1. Fundamental Operations with Numbers
1.1. FOUR OPERATIONS
Four operations are fundamental in algebra, as in arithmetic. These are addition, subtraction, multiplication, and division.
When two numbers a and b are added, their sum is indicated by a + b. Thus 3 + 2 = 5.
When a number b is subtracted from a number a, the difference is indicated by a − b. Thus 6 − 2 = 4.
Subtraction may be defined in terms of addition. That is, we may define a − b to represent that number x such that x added to b
yields a, or x + b = a. For example, 8 − 3 is that number x which when added to 3 yields 8, i.e.,x + 3 = 8; thus 8 − 3 = 5.
The product of two numbers a and b is a number c such that a × b = c. The operation of multiplication may be indicated by a
cross, a dot or parentheses. Thus 5 × 3 = 5 · 3 = 5(3) = (5)(3) = 15, where the factors are 5 and 3 and the product is 15. When
letters are used, as in algebra, the notation p × q is usually avoided since × may be confused with a letter representing a
number.
When a number a is divided by a number b, the quotient obtained is written
a ÷ b or
a
or a/b,
b
where a is called the dividend and b the divisor. The expression a/b is also called a fraction, having numerator a and
denominator b.
Division by zero is not defined. See Problems 1.1(b) and (e).
Division may be defined in terms of multiplication. That is, we may consider a/b as that number x which upon multiplication by
b yields a, or bx = a. For example, 6/3 is that number x such that 3 multiplied by x yields 6, or 3x = 6; thus 6/3 = 2.
1.2. SYSTEM OF REAL NUMBERS
The system of real numbers as we know it today is a result of gradual progress, as the following indicates.
1. Natural numbers 1, 2, 3, 4, ... (three dots mean "and so on") used in counting are also known as the positive integers. If two
such numbers are added or multiplied, the result is always a natural number.
2. Positive rational numbers or positive fractions are the quotients of two positive integers, such as 2/3, 8/5, 121/17. The
positive rational numbers include the set of natural numbers. Thus the rational number 3/1 is the natural number 3.
3. Positive irrational numbers are positive real numbers which are not rational, such as
√2, π.
4. Zero, written 0, arose in order to enlarge the number system so as to permit such operations as 6 − 6 or 10 − 10. Zero has
the property that any number multiplied by zero is zero. Zero divided by any number ≠ 0 (i.e., not equal to zero) is zero.
5. Negative integers, negative rational numbers and negative irrational numbers such as −3, −2/3, and
−√2, arose in order to enlarge the number system so as to permit such operations as 2 − 8,
π − 3π
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π − 3π or
2 − 2√2.
When no sign is placed before a number, a plus sign is understood. Thus 5 is +5,
√2 is
+√2. Zero is considered a rational number without sign.
The real number system consists of the collection of positive and negative rational and irrational numbers and zero.
Note. The word "real" is used in contradiction to still other numbers involving
√−1, which will be taken up later and which are known as imaginary, although they are very useful in mathematics and the
sciences. Unless otherwise specified we shall deal with real numbers.
1.3. GRAPHICAL REPRESENTATION OF REAL NUMBERS
It is often useful to represent real numbers by points on a line. To do this, we choose a point on the line to represent the real
number zero and call this point the origin. The positive integers +1, +2, +3, ... are then associated with points on the line at
distances 1, 2, 3, ... units respectively to the right of the origin (see Fig. 1-1), while the negative integers −1, −2, −3, ... are
associated with points on the line at distances 1, 2, 3, ... units respectively to the left of the origin.
Figure 1-1
The rational number 1/2 is represented on this scale by a point P halfway between 0 and +1. The negative number −3/2 or
−1 12 is represented by a point R
1 12 units to the left of the origin.
It can be proved that corresponding to each real number there is one and only one point on the line; and conversely, to every
point on the line there corresponds one and only one real number.
The position of real numbers on a line establishes an order to the real number system. If a point A lies to the right of another
point B on the line we say that the number corresponding to A is greater or larger than the number corresponding to B, or that
the number corresponding to B is less or smaller than the number corresponding to A. The symbols for "greater than" and "less
than" are > and < respectively. These symbols are called "inequality signs."
