**MATHEMATICS**

The Department of Mathematics provides an engaging and intellectually rigorous curriculum that seeks to assist students in preparing for a career in which mathematics is directly or indirectly involved and to foster an appreciation of the vital role of mathematics in society.

In all our courses we encourage critical thinking and emphasize problem solving. Students thereby acquire and strengthen essential skills for their science and liberal arts courses.

We offer more than forty courses and two undergraduate degree programs – the Bachelor of Science in the Mathematical Sciences and the Associate of Science with a Mathematics Concentration – in addition to the Minor in Mathematics.

We support math minors, majors and all students taking math classes by promoting a “mathematical community” through our tutorial program, undergraduate research, guest lectures, research presentations, the math club and other extra-curricular activities.

Department of Mathematics faculty members conduct research in various areas of pure and applied mathematics, including algebra, geometry, representation theory, mathematical physics, fluid dynamics, number theory and topology, and on the teaching of mathematics.

This course is designed to provide a foundation in the principles and methods of arithmetic and an introduction to basic algebra. Topics include number line concepts and diagrams, the arithmetic of whole numbers, integers, common fractions, decimals and percents, applications of integers, common factors, basic geometry, formulas, evaluation, ratio and proportions and solving simple equations in one variable. Furthermore, applications of fractions and percents to everyday problems as well as introduction to word problems are included. Computer aided instruction and calculators will be integrated into the classes; however, no calculators are used during exams.

- Pre-Requisites: Incoming Student and Placement by CUNY/COMPASS Assessment Test

The course is designed to provide students with a solid background in real number algebra. Topics include elementary properties of the real number system and number line diagrams, exponents, polynomials, coordinate systems, graphs, factoring and algebraic fractions, linear and quadratic equations and their applications. Computer aided instruction and calculators will be integrated into the classes.

- Pre-Requisites: MTHP 009 or Exemption by CUNY/COMPASS Assessment Test

This course is designed to provide the students in the liberal arts with an introduction to some of the major concepts of modern mathematics. Topics include why and how numbers were invented, history of mathematics; set theory and Venn diagrams, comparative study of algebra of sets, and algebra of numbers, applications to logic circuits; selected topics from number theory; counting and elementary probability; compound interest, discrete and continuous. Calculators and computers will be used to do experiments and illustrate mathematical concepts. Writing assignments will be given culminating in a term paper.

- Pre-Requisites: MTH 010 or Placement by CUNY COMPASS Assessment Tests for Incoming Students

This course is designed to provide students in general with the knowledge and skills needed for further studies in the mathematical and physical sciences as well as in such fields as accounting and finance, marketing and management. The topics to be discussed in this course include rational and polynomial expressions, graphical methods, solving equations and systems of equations by Cramer's Rule, principles of analytic trigonometry, exponentials and logarithms, induction, the binomial theorem, progressions, conic sections. Applications to various fields will be emphasized. Electronic calculators will be used throughout the course to perform detailed numerical calculations.

- Pre-Requisites: Exit from Initial Placement by CUNY COMPASS Assessment Tests

This course is designed to provide initial preparation in mathematics for students who are majoring in, or who intend to major in, the mathematical sciences, computer science, or environmental science. It is also for those in other science programs whose course of study requires advanced mathematical skills and training. A thorough understanding of the topics to be studied in this course will form the essential background for further studies in the mathematical and physical sciences and related fields. The topics to be discussed include solutions of compound statements including absolute value equations and inequalities, rational and radical equations and inequalities, the algebra of functions, modeling with exponential and logarithmic functions, systems of linear equations by the Gaussian and Gauss-Jordan elimination methods, nonlinear systems of equations and inequalities, conic sections and parametric equations, modeling with exponentials and logarithms, sequence and series, the binomial theorem, and mathematical induction. Topics from trigonometry include trigonometric functions and their inverses, graphs, identities and equations, the laws of sines and cosines with applications, polar coordinates and De Moivre's theorem. Electronic calculators and computers (based on availability) will be used throughout the course to perform detailed numerical calculations, and graphical presentations.

- Pre-Requisites: Initial Placement by CUNY COMPASS Assessment Tests

This course is designed to provide non-science majors with the mathematical background required for the applications of elementary quantitative methods to problems in business and the social sciences. The topics covered include basic probability theory and its applications, introductory statistics, matrices and linear programming, as well as concepts from precalculus and calculus such as set relations and functions, introduction to limits, and the rule for differentiating simple algebraic functions. Whenever appropriate, computers and calculators will be integrated into the course.

- Pre-Requisites: MTH 136 or MTH 138

This course is designed to provide students with the mathematical preparation necessary for a successful study of calculus. It also gives students in general education an opportunity to fulfill their desire for a mature investigation and understanding of the level of mathematics beyond the Algebra sequence through the study of real functions. In the study of the properties of real functions, both analytical and graphical methods will be used. Whenever possible, an effort will be made to apply mathematics to problems in the sciences and other disciplines. Topics include absolute value equations and inequalities; polynomial, rational, trigonometric, exponential and logarithmic functions and composite and inverse functions. Computers and calculators will be utilized throughout the course to enhance understanding of mathematics concepts.

