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 relationships.
- Geometry: Study of shapes, sizes, and properties of space.
- Trigonometry: Study of relationships between angles and sides of triangles.
- Calculus: Study of change and motion, involving derivatives and integrals.
- Statistics: Collection, analysis, interpretation, and presentation of data.
- Probability: Analysis of random events and likelihood of occurrences.

Numbers:
- Natural Numbers: Positive integers (1, 2, 3, ...).
- Whole Numbers: Natural numbers plus zero (0, 1, 2, ...).
- Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).
- Rational Numbers: Numbers that can be expressed as fractions.
- Irrational Numbers: Numbers that cannot be expressed as simple fractions (e.g., π, √2).
Operations:
- Addition (+)
- Subtraction (−)
- Multiplication (×)
- Division (÷)

Expressions: Combinations of numbers and variables (e.g., 3x + 2).
Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
Functions: Relationships between sets that assign exactly one output for each input (e.g., f(x) = x²).

Points, Lines, and Planes: Fundamental elements of geometry.
Shapes:
- Polygons (triangles, quadrilaterals, etc.)
- Circles
Theorems: Statements proven based on previously established statements (e.g., Pythagorean theorem).

Limits: The value that a function approaches as the input approaches a point.
Derivatives: Measure of how a function changes as its input changes (slope of tangent).
Integrals: Measure of the area under a curve; reverse process of differentiation.

Descriptive Statistics: Summarizes and describes data (mean, median, mode).
Inferential Statistics: Makes inferences and predictions about a population based on sample data.

Events: Results of trials (e.g., flipping a coin).
Probability Formula: P(A) = Number of favorable outcomes / Total number of outcomes.
Law of Large Numbers: As trials increase, the experimental probability approaches the theoretical probability.

Mathematics encompasses the study of numbers, quantities, shapes, and patterns.
It has numerous branches including arithmetic, algebra, geometry, trigonometry, calculus, statistics, and probability.

Numbers form the foundation of mathematics, with various classifications including:
- Natural numbers are positive integers starting from 1 (1, 2, 3, ...).
- Whole numbers include natural numbers plus zero (0, 1, 2, ...).
- Integers encompass both positive and negative whole numbers along with zero (..., -2, -1, 0, 1, 2, ...).
- Rational numbers can be expressed as fractions, while irrational numbers cannot be represented as simple fractions, such as π and √2.
Operations are fundamental to manipulating numbers, including addition (+), subtraction (-), multiplication (×), and division (÷).

Expressions combine numbers and variables using operations (e.g., 3x + 2).
Equations state that two expressions are equal (e.g., 2x + 3 = 7).
Functions represent relationships between sets, assigning a unique output for each input (e.g., f(x) = x²).

Points, lines, and planes are fundamental elements of geometry, defining shapes and relationships in space.
Shapes encompass polygons (like triangles and quadrilaterals) and circles.
Theorems are proven statements in geometry, built on previously established facts (e.g., the Pythagorean theorem).

Trigonometric functions (sine, cosine, tangent, and their inverses) relate angles and sides of triangles.
Trigonometric identity: Sin²(x) + Cos²(x) = 1.
Applications: Trigonometry is used in measuring angles and modeling periodic phenomena.

Limits represent the value a function approaches as its input nears a specific point.
Derivatives measure how a function changes as its input changes, representing the slope of the tangent line.
Integrals calculate the area under a curve, representing the reverse process of differentiation.

Descriptive statistics summarizes and describes data using measures like mean, median, and mode.
Inferential statistics draws inferences and predictions about a population based on sample data.

Events represent possible outcomes of trials (e.g., flipping a coin).
Probability formula: P(A) = number of favorable outcomes / total number of outcomes.
Law of large numbers: As trials increase, the experimental probability approaches the theoretical probability.

Science and engineering utilize mathematics for problem-solving and modeling physical systems.
Finance relies on mathematics for calculating interest, investments, and risk assessment.
Computer science uses mathematics for algorithms, data structures, and complexity analysis.