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My math file...
This file describes my understanding of math.
Math as I understand it consits of three different ways of an understanding the math of things.
In math there is the graphical, the algebraic and the numeric ways to understand the math.
The graphical way be used to understand the math of something using a picture.
The algebraic uses numbers, identifiers and operators to produce a writing that
explains the math of something in an exact way.
The numeric is a table of numbers used to understand the math of something.
[ALGEBRA]
ADDITIVE INVERSE
a + b = b + a
MULTIPLICATIVE INVERSE
a * b = b * a
ACTIONS USED TO SOLVE EQUATIONS
a - b = x
x + b = a
a + b = y
y - a = b
y - b = a
root = N-th root (like square root with N = 2)...
xy = z
root(xy, y) = root(z, y)
x = root(z, y)
x2 = y
root(x2, 2) = root(y, 2)
x = root(y, 2)
root(x, 2) = y
root(x, 2)2 = y2
x = y2
x2 + 1 / x = y
x3 + x * (1 / x) = x * y
x3 + 1 = x * y
x3 + 1 = y * x
x3 = y * x - 1
x2 = y - 1
square_root(x2) = square_root(y - 1)
x = square_root(y - 1)
a * a = a2
x3 * x2 = x5
x + 3 * x = y
4 * x = y
x = y / 4
a * b = c
a = c / b
a / b = c
a * (1 / b) = c
(1 / b) = c / a
(1 / b)1/2 = (c / a)1/2
b = a / c
1/a = b
(1/a)(1/2) = (b)(1/2)
a = b(1/2)
a = 1/b
x + y = z + w
x = z + w - y
x2 = y
square_root(x2) = square_root(y)
x = square_root(y)
square_root(x) = y
square_root(x)2 = y2
x = y2
(a + b) / (c + d) = (x + y) / (z + v)
a / (c + d) + b / (c + d) = x / (z + v) + y / (z + v)
a * 1 / (c + b) + b * 1 / (c + d) = x * (1 / (z + v)) + y * (1 / (z + v))
a * 1 / (c + b) = x * (1 / (z + v)) + y * (1 / (z + v)) - b * (1 / (c + d))
a = (x * (1 / (z + v)) + y * (1 / (z + v)) - b * (1 / (c + d))) / (1 / (c + b)))
z = (1 / (a / b))
z = (b / a)
2 + a = b
a = b - 2
2 * a = b
a = b / 2
y = Ak
ln Ak = ln y
k * ln A = ln y
k = ln y / ln A
log x * y = log x + log y
log x / y = log x - log y
log xy = y * log x
log square_root(x) = log x1/y = (1/y) * log x
ln x * y = ln x + ln y
ln x / y = ln x - ln y
ln xy = y * ln x
ln square_root(x) = ln x1/y = (1/y) * ln x
FACTORIAL
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
TRIGONOMETRY
y = sin(x)
asin(y) = asin(sin(x))
x = asin(y)
[PROBABILITY AND STATISTICS]
COUNTING PROBLEMS
n = a number of choices possable
r = the number of choices made
n = 10
r = 5
4 counting problems
- order does count with replacement = rn
- order does count without replacement = n * (n - 1) * (n - 2) * (n - 3) * (n - 4)
- order does not count with replacement = nCr = n! / (r! * (n - r)!)
- order does not count without replacement = ((n - 1 + r)!) / ((n-1)! * r!)
PROBLABLILITY FUNCTION
number between (0 and 1.0) used like percentage value (0% and 100%)
educated guess (find probability value)
counting equasions used to find a probability value
a probability function that returns a value between 0 and 1.0
[INEQUALITY]
= a is equal to b
/= a is not equal to b
< a is less then b
<= a is less then or equal to b
> a is grater then b
>= a is grater then or equal to b
a + b < c + d
a < c + d - b
a * b < c * d
a < (c * d) / b
(a / b) < (c / d)
(a * (1 / b)) < (c / d)
a < ((c / d) / (1 / b))
[BASIC FUNCTIONS]
f(x) = a function
C = a constant value
f(x) = C
f(x) = x
f(x) = xC
f(x) = Cx
f(x) = xx
f(x) = square_root(x)
f(x) = root(x,n)
f(x) = sin(x)
f(x) = cos(x)
f(x) = tan(x)
f(x) = ln x
f(x) = ex
f(x) = x / C
f(x) = 1 / x
f(x) = C * x
f(x) = C + x
f(x) = logC x
f(x) =
[COMPOSITE FUNCTION]
A composit function is a composed of a function that
gets passed another function as a parameter to the function.