- Pre-Requisites: MTH 136 or MTH 138 or Initial Placement by CUNY COMPASS Assessment Tests

The analysis of functions numerically, graphically, and algebraically, aided by technology; velocity and distance; Riemann sums; the integral assigned area; Fundamental Theorem of Calculus; antiderivatives and the indefinite integral; basic properties of integrals; integrals tables; techniques of closed form integration; numerical integrations; Taylor series; applications of integrals to problems in geometry and the sciences.

- Pre-Requisites: MTH 151 with a Grade of "C" or Better

The analysis of functions numerically, graphically, and algebraically, aided by technology; velocity and distance; Riemann sums; the integral assigned area; Fundamental Theorem of Calculus; antiderivatives and the indefinite integral; basic properties of integrals; integrals tables; techniques of closed form integration; numerical integrations; Taylor series; applications of integrals to problems in geometry and the sciences.

- Pre-Requisites: MTH 202 with a Grade of "C" or Better

This course generalizes the concepts and applications of the differential and integral calculus of functions of one variable to higher dimensions. The analysis of multivariable functions numerically, graphically, and algebraically aided by technology; partial derivatives, directional derivative; Taylor approximations; optimization, the quadric surfaces, polar, cylindrical, spherical coordinates; vector fields, line and surface integrals; multiple integrals.

- Pre-Requisites: MTH 203 with a Grade of "C" or Better

This course is designed to introduce students to the idea and nature of ordinary differential equations. Computers will be integrated in teaching the theory and applications in gaining insight into the solution of both linear and nonlinear differential equations. Topics covered include direction fields, phase planes and phase portraits; first order equations, higher order equations, systems of first order differential equations, the Laplace transform; and series solutions.

- Pre-Requisites: MTH 203 with a grade of "C" or better.

This course is designed to provide students in the mathematical sciences degree program with a general introduction to the formal language and methods of proof and argument that are universally applied in the mathematical sciences. The close relationship between language (both natural and symbolic) and mathematical abstractions will be discussed in detail. The roles of undefined terms and defined terms in mathematics as well as the distinctions between them will be presented and illustrated. The basic mathematical terminology and standard notational systems will be presented, and students will be shown how to devise acceptable and efficient descriptive notation and symbols that may be required for specific mathematical tasks. The concepts of logical truth and consistency, along with the qualifiers and their use will be analyzed in detail. The construction method, the choose method, and the first and second principles of mathematical induction will be discussed in detail as will the indirect methods of proof by contradiction and proof by contrapositives. Proofs based on arguments by the method of exhaustion along with arguments based on the exhibition of a counterexample will be presented and illustrated. The distinction between general proofs and specific illustrations (examples) will be emphasized. Existence and uniqueness arguments from various branches of mathematics will be presented. Writing original proofs and detailed analyses will be emphasized throughout the course. When appropriate, computers will be used to test specific cases of general principles.

- Pre-Requisites: MTH 202

The course is designed to introduce students to elements of finite dimensional vector spaces over real numbers; linear transformations and applications; system of matrices; independence of vectors, bases, dimension; dot product; projections; linear transformations, matrix representation; eigenvalues and eigenvectors, diagonalization.

- Pre-Requisites: MTH 202

This course is designed to provide students with the basic statistical techniques commonly used in data collection, analysis and interpretation. Familiarity with such techniques is essential for any program of study and is vital for the nursing program. Topics include tabulation and presentation of data by charts and graphs; description of data using numerical measures: mean, median, mode, percentiles, variance and standard deviation; description of bi-variate data by scatter diagram, correlation co-efficient and regression line; intuitive development of probability for studying binomial and normal distributions; and applications to statistical inference such as estimation and tests of hypotheses. Required for nursing students. Whenever appropriate, computers and calculators will be integrated into the course. Not open to Science and Business students.

- Pre-Requisites: MTH 136 or MTH 138

This course is designed to provide students with an introduction to basic statistical techniques commonly used in data analysis and business operations. This course focuses on the use of statistics as a tool to navigate and make sensible decisions in an increasingly quantitative world. Topics include tabulation and presentation of data; descriptive statistics; elementary probability theory; binomial and normal distributions with applications to sampling theory; the Central Limit Theorem; confidence intervals; hypothesis testing; correlation; linear regression. Statistical computer programs will be integrated into the course and will be used extensively. Not open to Science and Nursing students.

