Mathematics

Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns,formulate new conjectures, and establish truthby rigorous deduction from appropriately chosen axioms and definitions.
Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapesand motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world in AD 800, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.
There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions "ALBERT EINSTEIN on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine and the social sciences ,applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.
Algebric basic Identities:

Closure Property of Addition

Sum (or difference) of 2 real numbers equals a real number
Additive Identity

a + 0 = a
Additive Inverse

a + (-a) = 0
Associative of Addition

(a + b) + c = a + (b + c)
Commutative of Addition

a + b = b + a
Definition of Subtraction

a - b = a + (-b)

Closure Property of Multiplication

Product (or quotient if denominator 0) of 2 reals equals a real number
Multiplicative Identity

a * 1 = a
Multiplicative Inverse

a * (1/a) = 1 (a 0)
(Multiplication times 0)

a * 0 = 0
Associative of Multiplication

(a * b) * c = a * (b * c)
Commutative of Multiplication

a * b = b * a
Distributive Law

a(b + c) = ab + ac
Definition of Division

a / b = a(1/b)


Polynomial Identities
(Math | Algebra | Polynomials)


(a+b) 2 = a 2 + 2ab + b 2

(a+b)(c+d) = ac + ad + bc + bd

a 2 - b 2 = (a+b)(a-b) (Difference of squares)

a 3 b 3 = (a b)(a 2 ab + b 2) (Sum and Difference of Cubes)

x 2 + (a+b)x + AB = (x + a)(x + b)

if ax 2 + bx + c = 0 then x = ( -b (b 2 - 4ac) ) / 2a (Quadratic Formula)


Exponential Identities
(Math | Algebra | Exponents)
Powers

x a x b = x (a + b)
x a y a = (xy) a

(x a) b = x (ab)

x (a/b) = bth root of (x a) = ( bth (x) ) a

x (-a) = 1 / x a

x (a - b) = x a / x b



Logarithms

y = logb(x) if and only if x=b y
logb(1) = 0

logb(b) = 1

logb(x*y) = logb(x) + logb(y)

logb(x/y) = logb(x) - logb(y)

logb(x n) = n logb(x)

logb(x) = logb(c) * logc(x) = logc(x) / logc(b)

Functions
(Math | Algebra | Functions)

Synonyms: correspondence, mapping, transformation
Definition: A function is a relation from a domain set to a range set, where each element of the domain set is related to exactly one element of the range set.

An equivalent definition: A function (f) is a relation from a set A to a set B (denoted f: A�B), such that for each element in the domain of A (Dom(A)), the f-relative set of A (f(A)) contains exactly one element.

Some common functions (with discussions)

trig functions
sine, sin(x)
cosine, cos(x)

Trigonometric Identities
(Math | Trig | Identities)

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
sin(-x) = -sin(x)
csc(-x) = -csc(x)
cos(-x) = cos(x)
sec(-x) = sec(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)
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 cosy sin x sin y
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 )

Given Triangle abc, with angles A,B,C; a is opposite to A, b opposite B, c opposite C:
a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)

c^2 = a^2 + b^2 - 2ab cos(C)
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 [(A-B)/2] / tan [(A+B)/2] (Law of Tangents)


Hyperbolic Trigonometric Identities
(Math | Trig | Hyperbolas)


Hyperbolic Definitions

sinh(x) = ( e x - e -x )/2
csch(x) = 1/sinh(x) = 2/( e x - e -x )

cosh(x) = ( e x + e -x )/2

sech(x) = 1/cosh(x) = 2/( e x + e -x )

tanh(x) = sinh(x)/cosh(x) = ( e x - e -x )/( e x + e -x )

coth(x) = 1/tanh(x) = ( e x + e -x)/( e x - e -x )



cosh 2(x) - sinh 2(x) = 1

tanh 2(x) + sech 2(x) = 1

coth 2(x) - csch 2(x) = 1



Inverse Hyperbolic Definitions

arcsinh(z) = ln( z + (z 2 + 1) )
arccosh(z) = ln( z (z 2 - 1) )

arctanh(z) = 1/2 ln( (1+z)/(1-z) )

arccsch(z) = ln( (1+(1+z 2) )/z )

arcsech(z) = ln( (1(1-z 2) )/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)
Calculus
Integration:

Power of x.
integration of x^n dx = x^(n+1) / (n+1) + C
(n not equal to -1) Proof 1/x dx = ln|x| + C

Exponential/logarithm.
integration of bx dx = bx / ln(b) + C
integration of ln(x) dx = x ln(x) - x + C
integration of sin x dx = -cos x + C
integration of COs x dx = sin x + C
integration of tan x dx = -ln|COs x| + C
integration of csc x dx = - ln|CSC x + cot x| + C
integration of sec x dx = ln|sec x + tan x| + C
integration of cot x dx = ln|sin x| + C
integration of arcsin x dx = x arcsin x + (1-x2) + C
integration of arccsc x dx = x arccos x - (1-x2) + C
integration of arctan x dx = x arctan x - (1/2) ln(1+x2) + C
integration of sinh x dx = cosh x + C
integration of csch x dx = ln |tanh(x/2)| + C
integration of cosh x dx = sinh x + C
integration of sech x dx = arctan (sinh x) + C
integration of tanh x dx = ln (cosh x) + C
integration of coth x dx = ln |sinh x| + C