Another way to view this is as a substitution that expands to
a more complex function.
C = a constant value
p(x) = a function
q(x) = a function
f(p(x), q(x)) = a composite function made from the functions p(x) and q(x)
f(p(x), q(x)) = p(x) * q(x)
f(p(x), q(x)) = p(x) / q(x)
f(p(x), q(x)) = p(x) + q(x)
f(p(x), q(x)) = p(x) - q(x)
f(p(x)) = p(x) * C
f(p(x)) = p(x) / C
f(p(x)) = p(x) + C
f(p(x)) = p(x) - C
f(p(x)) = C / p(x)
f(p(x)) = C - p(x)
f(p(x)) = square_root(p(x))
f(p(x)) = root(p(x), C)
f(p(x)) = log(p(x))
f(p(x)) = ln(p(x))
f(p(x)) = sin(p(x))
f(p(x)) = cos(p(x))
f(p(x)) = xp(x)
f(p(x)) = p(x)C
f(p(x), y) = p(x)y
f(p(x), y) = yp(x)
f(p(x), y) = p(x) * y
f(p(x), y) = p(x) / y
f(p(x), y) = p(x) - y
f(p(x), y) = p(x) + y
f(p(x), y) = root(p(x), y)
[INFINITY]
C = a constant value
C + INFINITY = INFINITY
C - INFINITY = - INFINITY
C * INFINITY = INFINITY
C / INFINITY = INFINITY
INFINITY / C = INFINITY
INFINITY - C = INFINITY
-1 * INFINITY = - INFINITY
INFINITY + INFINITY = INFINITY
INFINITY - INFINITY = undefined
INFINITY * INFINITY = INFINITY
INFINITY / INFINITY = undefined
(1/2) * INFINITY = INFINITY
1 / INFINITY = 0
[BOOLEAN ALGEBRA]
AND
TRUE AND TRUE = TRUE
FALSE AND FALSE = FALSE
TRUE AND FALSE = FALSE
FALSE AND TRUE = FALSE
OR
TRUE OR TRUE = TRUE
TRUE OR FALSE = TRUE
FALSE OR TRUE = TRUE
FALSE OR FALSE = FALE
NOT
NOT TRUE = FALSE
NOT FALSE = TRUE
NOT NOT TRUE = TRUE
NOT NOT FALSE = FALSE
EXCLUSIVE OR
TRUE XOR TRUE = FALSE
TRUE XOR FALSE = TRUE
FALSE XOR TRUE = TRUE
FALSE XOR FALSE = FALSE
[2D GEOMETRY]
2D CARTESIAN PLOT
SQUARE
RECTANGLE
PARALLELOGRAM
CIRCLE
ELLIPSE
POLYGON
LINES THAT ARE PERPONDICULAR
LINES THAT ARE PARALLEL
RISE / RUN = SLOPE OF A LINE
TRIANGLES
a2 + b2 = c2
EQUILATERAL TRIANGLE
RIGHT TRIANGLE
ISOSCELES TRIANGLE
[3D GEOMETRY]
3D CARTESIAN PLOT
SPHEAR
CUBE
BOX
CYLINDER
CONE
PYRAMID
[TRIGONOMETRY]
TRIGONOMETRY FUNCTIONS
TRIGONOMETRY IDENTITIES
[SOUND]
5000 cycles per second cps (Hz)
q(t) = A0 + sum(Ai * sin(t * wi + pi), i, 0, N-1)
t = time
A0 = air pressure
Ai = amplitude
pi = phase shift
wi = period
[CALCULOUS]
The calculous is defined as a table for making computations.
