Intro to Math: Arithmetic and Algebra

• Mathematics is the abstract science of number, quantity, and space
• Math studies topics such as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis)
• It has no generally accepted definition

Arithmetic

• Arithmetic is the oldest and most elementary branch of mathematics
• It deals with basic operations on numbers
• Basic operations include addition, subtraction, multiplication, and division
• These operations are performed on real numbers, complex numbers, and integers
• Arithmetic is the foundation for more advanced mathematics, such as algebra

Algebra

• Algebra is a branch of mathematics that generalizes arithmetic
• It uses symbols (usually letters) to represent numbers and quantities
• Algebraic expressions are formed by combining these symbols with arithmetic operations
• Algebra is used to solve equations and inequalities
• Elementary algebra deals with solving simple equations involving one or more variables
• Abstract algebra studies algebraic structures such as groups, rings, and fields

Geometry

• Geometry is a branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs
• Euclidean geometry is the study of plane and solid figures based on a set of axioms and postulates
• Analytic geometry uses algebraic techniques to study geometric problems
• Differential geometry uses calculus to study the properties of curves and surfaces
• Topology studies the properties of geometric objects that are preserved under continuous deformations

Trigonometry

• Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles
• It is based on trigonometric functions such as sine, cosine, and tangent
• Trigonometry is used in surveying, navigation, and engineering
• Trigonometric identities are equations that are true for all values of the variables

Calculus

• Calculus is a branch of mathematics that deals with continuous change
• It has two main branches: differential calculus and integral calculus
• Differential calculus deals with the rate of change of functions (derivatives)
• Integral calculus deals with the accumulation of quantities (integrals)
• Calculus is used in physics, engineering, economics, and other fields

Statistics

• Statistics is the science of collecting, analyzing, interpreting, and presenting data
• Descriptive statistics involves summarizing and describing the main features of a dataset
• Inferential statistics involves using sample data to make inferences about a population
• Probability theory provides a mathematical framework for understanding randomness and uncertainty
• Statistical methods are used in many fields, including science, business, and government

Logic

• Logic is the study of reasoning
• Mathematical logic is concerned with the application of formal logic to mathematics
• Propositional logic deals with the logical relationships between statements
• Predicate logic extends propositional logic by introducing quantifiers and predicates
• Set theory is a branch of mathematical logic that studies sets, which are collections of objects

Number Theory

• Number theory is a branch of mathematics concerned with the properties of integers
• It deals with prime numbers, divisibility, and congruences
• Important topics include Fermat's Last Theorem and the Riemann Hypothesis
• Cryptography relies on number theory for secure communication

Discrete Mathematics

• Discrete mathematics is a branch of mathematics that deals with discrete objects
• It includes topics such as graph theory, combinatorics, and logic
• Graph theory studies the properties of graphs, which are mathematical structures used to model relationships between objects
• Combinatorics deals with counting and arranging objects
• Discrete mathematics is used in computer science, operations research, and engineering

Mathematical Analysis

• Mathematical analysis is a branch of mathematics that studies the concepts of limit, continuity, differentiation, and integration
• Real analysis studies the real numbers, sequences, and functions of real variables
• Complex analysis studies complex numbers and functions of complex variables
• Functional analysis studies vector spaces and operators

Applied Mathematics

• Applied mathematics involves the application of mathematical methods to solve problems in other fields
• Numerical analysis develops algorithms for solving mathematical problems on computers
• Optimization deals with finding the best solution to a problem
• Mathematical modeling involves creating mathematical representations of real-world phenomena