Path: Math :
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 [PROBABILITY AND STATISTICS] COUNTING PROBLEMS n = a number of choices possable r = the number of choices made n = 10 r = 5 4 counting problems
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 = 0 INFINITY * INFINITY = INFINITY INFINITY / INFINITY = 1 (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 = TRUE 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 sin(x) = sin.c cos(x) = sin(x + (1/2 * PI)) tan(x) = tan.c asin(x) = asin.c acos(x) = acos.c atan(x) = atan.c cosh(x) = cosh.c sinh(x) = sinh.c tanh(x) = tanh.c acosh(x) = asinh(x) = atanh(x) = 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(square_root(x)) = ((2 * (x(3/2))) / 3) derivative(C * f(x)) = C * F(x) derivative(C) = C * x derivative(f(x) + g(x)) = F(x) + G(x) derivative(f(x) - g(x)) = F(x) - G(x) derivative(f(x) * g(x)) = f(x) * G(x) + F(x) * g(x) derivative(f(x) / g(x)) = (g(x) * F(x) - f(x) * G(x)) / (g(x))2 THE ANTI-DERIVATIVE antiderivative(1 / x) = ln x antiderivative(cos(x)) = sin(x) antiderivative(sin(x)) = cos(x) antiderivative(tan(x)) = (tan(x))2 + 1 antiderivative(x2) = 2 * x antiderivative(root(x)) = 1 / (2 * square_root(x)) antiderivative(ln(x)) = 1 / x antiderivative(log(x)) = 1 / (ln(10) * x) antiderivative(F(x) + G(x)) = f(x) + g(x) antiderivative(F(x) - G(x)) = f(x) - g(x) antiderivative(C * F(x)) = C * f(x) antiderivative(C * x) = C 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 =
B =
A * B =
MATRIX INVERSE A =
B =
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 =
det(A) = a * d - b * c DETERMINENT OF A MTRIX A FOR R3 A =
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
TRANSPOSE OF A MATRIX A =
AT =
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 =
To solve, the system of equasions you can:
Repeat the above until the matrix of coefficients are in reduced row echelon format. The result is a matrix in the form: B =
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 =
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 =
adjunct(A) =
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 =
ROTATION MATRIX FOR R3 COUNTERCLOCKWISE ABOUT THE X-AXIS =
ROTATION MATRIX FOR R3 COUNTERCLOCKWISE ABOUT THE Y-AXIS =
ROTATION MATRIX FOR R3 COUNTERCLOCKWISE ABOUT THE Z-AXIS =
TRANSLATE A VECTOR translate vector v2 x offset it's current value resulting in vector v3 v1 =
v2 =
v3 =
v1 + v2 = v3 SCALE A VECTOR scale vector v by a factor of 2 using matrix A v =
A =
A * v =
PROJECT A VECTOR v = a vector in R3 v =
A =
vproj = A * v = c c = the projection of vector v c =
TRACE OF A MATRIX A =
trace(A) =
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. 3D Java Applet - this is my 3D box animation as a Java Applet. 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 MATIX MULTIPLY TO ENCRYPT A * p = c A-1 * c = p ADD A TRUE RANDOM NUMBER TO ENCRYPT p + k = c c - k = p MULITIPLY AND THEN ADD TO ENCRYPT p * k1 + k2 = c (c - k2) / k1 = p [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
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