SUMMATION
sum(2x, x, a, b) = 2a + 2a+1 + ... + 2b
This is the summation of each term with a and b as the begining and ending values for
LIMIT
lim(f(x), x, INFINITY) = the value of the function f(x) as x approaches INFINITY
THE Nth TERM
f(x) = 1*x1 + 2*x2 + 3*x3
The Nth term of f(x) is = N * xN
FUNDAMENTAL THERUM OF CALCULOUS
f(x) = a function
F(x) = the derivative of the function f(x)
integral(f(x), a, b) = (F(b) - F(a))
THE DERIVATIVE
derivative(xn) = n * xn - 1
derivative(ln(x)) = -1 / x
derivative(sin(x)) = cos(x)
derivative(cos(x)) = -sin(x)
derivative(tan(x)) = -ln(|cos(x)|)
derivative(sqrt(x)) = -(1/(2*sqrt(x)))
THE ANTI-DERIVATIVE
antiderivative(sin(x)) = -cos(x)
AREA UNDER A CURVE
The area under a curve is the integral of the curve function.
f(t) = speed traveling given time
d = integral(f(t), t, a, b)
F(t) = is the derivative of f(t)
d = F(b) - F(a)
d = the distance travled if f(t) is the speed you are taveling from time point a to time point b on the t-axis.
SLOPE LINE OF ANOTHER LINE IS ...
DIFFERNETIAL EQUASIONS
dy/dx = 20mph / 10mph
this is acceleration
change in x over a change in y
MULTI-VARIABLE FUNCTIONS
z = f(x, y) = x * y
z = f(x, y) = x / y
z = f(x, y) = x + y
z = f(x, y) = x - y
z = f(x, y) = xy
z = f(x, y) = C1 * x + C2 * y
z = f(x, y) = C1 * xC2 + C3 * yC4
[LINEAR ALGEBRA OR MATRIX ALGEBRA]
A matrix is a rectagular array of numbers or math expressions.
MATRIX MULTIPLY
A * B /= B * A
A =
| a | b |
| c | d |
B =
| f | g |
| p | q |
A * B =
| a*f + b*p | a*g + a*q |
| c*f + d*p | c*g + d*q |
MATRIX INVERSE
A =
| a | b |
| c | d |
B =
| d | -b |
| -c | a |
A-1 = the inverse of the matix A
A-1 = (1 / det(A)) * B
UNDO A LINEAR TRANSFORMATION WITH AN INVERSE MATRIX
A * v = c
A-1 * c = v
DETERMINENT OF A MTRIX A FOR R2
A =
| a | b |
| c | d |
det(A) = a * d - b * c
DETERMINENT OF A MTRIX A FOR R3
A =
| a11 | a12 | a13 |
| a21 | a22 | a23 |
| a31 | a32 | a33 |
det(A) = a11*a22*a33 + a12*a23*a31 + a13*a21*a32 - a13*a22*a31 - a12*a21*a33 - a11*a23*a32
IF THE DETERMINENT OF A MATRIX LARGER THEN 3 * 3 IS NEEDED
-
Find the determinents of 2 * 2 or 3 * 3 matrices that fit into the larger matrix
with no overlaping and no matrix values left out of the computation. - Make a new matrix with the small matrix determinent values as the values of the matrix.
- Do this action until you are left with one value in a 1 * 1 matrix.
- This value of this computation is the determenent of the original matrix
TRANSPOSE OF A MATRIX
A =
| 1 | 2 | 3 |
| 4 | 5 | 6 |
| 7 | 8 | 9 |
AT =
| 9 | 8 | 7 |
| 4 | 5 | 6 |
| 1 | 2 | 3 |
GOUS JORDAN ROW REDUCTION ALGORITHM FOR SOLVING A SYSTEM OF LINEAR EQUATIONS
To solve, a linear system of equations with 3 equations and 3 unknowns
c1 = c4 * x + c5 * y + c6 * z
c2 = c7 * x + c8 * y + c9 * z
c3 = c10 * x + c11 * y + c12 * z
Use the coefficients to the equisons as a matrix.
A =
| x | y | z | intersection |
| c4 | c5 | c6 | c1 |
| c7 | c8 | c9 | c2 |
| c10 | c11 | c12 | c3 |
To solve, the system of equasions you can:
- Add or subtract a value to a row
- Multiply or divide a value by a row
- Add or subtract a multiple of a row with a row
- Multiply or divide a multiple of a row by a row
- Swap two rows
Repeat the above until the matrix of coefficients are in reduced row echelon format.
