Branches of Mathematics Overview

Arithmetic: Study of numbers and basic operations (addition, subtraction, multiplication, division).
Algebra: Involves symbols and letters to represent numbers in equations.
Geometry: Deals with shapes, sizes, and properties of space.
Trigonometry: Focuses on relationships between angles and sides of triangles.
Calculus: Studies change, involving derivatives and integrals.
Statistics: Analyzes data, involves collection, interpretation, and presentation of data.
Probability: Measures the likelihood of events occurring.

Numbers:
- Natural numbers (1, 2, 3,…)
- Integers (...-2, -1, 0, 1, 2,…)
- Rational numbers (fractions)
- Irrational numbers (non-repeating, non-terminating decimals)
- Real numbers (all rational and irrational numbers)
Operations:
- Addition (+)
- Subtraction (−)
- Multiplication (×)
- Division (÷)

Expressions: Combinations of numbers, variables, and operations (e.g., 3x + 5).
Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
Functions: Relationships between sets of inputs and outputs (e.g., f(x) = x^2).

Shapes:
- 2D (e.g., squares, circles, triangles)
- 3D (e.g., cubes, spheres, cylinders)
Theorems:
- Pythagorean theorem (a² + b² = c² for right triangles)
- Area and perimeter calculations.

Functions: Sine (sin), Cosine (cos), Tangent (tan).
Unit Circle: A circle with a radius of 1 used to define trigonometric functions.
Right Triangle Ratios: Relationships between angles and sides.

Limits: The value a function approaches as the input approaches a certain point.
Derivatives: Measure the rate of change of a function.
Integrals: Calculate the area under a curve.

Descriptive statistics: Summarizes data (mean, median, mode, range).
Inferential statistics: Makes predictions or inferences about a population based on a sample.
Probability distributions: Describes how probabilities are distributed over values.

Fundamental Theorem of Algebra: Every non-constant polynomial equation has at least one complex root.
Theorems of Calculus: Include the Mean Value Theorem and Fundamental Theorem of Calculus.

Arithmetic: Concerned with basic operations on numbers including addition, subtraction, multiplication, and division.
Algebra: Uses symbols and letters as substitutes for numbers to create equations and expressions.
Geometry: Studies shapes, sizes, and properties of figures and spaces, both in two dimensions and three dimensions.
Trigonometry: Examines the relationships between the angles and sides of triangles, essential for understanding waves and oscillations.
Calculus: Focuses on concepts of change with applications in derivatives (rate of change) and integrals (area under curves).
Statistics: Involves the collection, analysis, interpretation, and presentation of data to understand and make predictions.
Probability: Quantifies the likelihood of events occurring, providing a foundation for statistics and risk assessment.

Types of Numbers:
- Natural numbers are counting numbers starting from 1.
- Integers include all whole numbers, both positive and negative, including zero.
- Rational numbers can be expressed as fractions of integers.
- Irrational numbers cannot be expressed as fractions, having non-repeating and non-terminating decimals.
- Real numbers encompass both rational and irrational numbers, representing all points on a number line.
Mathematical Operations: Essential for computations include addition (+), subtraction (−), multiplication (×), and division (÷).

Expressions: Combinations of variables and numbers where operations are performed, exemplified by expressions like 3x + 5.
Equations: Mathematical statements asserting equality between two expressions, such as 2x + 3 = 7.
Functions: Represent relationships that map inputs to outputs; for instance, f(x) = x² defines a quadratic function.

Shapes:
- Two-dimensional figures such as squares, circles, and triangles.
- Three-dimensional figures including cubes, spheres, and cylinders.
Theorems: Important principles such as the Pythagorean theorem (a² + b² = c²) for right-angle triangles, and methods to calculate area and perimeter.

Trigonometric Functions: Key functions include Sine (sin), Cosine (cos), and Tangent (tan) for calculating angles and sides in right triangles.
Unit Circle: A fundamental concept where a circle with a radius of one unit is used to define trigonometric properties.
Right Triangle Ratios: Defines specific relationships between the angles and lengths of the triangle's sides.

Limits: The foundational concept analyzing the value a function nears as its input approaches a specific point, crucial for understanding continuity.
Derivatives: Measure the instantaneous rate of change of a function, which has significant applications in various fields.
Integrals: Used to determine the accumulated area under a curve, with applications in physics and engineering.

Descriptive Statistics: Provides a summary of data sets through measures like mean, median, mode, and range.
Inferential Statistics: Enables predictions or generalizations about a population derived from samples.
Probability Distributions: Mathematical functions that describe how probabilities are assigned to values, helping analyze random events.

Area of a Circle: A = πr², where r is the radius.
Circumference of a Circle: C = 2πr, defining the perimeter of a circle.
Volume of a Cylinder: V = πr²h, where r is the radius and h is the height.
Quadratic Formula: Used to solve quadratic equations, given as x = (-b ± √(b² - 4ac)) / 2a.

Fundamental Theorem of Algebra: Guarantees that every non-constant polynomial has at least one complex root, linking algebra and complex numbers.
Calculus Theorems: Include the Mean Value Theorem and the Fundamental Theorem of Calculus, providing insights into behavior of functions.

Science and Engineering: Essential for modeling and solving practical problems, including motion and forces.
Finance: Used for calculating interest rates, analyzing investments, and managing financial risks.
Computer Science: Involves algorithms, data structure optimization, and cryptographic methods for securing data.