Pure vs. Applied Mathematics

• Math is the science and study of quantity, structure, space, and change
• It uses patterns to formulate new conjectures and establishes truth by rigorous deduction from suitably chosen axioms and definitions
• There is debate about whether mathematical objects exist independently of human experience (Platonism) or are human creations (constructivism)

Pure Mathematics

• Studies mathematical concepts independently of any application outside mathematics
• Abstract and theoretical
• Explores fundamental mathematical principles, relationships, and structures
• Development of mathematical knowledge for its own sake

Applied Mathematics

• Uses mathematical methods for practical problems in other domains
• Includes mathematical modeling, simulation, and optimization
• Used in engineering, physics, biology, finance, and computer science
• Focuses on using math to solve real-world problems

Arithmetic

• Basic operations: addition, subtraction, multiplication, and division
• Properties of numbers and their relationships
• Foundation for more advanced mathematics

Algebra

• Generalizes arithmetic using variables and symbols
• Solving equations and inequalities
• Study of mathematical structures like groups, rings, and fields

Geometry

• Study of shapes, sizes, positions, angles, and dimensions of objects
• Euclidean geometry focuses on flat, two-dimensional shapes and three-dimensional shapes
• Trigonometry deals with relationships between angles and sides of triangles

Calculus

• Study of continuous change
• Differential calculus: finding rates of change and slopes of curves
• Integral calculus: finding areas under curves and accumulation

Statistics

• Collection, analysis, interpretation, presentation, and organization of data
• Descriptive statistics summarizes data
• Inferential statistics makes predictions and generalizations

Logic

• Principles of valid reasoning and inference
• Used in mathematical proofs and computer science
• Propositional logic and predicate logic

Set Theory

• Study of sets, which are collections of objects
• Used as a foundation for other areas of mathematics
• Concepts like union, intersection, and complement

Number Theory

• Properties and relationships of numbers, especially integers
• Includes prime numbers, divisibility, and modular arithmetic
• Has applications in cryptography

Topology

• Properties of spaces that are preserved under continuous deformations
• Deals with connectedness, continuity, and boundaries
• Studies shapes undergoing stretching and bending

Discrete Mathematics

• Mathematical structures that are discrete rather than continuous
• Includes combinatorics, graph theory, and coding theory
• Important in computer science

Real Analysis

• Rigorous study of real numbers, sequences, series, and functions
• Provides a foundation for calculus

Complex Analysis

• Extension of calculus to complex numbers
• Properties of complex functions and their applications

Numerical Analysis

• Algorithms for solving mathematical problems approximately
• Used when exact solutions are difficult or impossible to obtain
• Important in scientific computing

Mathematical Modeling

• Creating mathematical representations of real-world phenomena
• Used to make predictions and understand complex systems

Optimization

• Finding the best solution to a problem from a set of possible solutions
• Linear programming, nonlinear programming, and dynamic programming

Financial Mathematics

• Application of mathematical methods to financial problems
• Includes pricing derivatives, managing risk, and portfolio optimization

Actuarial Science

• Assessing and managing risk in insurance and finance
• Uses probability, statistics, and financial mathematics

Cryptography

• Techniques for secure communication in the presence of adversaries
• Relies on number theory, algebra, and discrete mathematics

Game Theory

• Mathematical models of strategic interaction among rational agents
• Used in economics, political science, and biology

Chaos Theory

• Study of dynamical systems that are highly sensitive to initial conditions
• Small changes in initial conditions can lead to drastically different outcomes

Mathematical Physics

• Application of mathematical methods to problems in physics
• Includes classical mechanics, electromagnetism, and quantum mechanics
• Development of new mathematical techniques inspired by physics

Biomathematics

• Application of mathematical methods to biological problems
• Includes population modeling, epidemiology, and bioinformatics

Mathematical Education

• Study of teaching and learning mathematics
• Curriculum development and assessment methods
• Research in mathematics education

Mathematical Notation

• System of symbols and notations used to represent mathematical objects and concepts
• Standardized notation helps ensure clear communication
• Examples: +, -, ×, ÷, =, <, >, ∈, ⊆, ∃, ∀

Proofs

• Rigorous arguments that establish the truth of mathematical statements
• Deduction from axioms and previously proven theorems
• Different types of proofs: direct proof, proof by contradiction, proof by induction

Theorems

• Mathematical statements that have been proven to be true
• Fundamental building blocks of mathematical knowledge
• Examples: Pythagorean theorem, Fundamental Theorem of Calculus

Axioms

• Basic assumptions that are taken to be true without proof
• Starting point for mathematical theories
• Examples: axioms of Euclidean geometry, axioms of set theory

Conjectures

• Statements that are believed to be true but have not yet been proven
• Guide for mathematical research
• Examples: Goldbach's conjecture, Riemann hypothesis

Mathematical Software

• Computer programs used to perform mathematical calculations and simulations
• Examples: Mathematica, Maple, MATLAB
• Used in research, education, and industry

Role of Math in Technology

• Essential foundation for various technologies
• Algorithm design, data analysis, cryptography, and computer graphics
• Fuels advancements in AI, machine learning, and data science

Importance of Mathematical Literacy

• Ability to understand and apply mathematical concepts in everyday life
• Critical thinking, problem-solving, and decision-making
• Essential skill for modern society

Math Competitions

• Events where students solve mathematical problems
• Promote interest in math and develop problem-solving skills
• Examples: International Mathematical Olympiad (IMO), Putnam Competition

History of Mathematics

• Development of mathematical ideas and concepts over time
• Contributions from different cultures and civilizations
• Ancient Greek mathematics, Islamic mathematics, Renaissance mathematics