The result is a matrix in the form:
B =
| x | y | z | intersection |
| 1 | 0 | 0 | a |
| 0 | 1 | 0 | b |
| 0 | 0 | 1 | c |
Where the vector (a, b, c) is the solution to the intersection of the lines in the system
of lenear equison.
The result is a vector or vector space
v =
| a |
| b |
| c |
If and only if the columns x, y, z are ones and in echelon form.
Some of the compents of the vector can be a variable yelding a vector space for intersection.
rref(A) = v
rref stands for row reduced echelon form.
3 linear equisons with 3 variables (lines) can intersect at a point, a line or a plane.
THE ADJUNCT OF A MATRIX
A =
| a | b |
| c | d |
adjunct(A) =
| 0 | b |
| c | 0 |
VECTOR
A vector is a line from origin to a point denoting distance and direction.
v = a vector
v = (1, 2, 3)
VECTOR SPACE
A vector space is a set of infinitely meny vectors.
A vector space is offten represented by the variable s.
The numbers in each vector are called the components.
A vector space is can be an algebraic expression for the vector componenents.
s = (x, y, z)
VECTOR SET
A vector set is a finite number of vectors.
A table of values could be used to represent a vector set.
s = ((1, 2, 3), (4, 5, 6) ... (N1, N2, N3))
MATRICES AND VECOTR SPACES
A matrix can be used as an operator on a vector space.
A * s = s2
The basic operation on a vector space using matrix multiply are:
tralations, rotations, scaleing and projections.
To multiply, a matrix by a vector space you convert a vector
to a collum matrix with the first value the top value of the collum matrix and the last value
the bottom value of the collum matrix. The inbetween values are in order as they are found
in the vector from left to right. Then the n * m matrix is multiplyed by the n * 1 matrix.
ROTATION OPERATOR
ROTATION MATRIX FOR R2 =
| cos(angle) | -sin(angle) |
| sin(angle) | cos(angle) |
ROTATION MATRIX FOR R3 COUNTERCLOCKWISE ABOUT THE X-AXIS =
| 1 | 0 | 0 |
| 0 | cos(angle) | -sin(angle) |
| 0 | sin(angle) | cos(angle) |
ROTATION MATRIX FOR R3 COUNTERCLOCKWISE ABOUT THE Y-AXIS =
| cos(angle) | 0 | sin(angle) |
| 0 | 1 | 0 |
| -sin(angle) | 0 | cos(angle) |
ROTATION MATRIX FOR R3 COUNTERCLOCKWISE ABOUT THE Z-AXIS =
| cos(angle) | -sin(angle) | 0 |
| sin(angle) | cos(angle) | 0 |
| 0 | 0 | 1 |
TRANSLATE A VECTOR
translate vector v2 x offset it's current value resulting in vector v3
v1 =
| x |
| 0 |
| 0 |
v2 =
| a |
| b |
| c |
v3 =
| x+a |
| 0+b |
| 0+c |
v1 + v2 = v3
SCALE A VECTOR
scale vector v by a factor of 2 using matrix A
v =
| a |
| b |
A =
| 2 | 0 |
| 0 | 2 |
A * v =
| 2*a + 0*b |
| 0*a + 2*b |
PROJECT A VECTOR
v = a vector in R3
v =
| a |
| b |
| c |
A =
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 0 |
vproj = A * v = c
c = the projection of vector v
c =
| a |
| b |
| 0 |
TRACE OF A MATRIX
A =
| a | b |
| c | d |
trace(A) =
| 0 | b |
| c | 0 |
LINEAR TRANSFORMATION
A matrix that is an operator on a vector space
THE KERNEL
A * s = c
A = a linear transformation matrix
s = a vector space
c = a vector space as the result of the linear transformation
All the vectors in vector space s that map onto the zero vector
are the kernel of the linear transformation A on the vector space s.
The kernel is written as:
kern(A*s)
EQULIDEAN N SPACE RN
The number of vector componenets in a vector for a given vector space..
