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Click or touch headings below:
− 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) |
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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 |
2A → power set of A (set of all subsets) |
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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 |
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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
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− 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 |
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Basic Algebra: |
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(commutative) |
(commutative) |
(associative) |
(associative) |
(distributive) |
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Algebraic Equivalences: |
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− 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. |
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(even function) |
(odd function) |
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(composition) |
(non-commutative) |
(inverse function f−1) |
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Common Functions: |
→ natural logarithm (base e) |
→ exponent ex |
→ 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.
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(x squared) |
(x cubed) |
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Roots: |
Roots are denoted with the radical sign √ and are the result of taking fractional exponents. |
(square root) |
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(cube root) |
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(nth root) |
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− Logarithms
A logarithm is denoted logn x where n is called the base and is the inverse of exponent. |
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(natural log) |
(common log) |
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− 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. |
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(conjugate) |
(absolute value) |
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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: |
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+ 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 x-axis, and the primary angle as the angle between the x-axis and the hypotenuse.
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Basic Identities: |
(sine;cosine) |
(tangent;cotangent) |
(secant;cosecant) |
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Pythagorean Identities: |
(Pythagorean Theorem) |
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Reciprocal Identities: |
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Quotient Identities: |
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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) |
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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 |
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Exponential Forms(i2=-1): |
(Euler's formula) |
(Euler's identity) |
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Law of Cosines |
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where for any triangle: |
a and b = adjacent sides |
θ = angle between a and b |
c = opposite side of angle θ |
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Law of Sines |
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where for any triangle: |
A, B, and C are opposite angles of sides a, b, and c |
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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 |
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− 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. |
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Exponential Definitions: |
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Inverse Hyperbolic Functions: |
Where: |
; −∞ < x < ∞ |
; 1 ≤ x < ∞ |
; 1 ≤ x < ∞ |
; 0 ≤ x ≤ 1 |
; −∞ < x < ∞ |
; −1 ≤ x ≤ 1 |
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Definitions: |
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Identities: |
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Relation to Trig Functions: |
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− 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) |
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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) |
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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:
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| where n! denotes the factorial of n and ƒ(n)(a) denotes the nth derivative of function ƒ evaluated at point a. |
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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:
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| where n! denotes the factorial of n and ƒ{n}(0) denotes the nth derivative of function ƒ evaluated at 0. |
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Common Maclaurin Series: |
for |x| < 1 (geometric series) |
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− 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.
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σ = std. deviation of population S = std. deviation of sample xi = sample variable μ = arithmetic mean of population X = arithmetic mean of sample SE = Standard Error |
Normal Distribution: |
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Log-Normal Distribution: |
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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.
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This example shows a limit when approaching a divide-by-zero condition: |
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This example shows a limit when the input value approaches infinity: |
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Limit of sum/difference: |
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Limit of product: |
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Limit of quotient: |
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Multiplication by constant: |
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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.
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The derivative of the function f(x) at the point x0 is denoted: |
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Second derivative: |
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Derivative of Sum/Difference: |
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Derivative of Product: |
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Derivative of Quotient: |
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Chain Rule: |
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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.
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Where: F'(x) = f(x) |
- for a continuous function f(x), the indefinite integral is denoted: |
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- for a continuous function f(x) over interval [a,b], the definite integral is denoted: |
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Fundamental Theorem of Calculus: |
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Conventions: |
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If a ≤ b ≤ c : |
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Integral of Sum/Difference: |
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Multiplication by a constant: |
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− Table of Derivatives
Basics: |
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Trigonometric: |
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Inverse Trigonometric: |
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Hyperbolic: |
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Inverse Hyperbolic: |
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− Table of Integrals
(Constant of integration ...+C assumed) |
Basics: |
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Rational Functions: |
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Exponential: |
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Logarithmic: |
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Trigonometric: |
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Inverse Trigonometric: |
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Hyperbolic: |
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Inverse Hyperbolic: |
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