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 involves studying numbers and the basic operations that can be performed on them
Basic operations include addition, subtraction, multiplication, and division
Whole numbers are non-negative integers (0, 1, 2, 3...)
Integers include all whole numbers as well as their negative counterparts (...-3, -2, -1, 0, 1, 2, 3...)
Rational numbers can be expressed as a fraction p/q, where p and q are integers and q is not zero
Irrational numbers cannot be expressed as a fraction, examples include √2 and π

Algebra involves generalizing arithmetic operations and studying mathematical relationships using symbols
Variables are symbols (usually letters) that represent unknown or changeable quantities
Equations are mathematical statements showing the equality of two expressions
Solving an equation involves finding the value(s) of the variable(s) that make the equation true
Linear equations are equations where the highest power of the variable is 1
Quadratic equations are equations where the highest power of the variable is 2, a standard form is ax² + bx + c = 0
Functions are relationships that map each input value to a unique output value
The graph of a function is a visual representation of this relationship
Common functions: linear, quadratic, exponential, logarithmic, and trigonometric

Geometry deals with shapes, sizes, relative positions of figures, and the properties of space
Euclidean geometry studies shapes in a two-dimensional plane and three-dimensional space based on axioms and theorems
Points, lines, and planes are fundamental undefined objects in Euclidean geometry
Angles are formed by two rays sharing a common endpoint (vertex), measured in degrees or radians
Triangles are three-sided polygons, categorized by their angles (acute, obtuse, right) and sides (equilateral, isosceles, scalene)
Circles are sets of points equidistant from a center point
Coordinate geometry uses a coordinate system to represent geometric shapes and solve geometric problems algebraically
Transformations involve changing the position, size, or orientation of a shape, examples include translation, rotation, reflection, and scaling
Trigonometry is the study of relationships between angles and sides of triangles, particularly right triangles
Trigonometric functions (sine, cosine, tangent) relate an angle to ratios of sides in a right triangle

Calculus studies continuous change, encompassing two main branches: differential calculus and integral calculus
Differential calculus deals with rates of change and slopes of curves
Derivatives measure the instantaneous rate of change of a function
Integral calculus deals with accumulation of quantities and areas under curves
Integrals are the reverse process of differentiation, used to find areas, volumes, and other accumulated quantities
Limits are the values that a function approaches as the input approaches some value
Continuity describes functions that have no abrupt changes or breaks in their graphs
Applications includes optimization problems, physics, engineering, economics, and statistics

Statistics involves collecting, analyzing, interpreting, and presenting data
Descriptive statistics summarize and describe the main features of a dataset, including measures of central tendency (mean, median, mode) and measures of dispersion (variance, standard deviation)
Inferential statistics uses sample data to make inferences and generalizations about a larger population
Probability is the measure of the likelihood that an event will occur
Probability ranges from 0 (impossible) to 1 (certain)
Probability distributions describe the probabilities of different outcomes in a random experiment
Common distributions include normal, binomial, and Poisson
Statistical hypothesis testing involves using data to assess the validity of a hypothesis about a population
Regression analysis examines the relationship between variables to make predictions

Discrete mathematics studies mathematical structures that are fundamentally discrete rather than continuous
Logic deals with reasoning and argumentation, including propositional logic and predicate logic
Set theory studies sets, which are collections of objects
Relations and functions formalize relationships between sets
Combinatorics involves counting and arranging objects, including permutations and combinations
Graph theory studies graphs, which are structures consisting of vertices and edges connecting pairs of vertices
Applications includes computer science, cryptography, and operations research

Mathematical analysis provides a rigorous foundation for calculus and relates areas of mathematics
Real analysis studies the real number system, sequences, series, limits, continuity, and differentiability of real-valued functions
Complex analysis extends concepts of calculus to complex numbers
Functional analysis studies vector spaces and linear operators
Topology studies properties of spaces that are preserved under continuous deformations