Overview of Mathematics

Definition: The study of numbers, quantities, shapes, and patterns.
Branches:
- Arithmetic: Basic operations (addition, subtraction, multiplication, division).
- Algebra: Use of symbols and letters to represent numbers and quantities in formulas and equations.
- Geometry: Study of shapes, sizes, and properties of space.
- Trigonometry: Relationship between angles and sides of triangles.
- Calculus: Study of change, limits, derivatives, and integrals.
- Statistics: Collection, analysis, interpretation, and presentation of data.
- Probability: Measure of the likelihood of events occurring.

Numbers:
- Natural Numbers: Positive integers (1, 2, 3,...).
- Whole Numbers: Natural numbers including zero (0, 1, 2,...).
- Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2,...).
- Rational Numbers: Numbers that can be expressed as a fraction (e.g., 1/2, 0.75).
- Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., √2, π).
Fundamental Operations:
- Addition: Combining quantities.
- Subtraction: Finding the difference between quantities.
- Multiplication: Repeated addition.
- Division: Splitting into equal parts.

Variables: Symbols used to represent unknown values.
Equations: Mathematical statements asserting equality (e.g., 2x + 3 = 7).
Functions: Relationships that assign exactly one output for each input (e.g., f(x) = x²).

Shapes:
- Triangles: Sum of angles equals 180°.
- Quadrilaterals: Sum of angles equals 360°.
- Circles: Defined by radius, diameter, circumference, and area.
Theorems:
- Pythagorean Theorem: In a right triangle, a² + b² = c².

Functions: Sine, Cosine, Tangent and their relationships in right triangles.
Unit Circle: Circle with a radius of 1, used to define trigonometric functions.

Descriptive Statistics: Summarizes data (mean, median, mode).
Inferential Statistics: Draws conclusions from sample data (hypothesis testing).

Basic Concepts:
- Sample Space: Set of all possible outcomes.
- Event: A specific outcome or set of outcomes.
Rules:
- Addition Rule: For mutually exclusive events.
- Multiplication Rule: For independent events.

Logical Reasoning: Use of logic to deduce conclusions.
Proofs: Demonstrations that certain statements are true based on axioms and theorems.

Mathematics explores numbers, quantities, shapes, and the patterns that connect them across various branches.
Branches:
- Arithmetic involves basic operations such as addition, subtraction, multiplication, and division.
- Algebra utilizes symbols and letters to express numbers and relationships through equations and formulas.
- Geometry focuses on the properties and spatial relations of shapes and sizes.
- Trigonometry examines the angles and sides of triangles, providing essential tools for various applications.
- Calculus analyzes change and motion through concepts of limits, derivatives, and integrals.
- Statistics deals with data collection, analysis, interpretation, and presentation.
- Probability quantifies the chances of specific events occurring.

Numbers:
- Natural numbers are the set of positive integers starting from 1.
- Whole numbers include natural numbers and zero, ranging from 0 upwards.
- Integers comprise all positive and negative whole numbers, including zero.
- Rational numbers can be represented as fractions, such as 1/2 or 0.75.
- Irrational numbers cannot be expressed as simple fractions and include values like √2 and π.
Fundamental Operations:
- Addition combines two or more quantities.
- Subtraction determines the difference between values.
- Multiplication is repeated addition.
- Division divides quantities into equal parts.

Variables are symbols representing unknown values, crucial in forming equations.
Equations assert that two expressions are equal (e.g., 2x + 3 = 7).
Functions define a relationship giving one output for each input, exemplified by f(x) = x².

Shapes:
- In triangles, the sum of internal angles is always 180°.
- Quadrilaterals have internal angles that add up to 360°.
- Circles are defined by their radius, diameter, circumference, and area.
Theorems:
- The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a² + b² = c²).

Functions such as sine, cosine, and tangent relate the angles of right triangles to the ratios of their sides.
The Unit Circle, with a radius of 1, is foundational for defining trigonometric functions.

Descriptive Statistics provides summaries of data through measures like mean, median, and mode.
Inferential Statistics involves drawing conclusions based on sample data, utilizing concepts such as hypothesis testing.

Basic Concepts:
- The Sample Space encompasses all possible outcomes of an event.
- An Event is a specific outcome or a collection of outcomes.
Rules:
- The Addition Rule applies to mutually exclusive events, allowing the addition of probabilities.
- The Multiplication Rule pertains to independent events, facilitating the calculation of joint probabilities.

Logical Reasoning applies logic to arrive at conclusions through structured thought processes.
Proofs are formal demonstrations validating statements based on established axioms and theorems.