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1Elementary Algebra and Geometry
1.1 Fundamental Properties (Real Numbers)
a + b = b + a Commutative Law for Addition
(a + b) + c = a + (b + c) Associative Law for Addition
a + 0 = 0 + a Identity Law for Addition
a + (−a) = (−a) + a = 0 Inverse Law for Addition
a(bc) = (ab)c Associative Law for Multiplication
aa a
a a1 1
1 0æèç
öø÷ = æ
èç
öø÷ = ¹, Inverse Law for
Multiplication
(a)(1) = (1)(a) = a Identity Law for Multiplication
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2 Integrals and Mathematical Formulas
ab = ba Commutative Law for Multiplication
a(b + c) = ab + ac Distributive Law
Division by zero is not defined.
1.2 Exponents
For integers m and n,
a a a
a a a
a a
ab a b
a b a b
n m n m
n m n m
n m nm
m m m
m m m
=
=
=
=
=
+
-/
/ /
( )
( )
( )
1.3 Fractional Exponents
a ap q q p/ /( )= 1
where a1/q is the positive qth root of a if a > 0 and the negative qth root of a if a is negative and q is odd. Accordingly, the five rules of exponents given above (for integers) are also valid if m and n are fractions, provided a and b are positive.
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3Elementary Algebra and Geometry
1.4 Irrational Exponents
If an exponent is irrational, e.g., 2 , the quantity, such as a 2 , is the limit of the sequence a1.4, a1.41, a1.414, ….
• Operations with Zero
0 0 10m a= =;
1.5 Logarithms
If x, y, and b are positive and b ≠ 1,
log log log
log / log log
log log
log
b b b
b b b
bp
b
b
xy x y
x y x y
x p x
( )
( )
= +
= -
=
(( )
.log
1
1
1 0
/ log
log
log :
x x
b
Note b x
b
b
bxb
= -
=
= =
• Change of Base (a ≠ 1)
log log logb a bx x a=
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4 Integrals and Mathematical Formulas
1.6 Factorials
The factorial of a positive integer n is the product of all the positive integers less than or equal to the integer n and is denoted n! Thus,
n n! .= × × ×¼×1 2 3
Factorial 0 is defined 0! = 1.
• Stirling’s Approximation
lim /n
nn e n n®¥
=( ) !2p
(See also Section 9.2.)
1.7 Binomial Theorem
For positive integer n,
( )( )
!
( )( )!
x y x nx yn n
x y
n n nx y
nxy
n n n n
n
n
+ = + + -
+ - - +
+
- -
-
1 2 2
3 3
12
1 23
�
-- +1 yn.
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5Elementary Algebra and Geometry
1.8 Factors and Expansion
( )
( )
( )
(
a b
a b
a b
a b
a ab b
a ab b
a a b ab b
+ =
- = -
+ =
-
+ +
+
+ + +
2
2
3
2 2
2 2
3 2 2 3
2
2
3 3
))
( ) ( )( )
( ) ( )(
3
2 2
3 3 2
3 2 2 33 3= - -
- = - +
- = - + +
+a a b ab b
a b a b a b
a b a b a ab b22
3 3 2 2
)
( ) ( )( )a b a b a ab b+ = + - +
1.9 Progression
An arithmetic progression is a sequence in which the difference between any term and the preced-ing term is a constant (d):
a a d a d a n d, , , , ( ) .+ + + -2 1…
If the last term is denoted l[= a + (n−1)d], then the sum is
s
na l= +
2( ).
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6 Integrals and Mathematical Formulas
A geometric progression is a sequence in which the ratio of any term to the preceding terms is a con-stant r. Thus, for n terms,
a ar ar arn, , ,,2 1… -
The sum is
S
a arr
n
= --1
1.10 Complex Numbers
A complex number is an ordered pair of real numbers (a, b).
Equality: (a, b) = (c, d) if and only if a = c and b = d
Addition: (a, b) + (c, d) = (a + c, b + d)
Multiplication: (a, b)(c, d) = (ac −.bd, ad + bc)
The first element of (a, b) is called the real part; the second, the imaginary part. An alternate notation for (a, b) is a + bi, where i2 = (−1, 0), and i(0, 1) or 0 + 1i is written for this complex number as a con-venience. With this understanding, i behaves as a number, i.e., (2 −.3i)(4 + i) = 8 −.12i + 2i −.3i2 = 11 −.10i. The conjugate of a + bi is a −.bi, and the product of a complex number and its conjugate is a2 + b2.
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7Elementary Algebra and Geometry
Thus, quotients are computed by multiplying numerator and denominator by the conjugate of the denominator, as illustrated below:
2 34 2
4 2 2 34 2 4 2
14 820
7 410
++
=- +- +
=+
=+i
ii ii i
i i( )( )( )( )
1.11 Polar Form
The complex number x + iy may be represented by a plane vector with components x and y:
x iy r i+ = +( )cos sinq q
(see Figure 1.1). Then, given two complex num-bers z1 = r1(cosθ1 + i sinθ1) and z2 = r2(cosθ2 + i sinθ2), the product and quotient are:
Product: z1z2 = r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)]
Quotient: z z r r i1 2 1 2 1 2 1 2/ / cos sin= - + -( )[ ( ) ( )]q q q q
Powers: z r i
r n i n
n n
n
= +( )éë ùû= +éë ùû
cos sin
sin
q q
q qcos
Roots: z r i
rkn
ikn
n
n
1 1
1 360 360
/ /
/
cos sin
. .
