Fundamentals of Mathematics

• Mathematics is the abstract science of number, quantity, and space.
• Mathematics can be used as a pure science, or applied to other disciplines.

Arithmetic

• Arithmetic involves studying numbers, mainly the properties of traditional operations like addition, subtraction, multiplication, division, exponentiation, and extraction of roots.
• A field is any set of objects upon which all of these operations can be performed.
• Arithmetic operations follow a specific order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS).

Algebra

• Algebra is a branch of mathematics that generalizes arithmetic, using variables to represent numbers.
• Elementary algebra deals with solving equations and inequalities.
• Linear equations can be solved by isolating the variable on one side.
• Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
• Abstract algebra extends the concepts found in elementary algebra to more general structures, such as groups, rings, and fields.
• Boolean algebra operates on binary values (0 and 1) and is fundamental to digital logic.

Geometry

• Geometry focuses on the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Euclidean geometry, based on Euclid's axioms, deals with the properties of space, distances, angles, and shapes.
• Analytic geometry uses algebraic techniques to study geometric problems, often using coordinate systems like the Cartesian coordinate system.
• Trigonometry studies the relationships between the sides and angles of triangles, particularly right triangles, and uses trigonometric functions such as sine, cosine, and tangent.
• Differential geometry employs calculus to study the properties of curves and surfaces.
• Topology studies the properties of geometric objects that remain unchanged under continuous deformations such as stretching and bending.

Calculus

• Calculus is concerned with the study of continuous change.
• Differential calculus deals with the rate of change (derivatives) and slopes of curves.
• Integral calculus deals with the accumulation of quantities (integrals) and the areas under and between curves.
• The fundamental theorem of calculus connects differentiation and integration.
• Calculus is used in physics, engineering, economics, and computer science.

Statistics and Probability

• Statistics involves the science of collecting, analyzing, presenting, and interpreting data.
• Descriptive statistics focuses on summarizing and presenting data through measures such as mean, median, mode, variance, and standard deviation.
• Inferential statistics involves drawing conclusions and making predictions based on a sample of data.
• Probability theory is the study of randomness and uncertainty, providing a framework for calculating the likelihood of events.
• Random variables are variables whose values are numerical outcomes of a random phenomenon.
• Probability distributions describe the likelihood of different outcomes for random variables.
• Common distributions include the normal distribution, binomial distribution, and Poisson distribution.

Discrete Mathematics

• Discrete mathematics studies mathematical structures that are discrete rather than continuous.
• Logic deals with reasoning and proof techniques, and it is fundamental to computer science.
• Set theory deals with collections of objects and operations on them.
• Graph theory studies graphs, which are mathematical structures used to model pairwise relations between objects.
• Combinatorics is concerned with counting and arranging objects.

Number Theory

• Number theory is a branch of mathematics devoted primarily to studying integers and their properties.
• Prime numbers, divisible only by 1 and themselves, are fundamental to number theory.
• Divisibility rules provide shortcuts for determining whether a number is divisible by another number.
• Modular arithmetic deals with the remainders of numbers after division.
• Cryptography relies heavily on number theory for secure communication.

Mathematical Analysis

• Deals with the rigorous study of calculus concepts.
• Focuses on limits, continuity, differentiation, integration, and infinite series.
• Real analysis studies real numbers and functions.
• Complex analysis extends calculus to complex numbers.
• Functional analysis studies vector spaces and linear operators.

Numerical Analysis

• Develops algorithms for solving mathematical problems.
• Approximates solutions to problems that are difficult or impossible to solve analytically.
• Key areas: root finding, interpolation, numerical integration, and solving differential equations.