7b.
The inequality symbol must point toward the
lesser number.
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R.2 Real Numbers and Their Properties
EXAMPLE 8 Measuring Blood Pressure Difference
Systolic blood pressure is the maximum pressure produced by each heartbeat.
Both low blood pressure and high blood pressure may be cause for medical con- cern. Therefore, health care professionals are interested in a patient’s “pressure difference from normal,” or Pd.
If 120 is considered a normal systolic pressure, then
Pd = 0P- 1200, where P is the patient’s recorded systolic pressure.
Find Pd for a patient with a systolic pressure, P, of 113.
SOLUTION Pd = P- 120
= 0113- 1200 Let P=113.
= 0 -70 Subtract.
= 7 Definition of absolute value
■✔ Now Try Exercise 89.
Properties of Absolute Value Let a and b represent real numbers.
Property Description
1. 0a0 #0 The absolute value of a real number is positive or 0.
2. 0−a0 = 0a0 The absolute values of a real number and its opposite are equal.
3. 0a0 # 0b0 = 0ab0 The product of the absolute values of two real numbers equals the absolute value of their product.
4. 0a0
0b0 = `
a
b ` 1b 302 The quotient of the absolute values of two real numbers equals the absolute value of their quotient.
5. 0a +b0 " 0a0 + 0b0 (the triangle inequality)
The absolute value of the sum of two real numbers is less than or equal to the sum of their absolute values.
Examples of Properties 1–4:
0-150 =15 and 15 Ú 0. Property 1
0-100 =10 and 0100 = 10, so 0-100 = 0100. Property 2 050 # 0-40 = 5 # 4 = 20 and 051-420 = 0-200 = 20,
so 050 # 0-40 = 051-420. Property 3 020
030 =
2
3 and `2 3 ` = 2
3 , so 020
030 = `
2
3 `. Property 4
Example of the triangle inequality:
0a + b0 = 03+ 1-720 = 0-40 =4
Let a=3 and b= -7.
0a0 + 0b0 = 030 + 0-70 = 3+ 7= 10
Thus, 0a+ b0 … 0a0 + 0b0. Property 5
LOOKING AHEAD TO CALCULUS One of the most important definitions in calculus, that of the limit, uses absolute value. The symbols P (epsilon) and d (delta) are often used to represent small quantities in mathematics.
Suppose that a function ƒ is defined at every number in an open interval I containing a, except perhaps at a itself.
Then the limit of ƒ1x2 as x approaches a is L, written
limxSa ƒ1x2=L,
if for every P 70 there exists a d70 such that 0ƒ1x2-L0 6 P whenever 060x-a06d.
NOTE As seen in Example 9(b), absolute value bars can also act as grouping symbols. Remember this when applying the rules for order of operations.
EXAMPLE 9 Evaluating Absolute Value Expressions Let x = -6 and y = 10. Evaluate each expression.
(a) 02x- 3y0 (b) 20x0 - 03y0
0xy0
SOLUTION (a) 02x- 3y0
= 021-62 -311020 Substitute: x= -6, y=10.
= 0 -12 - 300 Work inside the absolute value bars. Multiply.
= 0 -420 Subtract.
= 42 Definition of absolute value (b) 20x0 - 03y0
0xy0
= 20-60 - 0311020
0-611020 Substitute: x= -6, y=10.
= 2 # 6- 0300
0-600 0-60 =6; Multiply.
= 12- 30
60 Multiply; 0300 =30, 0-600 =60.
= -18
60 Subtract.
= - 3
10 Write in lowest terms; -ba= -ab .
■✔ Now Try Exercises 93 and 95.
Distance between Points on a Number Line
If P and Q are points on a number line with coordinates a and b, respec- tively, then the distance d1P, Q2 between them is given by the following.
d1P, Q2 = 0b −a0 or d1P, Q2 = 0a− b0
Figure 12 P
a
Q
d(P, Q) b That is, the distance between two points on a number line is the absolute value of the difference between their coordinates in either order. See
Figure 12.
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R.2 Real Numbers and Their Properties
EXAMPLE 10 Finding the Distance between Two Points Find the distance between -5 and 8.
SOLUTION Use the first formula in the preceding box, with a= -5 and b= 8.
d1P, Q2 = 0b - a0 = 08- 1-520 = 08+ 50 = 0130 = 13 Using the second formula in the box, we obtain the same result.
d1P, Q2 = 0a -b0 = 01-52 -80 = 0-130 = 13
■✔ Now Try Exercise 105.