- Pre-Requisites: MTH 136 or MTH 138

This course is a continuation of MTH 115 and is designed to provide the students in the liberal arts with additional major concepts of modern mathematics including the design of mathematical models that describe real world situations and how these models can be used to obtain solutions to a wide variety of practical problems. Topics include interest, annuities and amortization, inferential statistics, application of symbolic logic and predicate calculus to switching circuits, graph theory and its applications.

- Pre-Requisites: MTH 115

This course is designed to provide students with a survey of geometry and geometric methods. Students will be introduced to axiomatic systems and will be shown how different systems result in different geometries. The relationship between algebra and geometry will be examined in terms of coordinates in the plane and space. The perimeter, area and volume formulas for elementary plane and solid figures will be derived and applied to practical problems. The nature of proofs and their development from basic principles will be emphasized as will computational methods and compass and straightedge constructions. Non-Euclidean geometry will be investigated.

- Pre-Requisites: MTH 136 or MTH 138

This course is designed to help prospective teachers of elementary school mathematics to view elementary mathematics from a higher standpoint. Students will be provided with an enriched background in a broad selection of topics from advanced mathematics through exploration, conjecture and reasoning. The topics covered include numeration system, number theory, decimals, ratios, and percents, intuitive plane and solid geometry and mensuration, construction, magic square and modular arithmetic, probability and statistics. Whenever appropriate, computers and calculators will be integrated into the course.

- Pre-Requisites: MTH 136 or MTH 138

This course will provide a calculus-based introduction of probability theory and applications to statistical inference. Topics will include discrete and continuous probability distributions, moment generating functions, laws of large numbers, limit theorems, sampling distributions, and statistical inference using z, t, f and c2 distributions.

- Pre-Requisites: MTH 203

This course is intended to introduce students to classical number theory, including its proof techniques and history. Topics include divisibility, primes and their distribution, congruence, quadratic residues, Diophantine equations, continued fractions, and numbertheoretic functions.

- Pre-Requisites: MTH 206

The course is designed to provide an introduction to modern, abstract algebra through concrete structures. Topics include congruence in integers; groups; rings; fields and field extensions; and applications.

- Pre-Requisites: MTH 206

This course is designed to provide a deeper investigation of the structures and proof techniques introduced in MTH 308. Among the topics to be discussed will be the Sylow theorems, algebraic free abelian groups, group representations, factor rings and ideals, modules, field extensions, Galois Theory, and selected applications of abstract algebra.

- Pre-Requisites: MTH 308

This course will offer an introduction to the rigorous analysis of functions of one and several variables that will provide students with the background needed for advanced study in pure and applied analysis. Topics will include properties of the real number system, limits, continuity, differentiability, vector analysis, and introductory differential geometry.

- Pre-Requisites: MTH 203 and MTH 206

This course will offer a continuation of the rigorous analysis of functions begun in MTH 311. Topics will include multiple integrals, line and surface integrals, Green's Theorem, Stokes' Theorem, infinite series and improper integrals.

- Pre-Requisites: MTH 311

The course is designed to provide a rigorous introduction to the theory and applications of functions of a complex variable. Among the topics to be discussed are complex numbers, complex functions, analytic and harmonic functions, the Cauchy-Riemann equations, complex integration, Cauchy's integral theorem, Liouville's Theorem, Taylor and Laurent series, singularities, residues, the Argument Principle, and Rouche's Theorem.

- Pre-Requisites: MTH 203

After a review of selected results from MTH 315, the student will be introduced to more advanced topics in classical complex function theory. Topics to be discussed may include conformal mappings, the Riemann mapping theorem, analytic continuation, infinite products, the gamma function, asymptotic series, Jensen's theorem, the Phragmen-Lindelof theorems, and various applications of complex function theory.

- Pre-Requisites: MTH 315

An introductory course that introduces students to the major topics in elementary discrete mathematics and builds skills in mathematical reasoning and proof techniques. The course will cover such topics as sets, algorithms, mathematical induction, recursion, counting techniques, relations, graphs, trees, Boolean algebra, and applications.

- Pre-Requisites: MTH 207

The content of this course will vary depending on the interests and needs of the students and the interests of the faculty. Selected topics in advanced mathematics will be discussed. The course will allow students to experience specialized areas of mathematics that are not a regular part of the curriculum.

- Pre-Requisites: Permission of chairperson

A minimum of 9 hours of conference and independent research per week will be required. Library and/or laboratory investigation of problems in mathematical science or related fields will be selected and pursued under the guidance of the faculty of the department. Regular meetings with advisor, presentation of findings at departmental seminars, and submission of a written report of research carried out will also be required.

- Pre-Requisites: Completion of all required 300 Level Courses or Permission of chairperson. Only 3 of these may be applied to the Bachelors degree.

This is a capstone course that builds upon the mathematical maturity developed in earlier courses. It will require the reading of current and classical articles in mathematics journals and will develop a student's ability to solve problems. The course will unify the students' previous course work and illustrate the power and usefulness of mathematics in the modern world.

- Pre-Requisites: Permission of chairperson