Branches of Mathematics Overview

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; includes points, lines, angles, surfaces, and solids.
Trigonometry: Study of relationships between angles and sides of triangles; includes concepts like sine, cosine, and tangent.
Calculus: Study of change and motion; involves derivatives (instantaneous rate of change) and integrals (accumulated change).
Statistics: Collection, analysis, interpretation, and presentation of data; includes measures of central tendency (mean, median, mode) and dispersion (variance, standard deviation).
Probability: Study of uncertainty; assesses the likelihood of events occurring.

Numbers:
- Natural numbers: Positive integers (1, 2, 3, ...).
- Whole numbers: Natural numbers plus zero (0, 1, 2, ...).
- Integers: Whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3, ...).
- Rational numbers: Numbers that can be expressed as a fraction (e.g., 1/2, 3/4).
- Irrational numbers: Numbers that cannot be expressed as a simple fraction (e.g., √2, π).
Equations:
- Linear equations: Form ax + b = 0; graphs as straight lines.
- Quadratic equations: Form ax² + bx + c = 0; graphs as parabolas.
Functions:
- Definition: A relation that assigns exactly one output for each input (e.g., f(x) = x²).
- Types: Linear, quadratic, polynomial, exponential, logarithmic.

Order of Operations: Follow PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Factoring: Breaking down an expression into simpler components (e.g., factoring x² - 5x + 6 results in (x - 2)(x - 3)).

Commutative Property:
- Addition: a + b = b + a
- Multiplication: ab = ba
Associative Property:
- Addition: (a + b) + c = a + (b + c)
- Multiplication: (ab)c = a(bc)
Distributive Property: a(b + c) = ab + ac

Real-world Applications:
- Finance (interest calculations, budgets)
- Engineering (design and analysis)
- Data Science (data analysis, machine learning)
- Architecture (designing structures)

Arithmetic: Involves basic operations like addition, subtraction, multiplication, and division that form the foundation of numerical calculations.
Algebra: Utilizes letters and symbols to represent numbers, enabling the formulation and solving of equations and expressions.
Geometry: Focuses on the properties and relationships of various shapes and sizes in space, covering fundamental concepts such as points, lines, angles, surfaces, and solids.
Trigonometry: Examines the angles and sides of triangles, introducing essential functions such as sine, cosine, and tangent for analyzing geometric relationships.
Calculus: Analyzes changes and motion through the study of derivatives (indicating instantaneous change) and integrals (representing cumulative change).
Statistics: Involves systematic collection and analysis of data with techniques to summarize information through measures such as mean, median, mode, variance, and standard deviation.
Probability: Evaluates the likelihood of events occurring, providing a mathematical framework to quantify uncertainty.

Numbers:
- Natural numbers: The set of positive integers (1, 2, 3,...).
- Whole numbers: Includes natural numbers plus zero (0, 1, 2,...).
- Integers: Comprises whole numbers along with their negatives (..., -3, -2, -1, 0, 1, 2, 3,...).
- Rational numbers: Can be expressed as fractions (for example, 1/2 or 3/4).
- Irrational numbers: Cannot be expressed as simple fractions (examples include √2 and π).
Equations:
- Linear equations: Generally represented as ax + b = 0, characterized by straight-line graphs.
- Quadratic equations: Expressed in the form ax² + bx + c = 0, producing parabolic graphs.
Functions:
- Defined as relations that assign exactly one output for each input (example: f(x) = x²).
- Types include linear, quadratic, polynomial, exponential, and logarithmic functions.

Order of Operations: Follow PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to correctly calculate expressions.
Factoring: Involves breaking down complex expressions into simpler factors; for instance, x² - 5x + 6 can be factored as (x - 2)(x - 3).

Commutative Property:
- Applies to addition (a + b = b + a) and multiplication (ab = ba).
Associative Property:
- Pertains to addition ((a + b) + c = a + (b + c)) and multiplication ((ab)c = a(bc)).
Distributive Property: Demonstrates that a(b + c) simplifies to ab + ac.

Finance: Used for computing interest rates and creating budgets.
Engineering: Essential for design processes and analytical evaluations.
Data Science: Key in analyzing datasets and developing machine learning algorithms.
Architecture: Crucial for designing and constructing safe, aesthetic structures.

Understand the Problem: Carefully read the problem description, pinpointing what is being asked.
Devise a Plan: Select an effective strategy, such as drawing diagrams or constructing tables.
Carry Out the Plan: Implement the chosen method methodically to reach a solution.
Review/Reflect: Verify results to ensure both accuracy and logical reasoning.