THE UNIT VECTORS FOR R3
u1 = (1, 0, 0)
u2 = (0, 1, 0)
u3 = (0, 0, 1)
VECTOR CROSS PRODUCT
how to find a vector that is perpendicual to two give vectors
u = a vector in R3
v = a vector in R3
u = (u1, u2, u3)
v = (v1, v2, v3)
u x v = ( u2*v3 - u3*v2, u3*v1 - u1*v3, u1*v2 - u2*v1 )
VECTOR DOT PRODUCT
how to find the angle between two vectors u and v
u = a vector
v = a vector
angle = angle between the two vectors u and v
u . v = ||u|| * ||v|| * cos(angle)
if u /= 0 and v /= 0
u . v = 0
if u = 0 or v = 0
DISTANCE FORMULA FROM THE ORIGIN TO A VECTOR
distance(x,y,z) = (1 / (square_root(x2 + y2 + z2)))
EIGENVECTORS
EIGENVALUES
ORTHOGONAL
In the context of matrix algebra a matrix row and collum corradenets match up to yeld a value
in a matrix making a matrix orthogonal.
[APPLICATIONS]
CALCULOUS OPTIMIZATION PROBLEMS
INSTANTANEOUS COMPOUNDING INTREST COMPUTATION
PROBABILITY FUNCTION EDUCATED GUESS
3D VIDEO GAMES
Matrix.java - my matrix object written in the java programming language.
Progject6.java - this is Java source code.
DTMF DETECT AND GENERATE
FAST FOURIER TRANSFER
LEAST SQUARES ERROR REDUCTION
TAYLOR POLYNOMIAL LINE APPROXIMATION
[CRYPTOGRAPHY]
p = plain text message
c = cypher text of the plain text message
MATRIX MULTIPLY TO ENCRYPT
A * p = c
A-1 * c = p
HillHack.java - this is the hack
Matrix.java
P = plain text 2x2 matrix
C = cypher text 2x2 matrix
R = CT|PT
R = rref(R)
H = right(R)
H = HT
H = the decrypt key
MULTIPLY AND THEN ADD TO ENCRYPT
p * k1 + k2 = c
(c - k2) / k1 = p
lin_reg.cpp - this is the hack
[PERFECT NUMBERS]
[COMPLEX NUMBERS]
[SET MATH AND LOGIC]
A = a set
A = (1, 2, 3)
B = a set
B = (3, 4, 5)
SET UNION
A OR B = (1, 2, 3, 4, 5)
SET INTERSECTION
A AND B = (3)
SET DIFFERENCE
A - B = (1, 2)
MAP ONE SET ONTO ANOTHER SET
A = a set
B = a set
A -> B
ONE TO ONE MAPING
Each element of a set B is maped onto by one value in set A
ONE TO MENY MAPING
Each element of a set A is maped onto meny values in set B
MENY TO ONE MAPING
Meny element of a set B are maped onto by meny values in set A
MENY TO MENY MAPING
Meny elements of a set A map onto a value or values in set B
ONTO MAPING
Every element of a set A map onto a value or values in set B
NO MAPING
No maping form A to B
CONDITIONAL
a if b
If a is exists then we learn that b exists.
If b is exists then we learn that a exists.
BICONDITIONAL
a if and only if b
a exists only if b already exists.
SYLLOGISM
if <condition>
therefore <statement>
If the <condition> is meet you learn the <statement>.
Where <condition> and <statement> are logic statements.
Statements about a broad set offten do not hold.
FALCES ARE LOGIC STATEMENTS THAT REJECT OTHER STATEMENTS
- BROKEN LOGIC
- ILL CONTRARY
- OVERLY NEGATIVE IN A SENTENCE
- HASTY JUDGEMENT
- OVERLY COMPLEX STATEMENT
- UN-JEWISH OR UN-CHRISTIAN ARGUMENT
- AMBIGUOUS STATEMENT
- UNCLEAR STATEMENT
- UN-STOIC ARUGUMENT
- UN-LOGOS ARUGUMENT
- ETHNOCENTRIC ARUGUMENT
- ANTISEMITIC ARUGUMENT