Basics of Math, Algebra, and Calculus

• 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.
• Math is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance, and social sciences.
• Applied math concerns itself with the use of math tools to solve problems in science, engineering, business, and other areas.
• Math is essential in the natural sciences for modeling phenomena.
• Math's abstract nature makes it a powerful tool applicable in diverse fields.

Algebra

• Algebra is the study of mathematical symbols and the rules for manipulating these symbols.
• It is a unifying thread of almost all of math.
• Elementary algebra deals with solving equations and inequalities.
• Abstract algebra studies algebraic structures such as groups, rings, and fields.

Calculus

• Calculus deals with the study of continuous change.
• It has two major branches: differential calculus and integral calculus.
• Differential calculus concerns instantaneous rates of change, and the slopes of curves.
• Integral calculus concerns accumulation of quantities, and the areas under and between curves.
• Calculus is used extensively in physics, engineering, economics, and computer science.

Geometry

• Geometry is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
• Euclidean geometry studies shapes in a plane and in three-dimensional space based on axioms by Euclid.
• Analytic geometry connects geometry to algebra using a coordinate system.
• Differential geometry uses calculus to study curves and surfaces.
• Topology studies properties of shapes that are preserved through continuous deformations.

Trigonometry

• Trigonometry studies relationships between angles and sides of triangles.
• Trigonometric functions (sine, cosine, tangent) relate angles to ratios of sides.
• Trigonometry is used in navigation, surveying, astronomy, and physics.

Number Theory

• Number theory is a branch of math devoted primarily to the study of integers.
• Prime numbers, divisibility, and modular arithmetic are central to number theory.
• Number theory has applications in cryptography and computer science.

Statistics and Probability

• Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data.
• Probability is the measure of the likelihood that an event will occur.
• Probability theory provides the theoretical foundation for statistics.
• Statistics is used in diverse fields such as science, business, and government.

Discrete Math

• Discrete math deals with mathematical structures that are fundamentally discrete rather than continuous.
• Logic, set theory, graph theory, combinatorics, and algorithms are important topics.
• Discrete math is essential in computer science.
• It provides the mathematical foundations for algorithms, data structures, and databases.

Logic

• Logic is the study of reasoning.
• It is used to determine whether an argument is valid, and to prove theorems.
• Propositional logic and predicate logic are two important branches of logic.
• Logic is used in computer science to design computer circuits and to write computer programs.

Set Theory

• Set theory is the branch of mathematical logic that studies sets, which are collections of objects.
• Set theory is fundamental to many other areas of math.
• Concepts like union, intersection, and complement of sets are important.

Graph Theory

• Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.
• Graphs consist of vertices (nodes) and edges that connect these vertices.
• Graph theory is used in computer science, operations research, and social network analysis.

Combinatorics

• Combinatorics is the branch of math concerning the study of finite or countable discrete structures.
• Counting, arrangement, and selection of objects are important topics.
• Combinatorics has applications in probability theory, statistics, and computer science.

Algorithms

• An algorithm is a step-by-step procedure for solving a problem or accomplishing some end.
• Algorithms are essential in computer science.
• The study of algorithms includes their design, analysis, and implementation.

Mathematical Analysis

• Mathematical analysis is a branch of math that deals with the concepts of limits, continuity, differentiation, and integration.
• It includes real analysis, complex analysis, and functional analysis.
• Mathematical analysis provides a rigorous foundation for calculus.

Numerical Analysis

• Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis.
• It is used to obtain approximate solutions to problems that are difficult or impossible to solve analytically.
• Numerical analysis is used in science, engineering, and finance.

Topology

• Topology is concerned with the properties of geometric objects that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending.
• It deals with concepts such as connectedness, continuity, and boundary.
• Point-set topology, algebraic topology, and differential topology are important areas.

Applications of Math

• Math is used extensively in science.
• Physics uses differential equations to model motion, electromagnetism, and gravity.
• Chemistry uses math to model chemical reactions, molecular structures, and thermodynamics.
• Biology uses statistics to analyze data from experiments.
• Math is used in engineering to design structures, circuits, and machines.
• Civil engineering uses math to design bridges and buildings.
• Electrical engineering uses math to design circuits.
• Mechanical engineering uses math to design machines.
• Math is used in finance to model financial markets, price derivatives, and manage risk.
• Actuarial science uses math to assess risk in insurance and finance.
• Math is used in computer science to design algorithms, data structures, and databases.
• Cryptography uses number theory to encrypt and decrypt messages.

This resource covers broad mathematical concepts. It introduces algebra, focusing on symbols, rules, equation solving, and algebraic structures. It also touches upon calculus and applied mathematics, highlighting math's use in diverse fields like science and engineering.