Math: Arithmetic and Algebra

• Mathematics is the abstract science of number, quantity, and space
• It may be studied in its own right (pure mathematics), or as it is applied to other disciplines such as physics and engineering (applied mathematics)

Arithmetic

• Arithmetic is the branch of mathematics dealing with numerical calculations
• Basic operations include addition, subtraction, multiplication, and division
• These operations are applied to different types of numbers, such as integers, fractions, decimals, and percentages
• Order of operations follows PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction

Algebra

• Algebra uses symbols and letters to represent numbers and quantities
• It involves manipulating these symbols to solve equations and inequalities
• Key concepts include variables, expressions, and equations
• Techniques for solving equations include simplification, factoring, and using the quadratic formula
• Linear equations can be represented graphically as straight lines

Geometry

• Geometry deals with the properties and relations of points, lines, surfaces, and solids
• Key concepts include angles, triangles, circles, and other geometric shapes
• Theorems such as the Pythagorean theorem are fundamental to geometry
• Coordinate geometry combines algebra and geometry to describe shapes using coordinates
• Transformations such as translations, rotations, and reflections are studied

Trigonometry

• Trigonometry focuses on the relationships between angles and sides of triangles
• Trigonometric functions such as sine, cosine, and tangent are used
• These functions are used to solve problems involving triangles and periodic phenomena
• The unit circle is a key tool for understanding trigonometric functions

Calculus

• Calculus deals with continuous change
• It includes differential calculus, which focuses on rates of change and slopes of curves, and integral calculus, which focuses on accumulation and areas under curves
• Key concepts include limits, derivatives, and integrals
• Calculus is used in many areas of science and engineering

Statistics

• Statistics involves collecting, analyzing, interpreting, and presenting data
• Descriptive statistics summarize and describe data using measures such as mean, median, and mode
• Inferential statistics use sample data to make inferences about populations
• Probability is a key concept in statistics, dealing with the likelihood of events

Number Theory

• Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions
• Prime numbers, divisibility, and congruences are central topics
• Theorems such as Fermat's Last Theorem have played a significant role in the development of number theory

Discrete Mathematics

• Discrete mathematics deals with mathematical structures that are fundamentally discrete rather than continuous
• Topics include logic, set theory, graph theory, and combinatorics
• Discrete mathematics is essential for computer science

Mathematical Logic

• Mathematical logic explores the applications of formal logic to mathematics
• It is closely related to metamathematics, the foundations of mathematics, and theoretical computer science
• Key concepts include propositional logic, predicate logic, and proof theory.

Topology

• Topology is concerned with the properties of geometric objects that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending, but not tearing or gluing
• It deals with concepts like connectedness, continuity, and boundaries
• Topology has applications in various fields, including physics and computer science

Mathematical Analysis

• Mathematical analysis is a branch of mathematics that deals with the rigorous study of calculus and related topics
• It includes real analysis, complex analysis, and functional analysis
• Key concepts include sequences, series, limits, continuity, differentiation, and integration

Numerical Analysis

• Numerical analysis studies algorithms for solving the problems of continuous mathematics
• It involves developing and analyzing numerical methods for approximation
• Numerical analysis is important for scientific computing and engineering

Abstract Algebra

• Abstract algebra extends the concepts found in elementary algebra and arithmetic of numbers to more general concepts
• Key concepts include groups, rings, fields, and modules
• It is crucial for many areas of mathematics, including number theory, algebraic geometry, and cryptography

Combinatorics

• Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures
• Aspects include counting the structures of a given kind and size, deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria
• Topics include permutations, combinations, and graph theory.