Math Formulas

Thinking Operations

General Information

Symbols, relationships

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

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

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 of right triangles

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

Triangle

Special Triangles

Special Triangles - Right Triangle, Equilateral Triangles, Isosceles Triangles

Quadrilateral

Quadrilateral

Squere

Squere

Rectangle

Rectangle

Rhombus

Rhombus

Parallelogram

Parallelogram

Kite

Kite

Trapezoid

Trapezoid

Circle

Circle

Circular sector

Circular sector

Circular segment

Circular segment

Three-dimensional geometric shapes

Cube

Cube

Gömb

Sphere

Cone

Cone

Prism

Prism

Pyramide

Pyramide

Platonic solids

Platonic solids

Cylinder

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

ddxfx=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}

Measures of Central Tendency