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IMAGINARY NUMBERS
Real numbers include both rational numbers, such
as 42 and −23/129, and irrational numbers, such as
pi and the square root of two. A real number can be
given by an infinite decimal representation, such as
2.4871773339..., where the digits continue in some
way. Real numbers may be thought of as points on an
infinitely long number line.
Imaginary numbers are in the form "bi" where "b" is
a non-zero, real number and "i" is defined by i² =
-1, called the "imaginary unit". An Imaginary
number bi can be added to a real number a, to form a
"complex number" of the form (a + bi), where a and b
are called respectively, the "real part" and the
"imaginary part" of the complex number. Imaginary
numbers can therefore be thought of as complex
numbers where the real part is zero. The square of
an imaginary number is a negative real number.
The complex conjugate of a complex number has the
reverse sign of the original imaginary part. The
complex conjugate of (a + bi) is (a - bi).
Real and imaginary parts are additive: (a + bi) + (c
+di) = (a + c) + (b + d)i, and are multiplicative:
(a + bi) * (c + di) = ac + bci +adi + bdi2
= (ac - bd) + (bc + ad)i.
Division is accomplished by multiplying the
numerator and denominator by the complex conjugate
of the denominator:
BASIC ALGEBRA
 Elementary
algebra is one of the main branches of pure mathematics and
introduces the concept of variables representing numbers. Statements
based on these variables are manipulated using the rules of
operations that apply to numbers, such as addition. This can be done
for a variety of reasons, including equation solving.
While the word
algebra comes from the Arabic languageand much of its methods from
Arabic/Islamic mathematics, its roots can be traced to earlier
traditions, most notably ancient Indian mathematics.
A polynomial is an expression that is constructed
from one or more variables and constants, using only
the operations of addition, subtraction,
multiplication, and exponentiation. For example, x2 + 2x − 3 is a polynomial in
the single variable x.
Factoring of polynomials is the process of
expressing a given polynomial as a product of other
polynomials. The example polynomial above can be
factored as (x − 1)(x + 3).
A
quadratic equation is one in which a term is raised
to the power of 2 and other terms are raised to a
lower power or a are constants:
x2 + 2x − 3 =0 is a
quadradic equation.
A
root, or zero, of a polynomial in a single
variable is the value that of x that forces the
polynomial to produce a result equal to zero. In the
example above, x = 1 and x = -3 are roots or zeros
of the polynomial
x2 + 2x − 3 =
(x
− 1)(x + 3) = 0.
All real polynomials of odd degree have a real
number as a root. Many real polynomials of even
degree do not have a real root.
A
polynomial of degree n will have n roots. Roots are
not always obvious. The roots of a quadratic
equation are found with:
Roots of higher order equations are solved by
graphing or iterative "solver" software.
Linear simultaneous equations - see
Cramer's Rule.
Statistical analysis methods - see
Statistics
BASIC GEOMETRY
 Plane eometry is the study of angles and triangles, perimeter, area and
volume. It differs from algebra in that it develops a logical
structure where mathematical relationships are proved and applied.
Euclid (c. 300 BCE) introduced certain axioms, or postulates,
expressing primary or self-evident properties of points, lines, and
planes. He proceeded to rigorously deduce other properties by
mathematical reasoning. Some of these are listed below.
A point shows position.
A line is
infinite and straight and is a set of continuous points that
extend indefinitely in either of its direction. A line segment is
part of the straight line between two points.
Parallel lines do not cross.
A ray is the part of the line which consists of a given point and
the set of all points on one side of that point.
A circle is a set of points
equidistant from another point, known as the center of the circle.
An angle can be defined as two rays or two line segments having a
common end point. The endpoint becomes known as the vertex.
An angle occurs when two rays meet or unite at the same
endpoint.
A plane is a flat surface containing three points that are not all
in a straight line.
The intersection of two plones is a
line.
Rectangle: A = a * b
Parallelogram: A = b * h
Trapezoid: A = h * (b1 + b2) / 2
Circle:
A = pi * r 2
Ellipse:
A = pi * r1 * r2
Triangle:
A = 0.5 * b * h 
Equilateral triangle A = 0.25 *
 (3) * a2
Triangle given SAS: A = 0.5 * a * b * sin C
Triangle given a,b,c: A =
[s
* (s - a) * (s - b) * (s - c)] when s = (a + b + c) / 2
Regular
polygon: A = 0.5 * n * sin(360°/n) * S2
where: n = # of sides and S = length from center to a corner
Volumes
Cube: V
= a3 
Rectangular prism: V = a * b * c
Irregular prism: V = b * h
Cylinder:
V = b * h = pi * r2
* h 
Pyramid: V = 0.5 * b * h
Cone: V
= (1/3) * b * h = (1/3) * pi * r2 * h
Sphere:
V = (4/3) * pi * r3
Ellipsoid: V= (4/3)
* pi * r1 * r2 *
r3 
Surface Areas
Cube:
S = 6 * a2
Prism:
Lateral Area S = perimeter(b) * L
Total Area S = perimeter(b) * L + 2 * b
Sphere:
S = 4 * pi * r2
BASIC TRIGONOMETRY
Trigonometry
is a branch of mathematics that studies triangles on a plane
surfaces and deals with relationships between the sides and the
angles of triangles and with the trigonometric functions, which
describe those relationship.
