Math: Arithmetic and Algebra

• Math is the science and study of quantity, structure, space, and change.
• Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.

Arithmetic

• Arithmetic is the oldest and most elementary branch of mathematics.
• It deals with basic operations on numbers, namely addition, subtraction, multiplication, and division.
• Arithmetic operations are performed according to a specific order of operations.
• The order of operations is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• Key concepts include: Different types of numbers (natural, integer, rational, irrational, real, complex), fractions, decimals, percentages, ratios, and proportions.

Algebra

• Algebra is a branch of mathematics that generalizes arithmetic.
• It uses symbols or letters to represent numbers and quantities in formulas and equations.
• Key concepts in algebra include: Variables, expressions, equations, inequalities, polynomials, factoring, solving equations (linear, quadratic, etc.), and graphing equations.

Geometry

• Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Euclidean geometry focuses on shapes that can be constructed using a compass and straightedge.
• Key concepts include: Points, lines, angles, planes, shapes (triangles, squares, circles, etc.), area, volume, and geometric theorems (Pythagorean theorem, etc.).
• Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.
• It is fundamental to surveying, navigation, and many branches of physics.

Calculus

• Calculus is the study of continuous change.
• It has two major branches: differential calculus and integral calculus.
• Differential calculus deals with rates of change and slopes of curves.
• Integral calculus deals with the accumulation of quantities and the areas under and between curves.
• Key concepts include: Limits, derivatives, integrals, functions, and theorems like the Fundamental Theorem of Calculus.

Statistics and Probability

• Statistics is the science of collecting, analyzing, interpreting, and presenting data.
• Probability is the measure of the likelihood that an event will occur.
• Key statistical concepts include: Data types, descriptive statistics (mean, median, mode, standard deviation), distributions, hypothesis testing, and regression analysis.
• Key probability concepts include: Events, sample spaces, probability distributions, conditional probability, and expectation.

Discrete Mathematics

• Discrete mathematics deals with mathematical structures that are fundamentally discrete rather than continuous.
• It is essential for computer science and includes topics like logic, set theory, graph theory, combinatorics, and number theory.

Mathematical Logic

• Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.
• It is closely related to metamathematics, the study of mathematics itself using mathematical methods.
• Key concepts include: Propositional logic, predicate logic, proof theory, model theory, and set theory.

Number Theory

• Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions.
• Key concepts include: Prime numbers, divisibility, congruences, Diophantine equations, and cryptography.

Topology

• Topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending.
• Key concepts include: Topological spaces, continuity, connectedness, compactness, and homotopy.

Applications of Math

• Mathematics is used in virtually every field, including science, engineering, medicine, finance, and social sciences.
• It provides tools for modeling, analyzing, and solving problems in these fields.
• Examples include: Physics (calculus for motion), computer science (discrete math for algorithms), and economics (statistics for econometrics).
• Mathematical modeling is the process of using mathematical concepts and language to describe a system.
• Numerical analysis provides algorithms for approximating solutions to mathematical problems, especially when analytical solutions are difficult or impossible to obtain.