Introduction to Mathematics

• Mathematics is the study of topics such as quantity, structure, space, and change
• It has no generally accepted definition
• Mathematicians seek and use patterns to formulate new conjectures
• They resolve the truth or falsity of conjectures by mathematical proofs
• Mathematical problems can be encountered throughout nature

Areas of Mathematics

• Arithmetic: Studies numbers and operations on them
• Algebra: Studies algebraic structures, relations, and quantities
• Geometry: Studies shapes, sizes, and positions of figures
• Trigonometry: Branch of mathematics that studies relationships between angles and sides of triangles
• Calculus: Studies continuous change

Arithmetic

• Deals with basic operations such as addition, subtraction, multiplication, and division
• Number theory is a branch of arithmetic that deals with the properties and relationships of numbers, especially integers

Algebra

• Uses symbols to represent numbers and quantities
• Includes solving equations and inequalities
• Abstract algebra deals with algebraic structures such as groups, rings, and fields

Geometry

• Concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs
• Euclidean geometry is based on postulates by the Greek mathematician Euclid
• Analytic geometry connects algebra and geometry using the concept of coordinate systems

Trigonometry

• Studies relationships between angles and sides of triangles
• Trigonometric functions like sine, cosine, and tangent are used to model periodic phenomena

Calculus

• Deals with continuous change
• Differential calculus: Concerns slopes and rates of change
• Integral calculus: Concerns accumulation of quantities and areas under curves
• Fundamental theorem of calculus connects differentiation and integration

Mathematical Proofs

• Used to establish the truth of mathematical statements
• Methods include direct proofs, indirect proofs, and proofs by induction
• A mathematical proof must be rigorous and based on accepted axioms and rules of inference

Mathematical Modeling

• Uses mathematical concepts and tools to describe and model real-world problems
• Used in physics, engineering, computer science, economics, and other disciplines
• Involves formulating a mathematical model, solving it, and interpreting the results

Discrete Mathematics

• Studies mathematical structures that are fundamentally discrete rather than continuous
• Includes topics such as combinatorics, graph theory, and logic
• Essential for computer science

Statistics and Probability

• Statistics: Collects, analyzes, interprets, and presents data
• Probability: Studies the likelihood of events occurring
• Used in various fields, including data science, finance, and social sciences

Topology

• Studies properties that are preserved through continuous deformations, such as stretching, twisting, crumpling and bending
• Considers properties that do not change under these deformations

Complex Analysis

• Studies functions of complex numbers
• Provides tools for solving problems in mathematical physics

Numerical Analysis

• Deals with developing and analyzing algorithms for solving continuous mathematical problems
• Used to approximate solutions where exact solutions are difficult or impossible to obtain

Mathematical Logic

• Studies the formal systems used in mathematics
• Includes topics such as set theory, model theory, recursion theory, and proof theory

Combinatorics

• Branch of mathematics dealing with counting, arrangement, and combination of objects
• Used to solve problems in probability and statistics

Game Theory

• Mathematical framework for analyzing strategic interactions between individuals or entities
• Used in economics, political science, and computer science

Set Theory

• Branch of mathematical logic that studies sets, which are collections of objects
• Provides a foundation for other areas of mathematics

Linear Algebra

• Studies vector spaces, linear transformations, and systems of linear equations
• Used in computer graphics, data analysis, and engineering

Differential Equations

• Equations containing derivatives
• Used to model phenomena involving rates of change

Fourier Analysis

• Studies the decomposition of functions into trigonometric series
• Used in signal processing and image analysis