− Numbers and Sets

Sets:

ℕ₀ = Natural Numbers: {0,1,2,3,...}

ℕ₁ = 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 [−4² = (−4)²]
xⁿ → 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)

(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)

(non-commutative)

(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 = (x-1)!

− Exponents and Roots

Exponents:

Exponents, also known as powers, are denoted with a superscript as in xⁿ 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)

− Complex Numbers

Imaginary numbers are denoted xi where i is the imaginary constant and i²=−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 real-number solutions such as square roots or logarithms of negative numbers. Complex numbers are typically represented as points on a plane where the x-axis is the real part and the y-axis 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 are defined as the ratios between two sides of a right triangle, denoted a and b called the legs, and c called the hypotenuse at a specific primary angle θ. All trig functions can be referenced to a unit circle with radius 1 on an X/Y axis, where the origin is defined as the center of the circle, the hypotenuse as the radius, the base as the x-axis, and the primary angle as the angle between the x-axis 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

; −π/2 ≤ θ ≤ π/2 radians

Exponential Forms(i²=-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 < ∞

; 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:

Log-Normal 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 divide-by-zero, and the limit is a finite quantity.

This example shows a limit when approaching a divide-by-zero 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 x₀ 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: