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# Mathematics, Minor

# Department

## Department of Mathematics

Rajee Amarasinghe, Chair

Peters Business Building, Room 381

559.278.2992

www.fresnostate.edu/csm/math/

### Degrees and Programs Offered

BA in Mathematics, B.A.

CRED in Single Subject Credential - Mathematics

CRED in Professional Clear Foundation Level Mathematics

MA in Mathematics, M.A.

MA in Mathematics - Teaching Option, M.A.

MN in Mathematics, Minor

Mathematics and related subjects play important dual roles in our culture. On the one hand, mathematics is a study in its own right; on the other hand, it is an indispensable tool for expressing and understanding ideas in the sciences, engineering, and an increasing number of other fields. As a consequence, employment opportunities for mathematicians have been expanding in recent years. The courses offered by the department are designed to develop skills in, and an appreciation and understanding of, both roles.

Because there are so many different areas in which a trained mathematician can find employment or continue studies, the department offers a large number of electives within the mathematics major. By selecting appropriate courses, students have considerable flexibility to accommodate their individual interests. Students should consult with a department adviser for specific recommendations as to which electives are suited to their career paths.

Electives in applied mathematics prepare students to assume positions in technical industries or government employment, or to continue advanced studies in the applied area.

Electives in pre-college teaching in mathematics provide students with the necessary background for obtaining a California Secondary Teaching Credential in mathematics. In order to complete the credential requirements, a fifth year of education courses, classroom observation, and practice teaching is needed. At the present time, there is an increasing demand for well-trained people in this area.

Electives in pure mathematics prepare students for the pursuit of graduate studies leading to advanced degrees and employment at the college or university level, or research in industries.

Electives in statistics and probability provide a foundation for students planning to work as statisticians for industry or government agencies. They also can enhance employment opportunities in the bioscience and health-related fields. Statistics courses (in addition to MATH 75 [or 75A and B], 76, and 77) are essential for the first two Actuarial Examinations offered by the Society of Actuaries.

# Courses

## Mathematics

###### CI 161. Content Area Methods and Materials in Secondary Teaching

Prerequisites: CI 152 AND CI 159 or concurrent enrollment; admission to the Single Subject Credential Program or teaching experience. Planning, delivering, and assessing content-specific instruction; academic and common core standards; identifying specific standards that require literacy strategies. (Instructional materials fee for Single Subject - Art Methods and Materials enrollees, $10)

Units: 3, Repeatable up to 999 units

Course Typically Offered: Fall, Spring

###### EHD 154B. Final Student Teaching Seminar - Mathematics

Prerequisites: Concurrent enrollment in EHD 155B. Seminar to accompany final student teaching that provides opportunities for candidates to investigate and discuss variety of topics and strategies and to reflect on issues that surface during their student teaching experience.

Units: 1

###### EHD 155B. Student Teaching in Secondary School - Math

Prerequisites: admission to student teaching, EHD 155A, CI 161 (or concurrently, depending on major departmental policy); senior or post baccalaureate standing; approval of major department including subject matter competency approval; completion of the subject matter preparation program or passing the subject matter examination(s) designated by the California Commission on Teacher Credentialing. Supervised teaching in single subject classroom; assignment is for the full day; five days per week. CR/NC grading only.

Units: 5-10, Repeatable up to 20 units

Course Typically Offered: Fall, Spring

###### ESM 1. Early Start Developmental Mathematics

Designed for students in the Early Start program. Review of topics in algebra and geometry: percentages, ratios, radicals, exponents, linear equations and inequalities, equations of lines, factoring, solving equations, area, volume, angles, and similar triangles. CR/NC/RP grading only; not applicable towards baccalaureate degree requirements.

Units: 1

Course Typically Offered: Summer

###### ESM 3. Early Start Algebra II

Designed for students in the Early Start program. Radicals, rational exponents, quadratic equations, simultaneous linear equations, graphing, inequalities, and complex numbers.

Units: 3

Course Typically Offered: Summer

###### MATH 1RA. Developmental Mathematics I

The first semester in a two semester sequence preparing students for college level mathematics. See the online Schedule of Courses for restrictions on enrollment based on the Entry Level Math test. Properties of ordinary arithmetic, integers, rational numbers and linear equations. CR/NC grading only; not applicable towards baccalaureate degree requirements.

Units: 3

Course Typically Offered: Fall

###### MATH 1RB. Developmental Mathematics II

Prerequisite: MATH 1RA. The second semester in a two semester sequence preparing students for college level mathematics. Systems of linear equations, exponents, rational expression, polynomials and quadratic equations. CR/NC grading only; not applicable toward baccalaureate degree requirements.

Units: 3

Course Typically Offered: Spring

###### MATH 3. College Algebra

Prerequisite: students must meet the ELM requirement. Equations and inequalities; rectangular coordinates; systems of equations and inequalities; polynomial, rational, exponential, and logarithmic functions and their graphs; complex numbers.