Thus since 5 is to the right of 3, 5 is greater than 3 or 5 > 3; we may also say 3 is less than 5 and write 3 < 5. Similarly, since −6
is to the left of −4, −6 is smaller than −4, i.e., −6 < −4; we may also write −4 > −6.
By the absolute value or numerical value of a number is meant the distance of the number from the origin on a number line.
Absolute value is indicated by two vertical lines surrounding the number. Thus
|−6| = 6,
|+4| = 4,
|−3/4| = 3/4.
1.4. PROPERTIES OF ADDITION AND MULTIPLICATION OF
REAL NUMBERS
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1. Commutative property for addition The order of addition of two numbers does not affect the result.
Thus
a + b = b + a,
5 + 3 = 3 + 5 = 8.
2. Associative property for addition The terms of a sum may be grouped in any manner without affecting the result.
a + b + c = a + (b + c) = (a + b) + c,
3 + 4 + 1 = 3 + (4 + 1) = (3 + 4) + 1 = 8
3. Commutative property for multiplication The order of the factors of a product does not affect the result.
a ⋅ b = b ⋅ a,
2 ⋅ 5 = 5 ⋅ 2 = 10
4. Associative property for multiplication The factors of a product may be grouped in any manner without affecting the
result.
abc = a(bc) = (ab)c,
3 ⋅ 4 ⋅ 6 = 3(4 ⋅ 6) = (3 ⋅ 4)6 = 72
5. Distributive property for multiplication over addition The product of a number a by the sum of two numbers (b + c) is equal
to the sum of the products ab and ac.
a(b + c) = ab + ac,
4(3 + 2) = 4 ⋅ 3 + 4 ⋅ 2 = 20
Extensions of these laws may be made. Thus we may add the numbers a, b, c, d, e by grouping in any order, as (a + b) + c + (d +
e), a + (b + c) + (d + e), etc. Similarly, in multiplication we may write (ab)c(de) or a(bc)(de), the result being independent of
order or grouping.
1.5. RULES OF SIGNS
1. To add two numbers with like signs, add their absolute values and prefix the common sign. Thus 3 + 4 = 7, (−3) + (−4) =
−7.
2. To add two numbers with unlike signs, find the difference between their absolute values and prefix the sign of the number
with greater absolute value.
Example
EXAMPLES 1.1.
17 + (−8) = 9,
(−6) + 4 = −2,
(−18) + 15 = −3
3. To subtract one number b from another number a, change the operation to addition and replace b by its opposite, −b.
Example
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EXAMPLES 1.2.
12 − (7) = 12 + (−7) = 5,
(−9) − (4) = −9 + (−4) = −13,
2 − (−8) = 2 + 8 = 10
4. To multiply (or divide) two numbers having like signs, multiply (or divide) their absolute values and prefix a plus sign (or
no sign).
Example
EXAMPLES 1.3.
(5)(3) = 15,
−6
=2
−3
(−5)(−3) = 15,
5. To multiply (or divide) two numbers having unlike signs, multiply (or divide) their absolute values and prefix a minus sign.
Example
EXAMPLES 1.4.
(−3)(6) = −18,
−12
= −3
4
(3)(−6) = −18,
1.6. EXPONENTS AND POWERS
When a number a is multiplied by itself n times, the product a · a · a · · · a (n times) is indicated by the symbol an which is
referred to as "the nth power of a" or "a to the nth power" or "a to the nth."
Example
EXAMPLES 1.5.
2 · 2 · 2 · 2 · 2 = 25 = 32,
2 · x · x · x = 2x3,
(−5)3 = (−5)(−5)(−5) = −125
a · a · a · b · b = a3b2,
(a − b)(a − b)(a − b) = (a − b) 3
In an, the number a is called the base and the positive integer n is the exponent.
If p and q are positive integers, then the following are laws of exponents.
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(1)
Thus:
(2)
35
ap ⋅ aq = ap+q
ap
1
= ap−q = q−p if a ≠ 0
aq
a
2 3 ⋅ 2 4 = 2 3+4 = 2 7
32
34
3
6
= 3 5−2 = 3 3 ,
=
3
1
6−4
=
32
3
(3)
(ap )q = apq
(4 2 ) = 4 6 ,
2
(3 4 ) = 3 8
(4)
(4 · 5)2 = 42 · 52
(ab )p = ap b p ,
a p ap
( ) = p if b ≠ 0
b
b
1
3
5
53
( ) = 3
2
2
1.7. OPERATIONS WITH FRACTIONS
Operations with fractions may be performed according to the following rules.