Basic Conversions
PI = 3.141 592... (approximately 22/7 = 3.1428)
radians = degress x PI / 180
degress = radians x 180 / PI
e = 2.718
282....
Trigonometry Functions
 The
trigonometry functions are defined by a right triangle with angle
theta, adjacent side b, oposite side a, and hypotenuse c.
sin(theta) = a / c
csc(theta) = 1 / sin(theta) = c / a
cos(theta) = b / c
sec(theta) = 1 / cos(theta) = c / b
tan(theta) = sin(theta) / cos(theta) = a / b
cot(theta) = 1/ tan(theta) = b / a
USEFUL EQUALITIES
sin^2(x) + cos^2(x) = 1
tan^2(x) + 1 = sec^2(x)
cot^2(x) + 1 = csc^2(x)
sin(x
 y)
= sin x cos y
cos
x sin y
cos(x
y)
= cos x cos y
 sin
x sin y
sin(-x) = -sin(x)
csc(-x) = -csc(x)
cos(-x) = cos(x)
sec(-x) = sec(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)
tan(x
y)
= (tan x
tan
y) / (1
tan
x tan y)
sin(2x) = 2 sin x cos x
cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x)
- 1 = 1 - 2 sin^2(x)
tan(2x) = 2 tan(x) / (1 - tan^2(x))
sin^2(x) = 1/2 - 1/2 cos(2x)
cos^2(x) = 1/2 + 1/2 cos(2x)
sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )
cos x - cos y = -2 sin( (x-y)/2 ) sin( (x + y)/2 )
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Table of Common
Angles
|
| Angle |
0 |
30 |
45 |
60 |
90 |
| sin^2(a)
|
0/4 |
1/4 |
2/4 |
3/4 |
4/4 |
| cos^2(a)
|
4/4 |
3/4 |
2/4 |
1/4 |
0/4 |
| tan^2(a)
|
0/4 |
1/3 |
2/2 |
3/1 |
4/0 |
Given
Triangle abc, with angles A,B,C; a is opposite to A, b oppositite B,
c opposite C:
Then:
a / sin(A) = b / sin(B) = c / sin(C)
c^2 =
a^2 + b^2 - 2ab cos(C) Law of Sines
b^2 = a^2
+ c^2 - 2ac cos(B)
a^2 = b^2
+ c^2 - 2bc cos(A) Law of Cosines
(a - b)/(a + b) = tan
1/2(A-B) / tan 1/2(A+B) (Law of
Tangents)
Hyperbolic Definitions
sinh(x) = ( ex - e-x )/2
csch(x) = 1/sinh(x) = 2/( ex - e-x )
cosh(x) = ( e x + e -x )/2
sech(x) = 1/cosh(x) = 2/( ex + e-x )
tanh(x) = sinh(x)/cosh(x) = ( ex - e-x )/( ex
+ e-x )
coth(x) = 1/tanh(x) = ( ex + e-x)/( ex
- e-x )
cosh2(x) - sinh2(x)
= 1
tanh2(x) + sech2(x) = 1
coth2(x) - csch2(x) = 1
Inverse Hyperbolic Defintions
arcsinh(z) = ln( z +
(z2
+ 1) )
arccosh(z) = ln( z
(z2
- 1) )
arctanh(z) = 1/2 ln( (1+z)/(1-z) )
arccsch(z) = ln( (1+(1+z2)
)/z )
arcsech(z) = ln( (1(1-z2)
)/z )
arccoth(z) = 1/2 ln( (z+1)/(z-1) )
Relations to Trigonometric Functions
sinh(z) = -i sin(iz)
csch(z) = i csc(iz)
cosh(z) = cos(iz)
sech(z) = sec(iz)
tanh(z) = -i tan(iz)
coth(z) = i cot(iz)
Spherical Trigonometry
Spherical trigonometry is a
branch of spherical geometry which deals with polygons (especially
triangles) on the sphere and the relationships between the sides and
the angles. This is of great importance for calculations in
astronomy and earth-surface, orbital and space navigation.
On the surface of a sphere, the
closest analogue to straight lines are great circles, i.e. circles
whose center coincide with the center of the sphere. As with a line
segment in a plane, an arc of a great circle (subtending less than
180°) on a sphere is the shortest path lying on the sphere between
its two endpoints. An area on the sphere, bounded by arcs of great
circles, is called a spherical polygon.
The sides of these polygons are
specified not by their lengths, but by the angles at the sphere's
center subtended by the endpoints of the sides. Note that this arc
angle, measured in radians, when multiplied by the sphere's radius
equals the arc length.
 The
sum of the vertex angles of spherical triangles is always larger
than the sum of the angles of plane triangles, which is exactly
180°. The amount E by which the sum of the angles exceeds 180° is
called spherical excess:
E = A + B + C - pi
where A, B, and C denote the
angles.
This surplus determines the surface area of any spherical triangle:
A = R2 * E
where R is the radius of the
sphere.
Identities
Click Here (mathworld.wolfram.com)
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