

Click or touch headings below:
− Numbers and Sets
Sets: 
ℕ_{0} = Natural Numbers: {0,1,2,3,...} 
ℕ_{1} = Natural Numbers: {1,2,3,...} 
ℤ = Integers: {...,3,2,1,0,1,2,3,...} 
ℚ = Rational Numbers: any value expressed as a ratio a/b where a and b are any integers 
ℝ = Real Numbers: any value that represents a quantity along a continuous line 
ℂ = Complex Numbers: any quantity in the form a+bi where a and b are real numbers and i is the imaginary constant 
∅ = the Empty Set with no elements: {} 
U = the Universal Set (all possible elements) 

Set Operators: 
A ∈ B → A is an element of set B 
A ∉ B → A is not an element of set B 
A ∪ B → union of sets A and B (all elments of both) 
A ∩ B → intersection of sets A and B (only elments in common) 
A ⊂ B → A is a proper subset of B (A ≠ B) 
A ⊆ B → A is a subset of B 
2^{A} → power set of A (set of all subsets) 

Constants: 
π = 3.141592653589793... ratio of circumference to diameter of circle 
e = 2.718281828459045... Euler's number 
i = the imaginary constant 
φ = 1.61803398874989... golden ratio 

Operator Precedence: 
 () → Operations inside parentheses
 −x → Negative numbers [−4^{2} = (−4)^{2}]
 x^{n} → Exponents
 xy → Terms like 2π [2π/4i = (2π)/(4i)]
 × / → Multiply/Divide
 + − → Add/Subtract

− Algebra
Relational Symbols: 
x = y → x is equal to y 
x ≠ y → x is not equal to y 
x <> y → x is not equal to y 
x ≈ y → x is approximately equal to y 
x > y → x is greater than y 
x ≫ y → x is much greater than y 
x ≥ y → x is greater than or equal to y 
x < y → x is less than y 
x ≪ y → x is much less than y 
x ≤ y → x is less than or equal to y 
x ∝ y → x is proportional to y 

Basic Algebra: 





(commutative) 
(commutative) 
(associative) 
(associative) 
(distributive) 









Algebraic Equivalences: 








− Functions
A function denoted as f(x) is a relation f between a set of inputs x and a set of permissible outputs with the property that each input is related to exactly one output. 

(even function) 
(odd function) 



(composition) 
(noncommutative) 
(inverse function f^{−1}) 

Common Functions: 
→ natural logarithm (base e) 
→ exponent e^{x} 
→ floor or nearest lesser integer 
→ ceiling or nearest greater integer 
→ real part of complex number 
→ imaginary part of complex number 
→ absolute value or distance from origin 
→ Gamma function = (x1)! 
− Exponents and Roots
Exponents: 
Exponents, also known as powers, are denoted with a superscript as in x^{n} which is described as 'x to the power n'.
This means 'x' multiplied by itself 'n' times.

(x squared) 
(x cubed) 











Roots: 
Roots are denoted with the radical sign √ and are the result of taking fractional exponents. 
(square root) 

(cube root) 

(n^{th} root) 



− Logarithms
A logarithm is denoted log_{n} x where n is called the base and is the inverse of exponent. 



(natural log) 
(common log) 





− Complex Numbers
Imaginary numbers are denoted xi where i is the imaginary constant and i^{2}=−1.
Complex numbers are denoted a+bi or a−bi; where i is the imaginary constant. This is usually called rectagular form.
Complex and imaginary numbers can result from operations that have no realnumber solutions such as square roots or logarithms of negative numbers.
Complex numbers are typically represented as points on a plane where the xaxis is the real part and the yaxis is imaginary. 








(conjugate) 
(absolute value) 

Alternatively, complex numbers can be represented in polar form denoted r∠θ
where r (modulus or amplitude) is the magnitude and θ (argument or phase) is the angle with respect to the real axis within the complex plane.
If z = a+bi then the polar form r∠θ is: 





+ Trigonometric Functions
− Trigonometric Functions
Trigonometric functions are defined as the ratios between two sides of a right triangle, denoted a, b, or c at a specific primary angle θ.
All trig functions can be referenced to a unit circle with radius 1, where the origin is defined as the center of the circle, the right triangle hypotenuse as the radius,
the base as the xaxis, and the primary angle as the angle between the xaxis and the hypotenuse.



Basic Identities: 
(sine;cosine) 
(tangent;cotangent) 
(secant;cosecant) 

Pythagorean Identities: 
(Pythagorean Theorem) 




Reciprocal Identities: 




Quotient Identities: 


Inverse Functions: 
Inverse trig functions can be denoted using any of the following forms with the example of (sin θ): 
(asin θ) or (sin^{1} θ) or (arcsin θ) 

Where: 
; −π/2 ≤ θ ≤ π/2 radians 
; 0 ≤ θ ≤ π radians 
; −π/2 ≤ θ ≤ π/2 radians 
; 0 ≤ θ ≤ π radians 
; 0 ≤ θ ≤ π radians 
; −π/2 ≤ θ ≤ π/2 radians 

Exponential Forms(i^{2}=1): 
(Euler's formula) 
(Euler's identity) 