1. The value of a fraction remains the same if its numerator and denominator are both multiplied or divided by the same
number provided the number is not zero.
Example
EXAMPLES 1.6.
3
3⋅2
6
=
= ,
4
4⋅2
8
15
15 ÷ 3
5
=
=
18
18 ÷ 3
6
2. Changing the sign of the numerator or denominator of a fraction changes the sign of the fraction.
Example
EXAMPLE 1.7.
−3
3
3
=− =
5
5
−5
3. Adding two fractions with a common denominator yields a fraction whose numerator is the sum of the numerators of the
given fractions and whose denominator is the common denominator.
Example
EXAMPLE 1.8.
3 4
7
3+4
+ =
=
5 5
5
5
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4. The sum or difference of two fractions having different denominators may be found by writing the fractions with a
common denominator.
Example
EXAMPLE 1.9.
1 2
3
8
11
+ =
+
=
4 3
12 12
12
5. The product of two fractions is a fraction whose numerator is the product of the numerators of the given fractions and
whose denominator is the product of the denominators of the fractions.
Example
EXAMPLES 1.10.
2
3
3
4
4
2⋅4
8
,
=
=
5
3⋅5
15
8
3⋅8
24
2
⋅ =
=
=
9
4⋅9
36
3
⋅
6. The reciprocal of a fraction is a fraction whose numerator is the denominator of the given fraction and whose denominator
is the numerator of the given fraction. Thus the reciprocal of 3 (i.e., 3/1) is 1/3. Similarly the reciprocals of 5/8 and −4/3
are 8/5 and 3/−4 or −3/4, respectively.
7. To divide two fractions, multiply the first by the reciprocal of the second.
Example
EXAMPLES 1.11.
a
÷
b
2
÷
3
c
a d
ad
,
= ⋅ =
d
b c
bc
4
2 5
10
5
= ⋅ =
=
5
3 4
12
6
This result may be established as follows:
a c
a/b
a/b ⋅ bd
ad
÷ =
=
=
.
b
d
bc
c /d
c/d ⋅ bd
1.8. SOLVED PROBLEMS
1.1
Write the sum S, difference D, product P, and quotient Q of each of the following pairs of numbers:
a. 48, 12;
b. 8, 0;
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c. 0, 12;
d. 10, 20;
e. 0, 0.
SOLUTION
a. S = 48 + 12 = 60, D = 48 − 12 = 36, P = 48(12) = 576,
Q = 48 ÷ 12 =
48
=4
12
b. S = 8 + 0 = 8,D = 8 − 0 = 8,P = 8(0) = 0, Q = 8 ÷ 0 or 8/0
But by definition 8/0 is that number x (if it exists) such that x(0) = 8. Clearly there is no such number, since any number
multiplied by 0 must yield 0.
c. S = 0 + 12 = 12, D = 0 − 12 = −12, P = 0(12) = 0,
Q=
0
=0
12
d. S = 10 + 20 = 30, D = 10 − 20 = −10, P = 10(20) = 200,
Q = 10 ÷ 20 =
10
1
=
20
2
e. S = 0 + 0 = 0,D = 0 − 0 = 0,P = 0(0) = 0, Q = 0 ÷ 0 or 0/0 is by definition that numberx (if it exists) such that x(0) = 0.
Since this is true for all numbers x there is no one number which 0/0 represents.
From (b) and (e) it is seen that division by zero is an undefined operation.
1.2
Perform each of the indicated operations.
a. 42 + 23, 23 + 42
b. 27 + (48 + 12), (27 + 48) + 12
c. 125 − (38 + 27)
d. 6 · 8, 8 · 6
e. 4(7 · 6), (4 · 7)6
f. 35 · 28
g. 756 ÷ 21
h.
(40 + 21)(72 − 38)
(32 − 15)
i. 72 ÷ 24 + 64 ÷ 16
j. 4 ÷ 2 + 6 ÷ 3 − 2 ÷ 2 + 3 · 4
k. 128 ÷ (2 · 4), (128 ÷ 2) · 4
SOLUTION
a. 42 + 23 = 65, 23 + 42 = 65. Thus 42 + 23 = 23 + 42.
