Basics of Mathematics: Arithmetic, Algebra and Geometry

• Mathematics is the abstract science of number, quantity, and space.
• It can 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 deals with operations on numbers, including addition, subtraction, multiplication, and division.
• These operations are fundamental to all other areas of mathematics.
• Key concepts include number systems (natural, integer, rational, real, and complex numbers), fractions, decimals, percentages, and ratios.

Algebra

• Algebra is a generalization of arithmetic that uses variables to represent numbers and express mathematical relationships.
• Key topics include solving equations and inequalities, working with polynomials, factoring, and simplifying expressions.
• Linear algebra extends these concepts to systems of linear equations, matrices, and vector spaces.

Geometry

• Geometry is concerned with the properties and relations of points, lines, surfaces, and solids.
• Euclidean geometry, based on a set of axioms by Euclid, deals with concepts like congruence, similarity, area, volume, and geometric constructions.
• Analytic geometry combines algebra and geometry using coordinate systems, such as the Cartesian plane, to represent geometric objects and solve geometric problems algebraically.

Trigonometry

• Trigonometry studies the relationships between angles and sides of triangles.
• Trigonometric functions like sine, cosine, and tangent are used to describe these relationships.
• Applications include solving triangles, modeling periodic phenomena, and understanding waves.

Calculus

• Calculus deals with continuous change and is divided into differential calculus and integral calculus.
• Differential calculus concerns itself with derivatives and rates of change.
• Integral calculus concerns itself with integrals and the accumulation of quantities.
• Key concepts include limits, derivatives, integrals, and the fundamental theorem of calculus, which connects differentiation and integration.

Statistics and Probability

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Key concepts include descriptive statistics (mean, median, mode, standard deviation), probability distributions, hypothesis testing, and regression analysis.
• Probability deals with the likelihood of events occurring.
• Key concepts include sample spaces, events, probability axioms, conditional probability, and independence.

Discrete Mathematics

• Discrete mathematics studies mathematical structures that are fundamentally discrete rather than continuous.
• Topics include logic, set theory, combinatorics, graph theory, and number theory.
• Logic deals with reasoning and proof.
• Set theory provides a foundation for mathematics with concepts like sets, relations, and functions.
• Combinatorics involves counting and arranging objects.
• Graph theory studies networks of vertices and edges.
• Number theory explores the properties of integers.

Topology

• Topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, bending, crumpling, and twisting, but not tearing or gluing.
• Key concepts include topological spaces, continuity, connectedness, and compactness.

Numerical Analysis

• Numerical analysis develops and analyzes algorithms for approximating solutions to mathematical problems.
• These problems arise in a variety of applications, including engineering, physics, and finance.
• Key topics include numerical methods for solving equations, approximating integrals, and solving differential equations.