Law of Cosines 

where for any triangle: 
a and b = adjacent sides 
θ = angle between a and b 
c = opposite side of angle θ 

Law of Sines 

where for any triangle: 
A, B, and C are opposite angles of sides a, b, and c 

Angles and Units: 
360 degrees = 2π radians = 400 grads 
Degrees 
Radians 
Grads 
30 
π/6 
33 1/3 
45 
π/4 
50 
60 
π/3 
66 2/3 
90 
π/2 
100 
120 
2π/3 
133 1/3 
135 
3π/4 
150 
150 
5π/6 
166 2/3 
180 
π 
200 

− Hyperbolic Functions
Hyperbolic functions are defined as analogs of trignometric functions except they are applied to hyperbolas instead of unit circles.
As a result, they exhibit similar identities.
Hyperbolic functions use the familiar trigonometric names with an 'h' suffix such as sinh, cosh, tanh. 

Exponential Definitions: 







Inverse Hyperbolic Functions: 
Where: 
; −∞ < x < ∞ 
; 1 ≤ x < ∞ 
; 1 ≤ x < ∞ 
; 0 ≤ x ≤ 1 
; −∞ < x < ∞ 
; −1 ≤ x ≤ 1 

Definitions: 







Identities: 













Relation to Trig Functions: 













− Series
Summation: 
Sums of discrete series are denoted using a capital Sigma (Σ), the lower expression defining the index variable and lower bound, the upper expression defining upper bound of integer values for the index. 
(definition) 







Products: 
Products of discrete series are denoted using a capital Pi (Π), the lower expression defining the index variable and lower bound, the upper expression defining upper bound of integer values for the index. 
(definition) 
(factorial) 



Taylor Series: 
The Taylor Series is a method that allows any differentiable function to be expressed as a series.
In many cases the Taylor series can have an infinite number of terms and the function can only be approximated via calculation.
Expressed as a sum, the Taylor series is denoted:

 where n! denotes the factorial of n and ƒ^{(n)}(a) denotes the n^{th} derivative of function ƒ evaluated at point a. 

Maclaurin Series: 
The Maclaurin Series is a special case of the Taylor Series that is evaluated at zero where the function has a solution at zero.
This case significantly simplifies the construction of the polynomial series.
A Maclaurin series can be constructed for many functions, which an infinite summation in the form:

 where n! denotes the factorial of n and ƒ^{{n}}(0) denotes the n^{th} derivative of function ƒ evaluated at 0. 

Common Maclaurin Series: 
for x < 1 (geometric series) 







− Statistics
Standard Deviation: 
In statistics or probability theory, standard deviation represented by lower case sigma (σ), shows how much variation exists in a data set from the average or mean value.
A small standard deviation indicates that the data points tend to be very close to the mean; a large standard deviation indicates that the data points are spread out over a wide range.





σ = std. deviation of population S = std. deviation of sample x_{i} = sample variable μ = arithmetic mean of population X = arithmetic mean of sample SE = Standard Error 
Normal Distribution: 


LogNormal Distribution: 


σ = std. deviation x = random variable μ = population mean CDF = Cumulative Distribution Function PDF = Probability Distribution Function 
− Calculus
Limits: 
A limit is the output value that a function or sequence approaches, or converges to, as the input approaches some specific value.
Limits are most useful when the input value approaches infinity, or approaches a dividebyzero, and the limit is a finite quantity.


This example shows a limit when approaching a dividebyzero condition: 

This example shows a limit when the input value approaches infinity: 

Limit of sum/difference: 

Limit of product: 

Limit of quotient: 

Multiplication by constant: 


Derivatives: 
A derivative can be described as the slope of a point on a continuous function.
Derivatives can represent instantaneous rates of change in any quantity that is described by a continuous function.
For any function f(x)=y derivative is defined using the limit of the ratio of the change in y over the change in x (Δy/Δx) as Δx approaches zero.


The derivative of the function f(x) at the point x_{0} is denoted: 

Second derivative: 

Derivative of Sum/Difference: 

Derivative of Product: 

Derivative of Quotient: 

Chain Rule: 


Integrals: 
An integral can be described as the area under a continuous curve on a graph.
Integrals can represent the sum of all values over any quantity that is described by a continuous function.
The integral is the inverse function of the derivative (antiderivative) as described by the Fundamental Theorem of Calculus.


Where: F'(x) = f(x) 
 for a continuous function f(x), the indefinite integral is denoted: 

 for a continuous function f(x) over interval [a,b], the definite integral is denoted: 


Fundamental Theorem of Calculus: 



Conventions: 


If a ≤ b ≤ c : 


Integral of Sum/Difference: 

Multiplication by a constant: 


− Table of Derivatives
Basics: 







Trigonometric: 






Inverse Trigonometric: 






Hyperbolic: 






Inverse Hyperbolic: 






− Table of Integrals
(Constant of integration ...+C assumed) 
Basics: 





Rational Functions: 







Exponential: 




Logarithmic: 




Trigonometric: 






Inverse Trigonometric: 



Hyperbolic: 






Inverse Hyperbolic: 