This illustrates the commutative law for addition.
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b. 27 + (48 + 12) = 27 + 60 = 87, (27 + 48) + 12 = 75 + 12 = 87. Thus 27 + (48 + 12) = (27 + 48) + 12.
This illustrates the associative law for addition.
c. 125 − (38 + 27) = 125 − 65 = 60
d. 6 · 8 = 48, 8 · 6 = 48. Thus 6 · 8 = 8 · 6, illustrating the commutative law for multiplication.
e. 4(7 · 6) = 4(42) = 168, (4 · 7)6 = (28)6 = 168. Thus 4(7 · 6) = (4 · 7)6.
This illustrates the associative law for multiplication.
f. (35)(28) = 35(20 + 8) = 35(20) + 35(8) = 700 + 280 = 980 by the distributive law for multiplication.
g.
h.
756
= 36 Check: 21 · 36 = 756
21
2
(40 + 21)(72 − 38)
(61)(34) 61 ⋅ 34
=
=
= 61 ⋅ 2 = 122
17
(32 − 15)
17
1
i. Computations in arithmetic, by convention, obey the following rule: Operations of multiplication and division precede
operations of addition and subtraction.
Thus 72 ÷ 24 + 64 ÷ 16 = 3 + 4 = 7.
j. The rule of (i) is applied here. Thus 4 ÷ 2 + 6 ÷ 3 − 2 ÷ 2 + 3 · 4 = 2 + 2 − 1 + 12 = 15.
k. 128 ÷ (2 · 4) = 128 ÷ 8 = 16, (128 ÷ 2) · 4 = 64 · 4 = 256
Hence if one wrote 128 ÷ 2 · 4 without parentheses we would do the operations of multiplication and division in the
order they occur from left to right, so 128 ÷ 2 · 4 = 64 · 4 = 256.
1.3 Classify each of the following numbers according to the categories: real number, positive integer, negative integer,
rational number, irrational number, none of the foregoing.
−5, 3/5, 3π, 2,−1/4, 6.3, 0, √5, √−1, 0.3782, √4,−18/7
SOLUTION
If the number belongs to one or more categories this is indicated by a check mark.
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Real number
Positive integer
Negative integer
√
Rational number
−5
√
3/5
√
3π
√
2
√
−1/4
√
√
6.3
√
√
0
√
√
√5
√
Irrational number
√
√
√
√
√
√
√−1
√
0.3782
√
√4
√
−18/7
√
1.4
None of foregoing
√
√
√
√
Represent (approximately) by a point on a graphical scale each of the real numbers in Problem 1.3.
Note: 3π is approximately 3(3.14) = 9.42, so that the corresponding point is between +9 and +10 as indicated.
√5 is between 2 and 3, its value to three decimal places being 2.236.
1.5
Place an appropriate inequality symbol (< or >) between each pair of real numbers.
a. 2, 5
b. 0, 2
c. 3, −1
d. −4, +2
e. −4, −3
f. π, 3
g.
√7, 3
h.
−√2,−1
i. −3/5, −1/2
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SOLUTION
a. 2 < 5 (or 5 > 2), i.e., 2 isless than 5 (or 5 is greater than 2)
b. 0 < 2 (or 2 > 0)
c. 3 > −1 (or −1 < 3)
d. −4 < +2 (or +2 > −4)
e. −4 < −3 (or −3 > −4)
f. π > 3 (or 3 < π)
g.
3 > √7 (or √7 < 3)
h.
−√2 < −1 (−1 > −√2)
i. −3/5 < −1/2 since −.6 < −.5
1.6
Arrange each of the following groups of real numbers in ascending order of magnitude.
a. −3, 22/7,
√5, −3.2, 0
b.
−√2,−√3, −1.6, −3/2
SOLUTION
a.
−3.2 < −3 < 0 < √5 < 22/7
b.
−√3 < −1.6 < −3/2 < −√2
1.7
Write the absolute value of each of the following real numbers.
