Introduction to Mathematics: Arithmetic and Algebra

Study Notes

Mathematics is the abstract science of number, quantity, and space
It may also be defined as the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects
Mathematics is used throughout the world as an essential took in many fields, including natural science, engineering, medicine, finance, and social sciences
Applied mathematics concerns itself with the use of mathematical tools to solve problems in natural science, engineering, medicine, finance, and social sciences

Arithmetic is the oldest and most elementary branch of mathematics
It deals with numerical calculations, including addition, subtraction, multiplication, and division

Algebra is a branch of mathematics that generalizes arithmetic
It uses letters or other symbols to represent numbers and quantities in formulas and equations
Includes solving equations and studying abstract structures like groups, rings, and fields

Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs
Includes Euclidean geometry (plane and solid), analytic geometry, differential geometry, and topology

Trigonometry studies relationships between angles and sides of triangles
Used to calculate distances and angles in various fields like surveying and navigation

Calculus is concerned with the study of continuous change
Divided into differential calculus (rates of change and slopes) and integral calculus (accumulation of quantities and areas)
Fundamental tool in physics, engineering, and economics

Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data
Used to make inferences and predictions about populations based on samples
Includes descriptive statistics, inferential statistics, and probability theory

Logic is the study of reasoning and argumentation
Provides a formal framework for constructing valid arguments and proving theorems
Important in computer science and philosophy

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions

Topology is concerned with the properties of geometric objects that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending, without tearing or gluing
Studies properties of space, connectivity, and continuity

Discrete mathematics deals with mathematical structures that are fundamentally discrete rather than continuous
Includes combinatorics, graph theory, and set theory
Important in computer science and cryptography

Mathematical analysis is a branch of mathematics that deals with the rigorous study of calculus and related topics
Includes real analysis, complex analysis, and functional analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis
Used extensively in engineering and scientific computing

Applied mathematics is a branch of mathematics that concerns itself with the application of mathematical knowledge to other fields
Includes mathematical modeling, optimization, and control theory

Mathematical physics is the application of mathematics to problems in physics
Involves developing mathematical models of physical phenomena and solving them using mathematical techniques

Financial mathematics applies mathematical methods to financial problems
Includes pricing derivatives, managing risk, and optimizing investment strategies

Mathematical biology, or biomathematics, is an interdisciplinary field that uses mathematical tools to study biological systems
Includes modeling population dynamics, biological processes, and spread of diseases

Game theory is a mathematical framework for analyzing strategic interactions between rational agents
Used in economics, political science, and computer science