Math Formulas

Thinking Operations

General Information

$|a|$

Symbols, and relationships in mathematics

Number theory

$ℕ \subset ℤ \subset ℚ \subset ℝ \subset ℂ$

Number Sets

ℙ = {2, 3, 5, 7, \dots}

Prime Numbers - Prime Factorization

a | m \Leftrightarrow n \cdot a = m

Divisibility Rules

(m; n) = l; [m; n] = k

Greatest Common Divisor (factor) - Least Common Multiple

\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}

Operations with rational numbers

Binomial theorem

{(a + b)}^{n}

Binomial theorem

Combinatorics

P_{n} = n!

Permutations

C_{n}^{k} = (\binom{n}{k})

Combination

V_{n}^{k} = \frac{n!}{(n - k)!}

Variation

Sets

A \cup B; A \cap B; A B

Operations on Sets

A \cup B = B \cup A

Fundamental laws of set algebra

|A \cup B \cup C|

Cardinality of sets

A \times B

Cartesian product

ρ \subseteq A x A

Relation

Logic

P \lor Q; P \Rightarrow Q

Logical Operations and Truth Tables

a \lor b = b \lor a

Properties of Logical Operators

Graph Theory

Graph Theory - definitions, relationships

Algebra

Polynomials

{(a + b)}^{2}

Special Binomials

Progressions

a_{n} = a_{a} + (n - 1) d

Arithmetic progression

a_{n} = a_{1} \cdot q^{n - 1}

Geometric progression

Logarithm

l o g_{a} b = c \Leftrightarrow a^{c} = b

Logarithm

l o g_{a} x = \frac{l o g_{b} x}{l o g_{b} a}

Changing the base of a logarithm

Exponents

a \cdot a \cdot a = a^{3}

Powers

Roots

\sqrt[n]{a} = b \Rightarrow b^{n} = a

Roots

Proportionality

a : b = c : d

Direct and Inversely Proportion

Inequalities

f (x) < g (x)

Inequalities

Equations

a x^{2} + b x + c = 0

Quadratic equation

a x^{3} + b x^{2} + c x + d = 0

Cubic Equations

\sqrt[n]{x} = a

Irrational and transcendental equations

Complex numbers

z = a + i b; i = \sqrt{- 1}

Complex Numbers

z^{n}

Power of complex numbers

z_{k} = \sqrt[n]{a + i b}

Roots of complex numbers

Kamatni račun

k = T \cdot \frac{p}{100}

Interest calculation,

Geometry

Trigonometry

\frac{a}{s i n α} = \frac{b}{s i n β} = \frac{c}{s i n γ}

Law of Sines

$c^{2} = a^{2} + b^{2} - 2 a b {c o s}^{2} γ$

Law of Cosines

s i n^{2} α + c o s^{2} α = 1

Trygonometry Identities of same angle

s i n (α \pm β)

Identities for the sum and difference of two angles

s i n (2 α)

Trigonometric identities of double angles

s i n \frac{α}{2}

Trigonometric identities of half angles

s i n α \pm s i n β

Sum and difference of trigonometric functions

s i n α = \frac{a}{c}

Trigonometry Special Angles

Two-dimensional geometric shapes

\frac{\bar{B_{1} B_{2}}}{\bar{B_{2} B_{3}}} = \frac{\bar{A_{1} A_{2}}}{\bar{A_{2} A_{3}}}

Basic Proportionality Theorem

Triangle

Special Triangles - Right Triangle, Equilateral Triangles, Isosceles Triangles

Quadrilateral

Squere

Rectangle

Rhombus

Parallelogram

Kite

Trapezoid

Circle

Circular sector

Circular segment

Three-dimensional geometric shapes

Cube

Sphere

Cone

Prism

Pyramide

Platonic solids

Cylinder

Vectors

\vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} | c o s α

Scalar Product of Vectors

\vec{c} = \vec{a} \times \vec{b}

Vector Product of Vectors

Analytical Geometry 2D

d = \bar{| P_{1} P_{2} |}

Distance between two points

x^{2} + y^{2} = r^{2}

Circle

y = m x + b

Linear equation

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

Ellipse

Mathematical analysis

Important functions

y = a x + b

Linear polynomial function

y = a x^{2} + b x + c

Quatratic polynomial function

y = a x^{3} + b x^{2} + c x + d

Cubic polynomial function

y = \frac{1}{x}

Rational function

y = \sqrt[2 k]{x}, k \in ℕ

Root function with even radical

y = \sqrt[2 k + 1]{x}, k \in ℕ

Root function with odd radical

y = a^{x}

Exponential function

y = \log_{a} x

Logarithmic function

y = \sin x

Sine function

y = \cos x

Cosine function

y = tg x

Tangent function

y = ctg x

Cotangent function

y = | x |

Absolute value functions

Function Transformations

f (x + c)

Variable and function value transformations

Limits

\underset{x \to x_{0}}{l i m} f (x) = a

Limit of a function

Dervation

\frac{d f (x)}{d x} = f^{'} (x)

Table of Derivatives

{(f \cdot g)}^{'} = f^{'} \cdot g \pm f \cdot g^{'}

Differentiation rules

T (x)

Taylor Series

Integration

\int f (x) d x = F (x) + C

Table of Integrals

\int c f (x) d x = c \int f (x) d x

Integration Rules

\int_{a}^{b} f (x) d x = F (b) - F (a)

Definite Integrals

Probability and Statistics

Probability

\sum_{i = 1}^{n} A_{i} = Ω

Events, Operations with events, Complete system of events

P (A) = \frac{|A|}{|Ω|}

Probability of an event

P (A | B) = \frac{P (A \cap B)}{P (B)}

Conditional Probability

P (A) = \sum_{i = 1}^{n} P (A | B_{i}) \cdot P (B_{i})

Law of Total Probability

P (B_{i} | A) = \frac{P (A | B_{i}) \cdot P (B_{i})}{P (A)}

Bayes’ Theorem

Statistics

A = \frac{{\sum x_{i}}_{i = 1}^{n}}{n}