−1,+3, 2/5,−√2,−3.14, 2.83,−3/8,−π,+5/7
SOLUTION
We may write the absolute values of these numbers as
|−1|, |+3|, |2/5|, ∣∣−√2∣∣ , |−3.14|, |2.83|, |−3/8|, |−π|, |+5/7|
which in turn may be written 1, 3, 2/5,
√2, 3.14, 2.83, 3/8, π, 5/7 respectively.
1.8
The following illustrate addition and subtraction of real numbers.
a. (−3) + (−8) = −11
b. (−2) + 3 = 1
c. (−6) + 3 = −3
d. −2 + 5 = 3
e. −15 + 8 = −7
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f. (−32) + 48 + (−10) = 6
g. 50 − 23 − 27 = 0
h. −3 − (−4) = −3 + 4 = 1
i. −(−14) + (−2) = 14 − 2 = 12
1.9
Write the sum S, difference D, product P, and quotient Q of each of the following pairs of real numbers:
a. −2, 2;
b. −3, 6;
c. 0, −5;
d. −5, 0
SOLUTION
a. S = −2 + 2 = 0, D = (−2) − 2 = −4, P = (−2)(2) = −4, Q = −2/2 = −1
b. S = (−3) + 6 = 3, D = (−3) − 6 = −9, P = (−3)(6) = −18, Q = −3/6 = −1/2
c. S = 0 + (−5) = −5, D = 0 − (−5) = 5,P = (0)(−5) = 0, Q = 0/−5 = 0
d. S = (−5) + 0 = −5, D = (−5) − 0 = −5, P = (−5)(0) = 0, Q = −5/0 (an undefined operation, so it is not a number).
1.10
Perform the indicated operations.
(5)(−3)(−2) = [(5)(−3)](−2) = (−15)(−2) = 30
= (5)[(−3)(−2)] = (5)(6)
= 30
a.
The arrangement of the factors of a product does not affect the result.
b. 8(−3)(10) = −240
8(−2) (−4)(−2)
−16 8
+
+ = 4+4 = 8
=
−4
2
−4
2
12(−40)(−12)
12(−40)(−12)
12(−40)(−12)
d.
=
=
= −960
−6
−15 − (−9)
5(−3) − 3(−3)
c.
1.11
Evaluate the following.
a.
23 = 2 ⋅ 2 ⋅ 2 = 8
b.
5(3)2 = 5 ⋅ 3 ⋅ 3 = 45
c.
24 ⋅ 26 = 24+6 = 210 = 1024
d.
25 ⋅ 52 = (32)(25) = 800
e.
34 ⋅ 33
f.
52 ⋅ 53
3
5
2
=
7
=
2
3
( 3)
37
3
2
5
7
55
= 37− 2 = 35 = 243
=
5
1
7−5
=
1
52
=
1
25
6
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2
g.
(23) = 23⋅2 = 26 = 64
h.
2 4 24
16
( ) =
=
4
3
81
3
3
4
i.
(34) ⋅ (32)
j.
38
15
(−3) ⋅ 3
35
1.12
−
4
42 ⋅ 24
26
=
312 ⋅ 38
15
−3 ⋅ 3
4
=−
+ 3(−2)3 = 33 −
320
3
19
42
22
= −31 = −3
+ 3(−8) = 27 − 4 − 24 = −1
Write each of the following fractions as an equivalent fraction having the indicated denominator.
a. 1/3; 6
b. 3/4; 20
c. 5/8; 48
d. −3/7; 63
e. −12/5; 75
SOLUTION
a. To obtain the denominator 6, multiply numerator and denominator of the fraction 1/3 by 2.
Then
1
1 2
2
= ⋅ = .
3
3 2
6
3
3⋅5
15
b.
=
=
4
4⋅5
20
5
5⋅6
30
c.
=
=
8
8⋅6
48
3
3⋅9
27
d. − = −
=−
7
7⋅9
63
12
12 ⋅ 15
180
e. −
=−
=−
5
5 ⋅ 15
75
1.13
Find the sum S, difference D, product P, and quotient Q of each of the following pairs of rational numbers:
a. 1/3, 1/6;
b. 2/5, 3/4;
c. −4/15, −11/24.
SOLUTION
a. 1/3 may be written as the equivalent fraction 2/6.