= +( )éë ùû
= + + +éëê
ù
q q
q q
n
cos sinûûú
= ¼
,
, , , ,k 0 1 2 1n -
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8 Integrals and Mathematical Formulas
1.12 Permutations
A permutation is an ordered arrangement (sequence) of all or part of a set of objects. The number of permutations of n objects taken r at a time is
p n r n n n n r
nn r
( , ) ( )( ) ( )
!( )!
= - - - +
=-
1 2 1�
A permutation of positive integers is even or odd if the total number of inversions is an even
0x
y
r
P (x, y)
θ
FIGURE 1.1Polar form of complex number.
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9Elementary Algebra and Geometry
integer or an odd integer, respectively. Inversions are counted relative to each integer j in the per-mutation by counting the number of integers that follow j and are less than j. These are summed to give the total number of inversions. For example, the permutation 4132 has four inversions: three relative to 4 and one relative to 3. This permuta-tion is therefore even.
1.13 Combinations
A combination is a selection of one or more objects from among a set of objects regardless of order. The number of combinations of n different objects taken r at a time is
C n r
P n rr
nr n r
( , )( , )
!!
!( )!= =
-
1.14 Algebraic Equations
• QuadraticIf ax2 + bx + c = 0, and a ≠ 0, then roots are
x
b b aca
= - ± -2 42
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10 Integrals and Mathematical Formulas
• CubicTo solve x3 + bx2 + cx + d = 0, let x = y −.b/3. Then the reduced cubic is obtained:
y py q3 0+ + =
where p = c − (1/3)b2 and q = d − (1/3)bc + (2/27)b3. Solutions of the original cubic are then in terms of the reduced cubic roots y1, y2, y3:
x y b x y b
x y b
1 1 2 2
3 3
1 3 1 3
1 3
= - = -
= -
( ) ( )
( )
/ /
/
The three roots of the reduced cubic are
y A B
y W A W B
y W A W B
11 3 1 3
21 3 2 1 3
32 1 3 1 3
= +
= +
= +
( ) ( )
( ) ( )
( ) ( )
/ /
/ /
/ /
where
A q p q= - + +1
21 27
14
3 2( ) ,/
B q p q= - - +1
21 27
14
3 2( ) ,/
W
iW
i= - + = - -1 32
1 32
2, .
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11Elementary Algebra and Geometry
When (1/27)p3 + (1/4)p2 is negative, A is complex; in this case, A should be expressed in trigono-metric form: A = r(cos θ + i sin θ), where θ is a first or second quadrant angle, as q is negative or positive. The three roots of the reduced cubic are
y r
y r
y r
11 3
21 3
1 3
2 3
23
120
23
= ( )
= ( ) + °æèç
öø÷
= ( )
/
/
/
( )cos /
cos
cos
q
q
q ++ °æèç
öø÷240
1.15 Geometry
Figures 1.2 through 1.12 are a collection of com-mon geometric figures. Area (A), volume (V), and other measurable features are indicated.
b
h
FIGURE 1.2Rectangle. A = bh.
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12 Integrals and Mathematical Formulas
b
h
FIGURE 1.3Parallelogram. A = bh.
h
b
FIGURE 1.4
Triangle. A bh=12
.
b
a
h
FIGURE 1.5
Trapezoid. A a b h= +12
( ) .
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13Elementary Algebra and Geometry
SR
θ
FIGURE 1.6Circle. A = πR2; circumference = 2πR; arc length S = Rθ (θ in radians).
θ
R
FIGURE 1.7
Sector of circle. A R A Rsector segment sin= = ( )12
12
2 2q q - q; .
R b
θ
FIGURE 1.8
Regular polygon of n sides. An
bn
Rb
n= =
4 22ctn csc
p p; .
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14 Integrals and Mathematical Formulas
h
R
FIGURE 1.9Right circular cylinder. V = πR2h; lateral surface area = 2πRh.
h
A
FIGURE 1.10Cylinder (or prism) with parallel bases. V = Ah.
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15Elementary Algebra and Geometry
R
FIGURE 1.12
Sphere. V R=43
3p ; surface area = 4πR2.
R
lh
FIGURE 1.11
Right circular cone. V R h=13
2p ; lateral surface area =
p pRl R R h= +2 2 .
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16 Integrals and Mathematical Formulas
1.16 Pythagorean Theorem
For any right triangle with perpendicular sides a and b, the hypotenuse c is related by the formula
c2 = a2 + b2
This famous result is central to many geometric relations, e.g., see Section 4.2.
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