Math Subjects

About Mathematics
     This subject does not contain any mathematics but rather articles about mathematics. Some popular articles include, Library of Math's Mission Statement, Math Quiz Guidelines, Math Course Syllabus, How to Read a Math Book, How Students Learn Math, ...

Analytic Geometry
    Analytic geometry is the study of geometry using the principles of algebra. Usually the Cartesian coordinate system is studied by manipulating equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. Topics covered mainly include vectors, lines and planes, distance problems, the dot and cross products, intersection problems, conic sections, and polar coordinates.

Business Algebra
    Business algebra is a collection of math articles introducing algebra and the mathematical applications that can be solved with its use. Functions, including linear, quadratic and then followed by polynomial and rational, are defined and analyzed including graphing, determining intervals of increasing and decreasing, domain and range, intercepts, and asymptotes. Exponential functions with emphasis on special properties, graphs and how to solve exponential equations. Simple interest, compound interest, compounding continuously, and annuities are worked on in detail.

Business Calculus
    Introducing the calculus and the mathematical applications that can be solved with its use is the instigating idea of business calculus. Usually following a course in business algebra, business calculus covers the notions of limits, continuity, derivatives, and integrals. The power of the calculus is shown by working on many applications for example, curve sketching, optimization techniques, and related rates. The ideas of cost, supply and demand, and of total cost, revenue, and profit are given throughout. Finally, the financial problems of simple and compounded interest and annuities are usually covered and an excellent course in business calculus usually ends by demonstrating the use of the fundamental theorems of the calculus in the world of business applications.

Business Mathematics
     Business mathematics is a collection of math articles covering algebra through calculus with special emphasis on financial applications. Polynomial, rational, exponential, logarithmic, and trigonometric functions are covered in detail. We motivate the notions of limits, continuity, derivatives, and integrals and demonstrate the power of the calculus by working on many applications; for example, curve sketching, optimization techniques, related rates, simple interest, compound interest, compounding continuously, and annuities.

Calculus
    Calculus consists of the concepts of limits, differentiation and integration and the applications of these concepts with single and multivariate functions. In general, the calculus is concerned with change and motion and it deals with quantities approaching other quantities (called limits). This level of mathematics is for high school and undergraduate students.

Calculus III
    Calculus III consists of the concepts of partial differentiation, multiple integrals (with applications), line integrals, Green's Theorem, surface integrals, Stokes' Theorem, and the divergence theorem. In Calculus III one learns how to use vectors in the treatment of three-dimensional analytic geometry because vectors give a particularly simple description of lines, planes, and curves in space. Also emphasized is the importance of vector-valued functions in the description of motion of objects through space. The idea of integration is extended to multidimensional Riemann integrals and is used to compute volume, surface area, mass, and centroids of more general regions than considered in Calculus I and II. Vector fields are studied cumulating in the theorems of Green, Stoke, and the divergence theorem, which are extended versions of the Fundamental Theorem of Calculus.

College Algebra
     College Algebra is the study of linear, quadratic and higher order polynomial equations and inequalities solved algebraically, graphically and numerically; graphs and operations on relations and functions; real and complex zeros of polynomials and rational functions; exponential and logarithmic functions; systems of linear equations; matrices.

Financial Mathematics
    Financial mathematics applies concepts from algebra and calculus to specific business problems, such as calculating the present or future value of a stream of cash flows (loans, annuities, bond coupon payments). Topics covered include simple and compound interest; nominal and effective rates of interest and discount; annuities and perpetuities; and loans and amortization. Once a framework has been developed for calculating present values of cash flows, the concept of duration is introduced along with a method for matching asset and liability cash flows.

Foundations of Geometry
    The concepts of incidence (i.e. a point lies on a line), betweeness (i.e. a point is between two other points), and congruence (i.e. line segments are congruent) have been the main improvements in Euclidean geometry since Euclid's time. Several great mathematicians including Pasch, Peano, Pieri, Veblen, Forder, Robinson, Levi, Hilbert, Birkhoff, and MacLane, and several groups including the School Mathematics Study Group (SMSG) and the University of Chicago School Mathematics Project (UCSMP) have made stunning improvements in Euclidean geometry as a mathematical (axiomatic) system. Eventually there evolved a consensus that the validity of a geometric axiomatic system was dependent on the consistency, independence, and completeness of the axiom set on which it is built and not on a physical or biased rationale.

Geometry
    This subject is the study of Euclidean and Non-Euclidean geometry, which has as its main ingredients points, lines, and planes. Incidence, betweeness, and congruence have been the main improvements in Euclidean geometry since Euclid's time. Various axiomatic systems can be used but Hilbert's axioms are detailed. This level of mathematics is for high school and undergraduate students.

Group Theory
    One of the main unifying themes in mathematics, group theory originated from concrete examples involving the study of polynomials in the early part of the 1800's. First we define mappings and then operations are introduced. Then we introduce a group by giving several concrete examples thus giving a firm understanding of what a group is. Immediately after the abstract group is introduced permutation groups are defined and explored, thus giving a concrete approach without loss of generality.

Number Theory
    Various topics in elementary number theory are detailed including divisibility, congruence, multiplicative functions, factoring, and polynomial congruence equations. Special theorems such as Wilson's theorem, Euler's theorem, and the law of quadratic reciprocity. This level of mathematics is for undergraduate students.

Precalculus
    Basically, precalculus is the study of algebra, trigonometry, and analytic geometry. A short list includes: algebraic equations and inequalities; graphs and operations on relations and functions; exponential, logarithmic, and trigonometric functions with applications; and systems of equations including matrices. This level of mathematics is for high school and undergraduate students.

Pre Calculus I
    This is the first semester of a two-semester sequence to prepare students for the study of calculus. An emphasis will be placed on introducing vocabulary, notation, and concepts encountered in calculus. Topics include: a review of fundamental algebra concepts, equations, inequalities, functions, graphs, exponential functions and logarithmic functions, and systems of equations.

Pre Calculus II
    This is the second semester of a two-semester sequence to further prepare students for the study of calculus. An emphasis will be placed on introducing vocabulary, notation, and concepts encountered in calculus. We introduce the trigonometric functions and their graphs followed by a detailed exposition of trigonometric equations, identities, and formulas. We also cover the inverse trigonometric functions, parametric curves, polar coordinates, and show how to graph in polar coordinates. Conic sections in Cartesian and polar coordinates are also detailed.

Trigonometry
    A detailed study of trigonometry usually starts with an understanding of angles, both in degrees and radians, and then defines the trigonometric functions of an acute angle. This allows a beginner time to understand the importance of the Pythagorean Theorem, the Basic Trigonometric Identities, and the Cofunction Theorem. Then the trigonometric functions of any angle are defined, thus allowing the solvability of any triangle with the laws of sines and cosines. Graphing the six trigonometric functions and investigating their transformations is usually followed by proving identities, solving equations, and using multiple angle formulas. An excellent course in trigonometry usually ends with an analysis of inverse trigonometric functions and an introduction to polar coordinates which demonstrates trigonometry with complex numbers and De Moivre's Theorem.

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