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1 1
2 1
3
1
+ = + = =
3 6
6 6
6
2
1 1
2 1
1
D= − = − =
3 6
6 6
6
S=
1
1
1
P = ( )( ) =
3
6
18
1/3
1 6
6
Q=
= ⋅ = =2
3 1
3
1/6
b. 2/5 and 3/4 may be expressed with denominator 20: 2/5 = 8/20, 3/4 = 15/20.
2 3
8
15
23
+ =
+
=
5 4
20 20
20
2 3
8
15
7
−
D= − =
=−
5 4
20 20
20
S=
2
3
6
3
P = ( )( ) =
=
5
4
20
10
2/5
2 4
8
Q=
= ⋅ =
5 3
15
3/4
c. −4/15 and −11/24 have a least common denominator 120: −4/15 = −32/120, −11/24 = −55/120.
4
11
32
55
87
29
) + (− ) = −
−
=−
=−
15
24
120 120
120
40
4
11
32
55
23
+
D = (− ) − (− ) = −
=
15
24
120 120
120
S = (−
1.14
4
11
11
) (− ) =
15
24
90
−4/15
4
24
32
Q=
= (− ) (− ) =
15
11
55
−11/24
P = (−
Evaluate the following expressions, given x = 2, y = −3, z = 5, a = 1/2, b = −2/3.
a. 2x + y = 2(2) + (−3) = 4 − 3 = 1
b. 3x − 2y − 4z = 3(2) − 2(−3) − 4(5) = 6 + 6 − 20 = −8
c. 4x2y = 4(2)2(−3) = 4 · 4 · (−3) = −48
d.
23 + 4(−3)
x3 + 4y
4
8 − 12
=
=
=−
3
2a − 3b
1+2
2(1/2) − 3(−2/3)
e.
−2/3
x 2
b
2 2
2 2
4
4
64
4 64
68
( ) − 3( ) = (
) − 3(
) = (− ) − 3(− ) = − 3 (− ) = +
=
y
a
−3
3
3
9
27
9
9
9
1/2
3
3
3
See Chapter 34 for more solved problems and more supplementary problems.
1.9. SUPPLEMENTARY PROBLEMS
1.15
Write the sum S, difference D, product P, and quotient Q of each of the following pairs of numbers:
a. 54, 18;
b. 4, 0;
c. 0, 4;
d. 12, 24;
e. 50, 75.
1.16
Perform each of the indicated operations.
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a. 38 + 57, 57 + 38
b. 15 + (33 + 8), (15 + 33) + 8
c. (23 + 64) − (41 + 12)
d. 12 · 8, 8 · 12
e. 6(4 · 8), (6 · 4)8
f. 42 · 68
g. 1296 ÷ 36
h.
(35 − 23)(28 + 17)
43 − 25
i. 45 ÷ 15 + 84 ÷ 12
j. 10 ÷ 5 − 4 ÷ 2 + 15 ÷ 3 + 2 · 5
k. 112 ÷ (4 · 7), (112 ÷ 4) · 7
l.
15 + 3 ⋅ 2
9−4÷2
1.17
Place an appropriate inequality symbol (< or >) between each of the following pairs of real numbers.
a. 4, 3
b. −2, 0
c. −1, 2
d. 3, −2
e. −8, −7
f.
1, √2
g.
−3, − √11
h.
−1/3, − 2/5
1.18
a.
Arrange each of the following groups of real numbers in ascending order of magnitude.
−√3, −2,
√6, −2.8, 4, 7/2
b. 2π, −6,
√8, −3π, 4.8, 19/3
1.19
Write the absolute value of each of the following real numbers: 2, −3/2,
−√6, +3.14, 0, 5/3,
√4, −0.001,
−π − 1.
1.20
Evaluate.
a. 6 + 5
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b. (−4) + (−6)
c. (−4) + 3
d. 6 + (−4)
e. −8 + 4
f. −4 + 8
g. (−18) + (−3) + 22
h. 40 − 12 + 4
i. −12 − (−8)
j. −(−16) − (−12) + (−5) − 15
1.21
Write the sum S, difference D, product P, and quotient Q of each of the following pairs of real numbers:
a. 12, 4;
b. −6, −3;
c. −8, 4;
d. 0, −4;
e. 3, −2.
1.22
Perform the indicated operations.
a. (−3)(2)(−6)
b. (6)(−8)(−2)
c. 4(−1)(5) + (−3)(2)(−4)
d.
(−4)(6) (−16)(−9)
+
−3
12
e. (−8) ÷ (−4) + (−3)(2)
f.
(−3)(8)(−2)
(−4)(−6) − (2)(−12)
1.23
Evaluate.
a. 33
b. 3(4)2
c. 24 · 23
d. 42 · 32
e.
56 ⋅ 53
f.
34 ⋅ 38
55
36 ⋅ 35
75
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g.
75
73 ⋅ 74
h. (32) 3
6
i.
j.
k.
l.
1
( ) ⋅ 25
2
(−2)3 ⋅ (2)3
2
3(22)
3(−3)2 + 4(−2)3
57
5
1.24
4
23 − 32
+
210
3
8 ⋅ (−2)
2
− 4(−3)4
Write each of the following fractions as an equivalent fraction having the indicated denominator.
a. 2/5; 15
b. −4/7; 28
c. 5/16; 64
d. −10/3; 42
e. 11/12; 132
f. 17/18; 90
1.25
Find the sum S, difference D, product P, and quotient Q of each of the following pairs of rational numbers:
a. 1/4, 3/8;
b. 1/3, 2/5;
c. −4, 2/3;
d. −2/3, −3/2.
1.26
Evaluate the following expressions, given x = −2, y = 4, z = 1/3, a = −1, b = 1/2.
a. 3x − 2y + 6z
b. 2xy + 6az
c. 4b2x3
d.
3y2 − 4x
ax + by
e.
x2y(x + y)
3x + 4y
f.
y 3
a 2 xy
( ) − 4( ) −
x
b
z2
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1.10. ANSWERS TO SUPPLEMENTARY PROBLEMS
1.15
a. S = 72, D = 36, P = 972, Q = 3
b. S = 4, D = 4, P = 0, Q undefined
c. S = 4, D = −4, P = 0, Q = 0
d. S = 36, D = −12, P = 288, Q = 1/2
e. S = 125, D = −25, P = 3750, Q = 2/3
1.16
a. 95, 95
b. 56, 56
c. 34
d. 96, 96
e. 192, 192
f. 2856
g. 36
h. 30
i. 10
j. 15
k. 4, 196
l. 3
1.17
a. 3 < 4 or 4 > 3
b. −2 < 0 or 0 > −2
c. −1 < 2 or 2 > −1
d. −2 < 3 or 3 > −2
e. −8 < −7 or −7 > −8
f.
1 < √2 or √2 > 1
g.
−√11 < −3 or − 3 > −√11
h. −2/5 < −1/3 or −1/3 > −2/5
1.18
a.
−2.8 < −2 < −√3 < √6 < 7/2 < 4
−3π < −6 < √8 < 4.8 < 2π < 19/3
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b.
−3π < −6 < √8 < 4.8 < 2π < 19/3
1.19
2, 3/2,
√6, 3.14, 0, 5/3,
√4, 0.001, π + 1
1.20
a. 11
b. −10
c. −1
d. 2
e. −4
f. 4
g. 1
h. 32
i. −4
j. 8
1.21
a. S = 16, D = 8, P = 48, Q = 3
b. S = −9, D = −3, P = 18, Q = 2
c. S = −4, D = −12, P = −32, Q = −2
d. S = −4, D = 4, P = 0, Q = 0
e. S = 1, D = 5, P = −6, Q = −3/2
1.22
a. 36
b. 96
c. 4
d. 20
e. −4
f. 1
1.23
a. 27
b. 48
c. 128
d. 144
4
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e. 54 = 625
f. 3
g. 1/49
h. 36 = 729
i. 1/2
j. −4/3
k. 5
l. −201
1.24
a. 6/15
b. −16/28
c. 20/64
d. −140/42
e. 121/132
f. 85/90
1.25
a. S = 5/8, D = −1/8, P = 3/32, Q = 2/3
b. S = 11/15, D = −1/15, P = 2/15, Q = 5/6
c. S = −10/3, D = −14/3, P = −8/3, Q = −6
d. S = −13/6, D = 5/6, P = 1, Q = 4/9
1.26
a. −12
b. −18
c. −8
d. 14
e. 16/5
f. 48
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