Units: 3

Course Typically Offered: Fall, Spring

###### MATH 4R. Transition to College Mathematics

Prerequisite: See the online Schedule of Courses for restrictions on enrollment based on the Entry Level Math test. Covers radicals, rational exponents, quadratic equations, simultaneous linear equations, graphing, inequalities, and complex numbers. Not applicable toward baccalaureate degree requirements. (3hr lecture, 2hr Activity)

Units: 4

###### MATH 4RA. Intermediate Algebra

Focuses on arithmetic review, linear equalities, and graphing. Note: MATH 4RA together with MATH 4RB is equivalent to MATH 4R. Enrollment is limited to first-time freshmen who score 30 or below on the ELM exam. CR/NC grading only; not applicable toward baccalaureate degree requirements.

Units: 3

###### MATH 5. Trigonometry

Prerequisite: students must meet the ELM requirement. Concept of a function, sine and cosine functions, tables and graphs, other trigonometric functions, identities and equations. Trigonometric functions of angles, solution of triangles. (See Duplication of Courses).

Units: 3

Course Typically Offered: Fall, Spring

###### MATH 6. Precalculus

Prerequisite: students must meet the ELM requirement. Basic algebraic properties of real numbers; linear and quadratic equations and inequalities; functions and graphs; polynomials; exponential and logarithmic functions; analytic trigonometry and functions; conics; sequences and series.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 10A. Structure and Concepts in Mathematics I

Prerequisite: students must meet the ELM requirement. Designed for prospective elementary school teachers. Development of real numbers including integers, rational and irrational numbers, computation, prime numbers and factorizations, and problem-solving strategies. Meets B4 G. E. requirement only for liberal studies majors.

Units: 3

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 10B. Structure and Concepts in Mathematics II

Prerequisite: MATH 10A. Designed for prospective elementary school teachers. Counting methods, elementary probability and statistics. Topics in geometry to include polygons, congruence and similarity, measurement, geometric transformations, coordinate geometry, and connections between numbers and geometry with selected applications.

Units: 3

Course Typically Offered: Fall, Spring

###### MATH 11. Elementary Statistics

Prerequisite: students must meet the ELM requirement. Illustration of statistical concepts: elementary probability models, sampling, descriptive measures, confidence intervals, testing hypotheses, chi-square, nonparametric methods, regression. It is recommended that students with credit in MATH 75 or MATH 75A and B take MATH 101.

Units: 3

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 45. What Is Mathematics?

Prerequisite: students must meet the ELM requirement. Covers topics from the following areas: (I) The Mathematics of Social Choice; (II) Management Science and Optimization; (III) The Mathematics of Growth and Symmetry; and (IV) Statistics and Probability. G. E. Foundation B4.

Units: 3

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 70. Calculus for Life Sciences

No credit if taken after MATH 75 or MATH 75A and B. Prerequisite: students must meet the ELM requirement. Functions and graphs, limits, derivatives, antiderivatives, differential equations, and partial derivatives with applications in Life Sciences.

Units: 4

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 75. Calculus I

Prerequisites: elementary geometry, intermediate algebra, and trigonometry; or precalculus. Passing score on the department's Calculus Readiness Test required prior to enrollment. In addition, students must meet the ELM requirement. Functions, graphs, limits, continuity, derivatives and applications, definite and indefinite integrals. G.E. Foundation B4. FS

Units: 4

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 75A. Calculus with Review IA

Prerequisites: elementary geometry, intermediate algebra, and trigonometry; or precalculus. Passing score on the department's Calculus Readiness Test required prior to enrollment. In addition, students must meet the ELM requirement. Functions, graphs, limits, continuity, derivatives, and applications, with extensive review of algebra and elementary functions. With MATH 75B, equivalent to MATH 75. G.E. Foundation B4. FS

Units: 4

Course Typically Offered: Fall, Spring

GE Area: B4

###### MATH 75B. Calculus with Review IB

Prerequisite: MATH 75A. Further applications of derivatives, and definite and indefinite integrals, with extensive review of algebra and elementary functions. With MATH 75A, equivalent to MATH 75.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 76. Calculus II

Prerequisite: MATH 75 or MATH 75A and B. Techniques and applications of integration, improper integrals, conic sections, polar coordinates, infinite series.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 77. Calculus III

Prerequisite: MATH 76. Vectors, three-dimensional calculus, partial derivatives, multiple integrals, Green's Theorem, Stokes' Theorem.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 81. Elementary Differential Equations with Linear Algebra

Prerequisite: MATH 77. Introduction to ordinary linear differential equations and linear systems of differential equations; solutions by Laplace transforms. Solution of linear systems of equations; introduction to vector spaces; eigenvalues and eigenvectors. Using computer software as an exploratory tool.

Units: 3

Course Typically Offered: Fall, Spring

###### MATH 90. Directed Study

Independently arranged course of study in some limited area of mathematics either to remove a deficiency or to investigate a topic in more depth. (1-3 hours, to be arranged)

Units: 1-3

###### MATH 100. Exploring Mathematics

Prerequisite: MATH 10B. The development of mathematical reasoning, problem solving, and communication skills for effective teaching of mathematics in elementary school.

Units: 3

Course Typically Offered: Fall, Spring

###### MATH 101. Statistical Methods

Prerequisite: MATH 70 or MATH 75, or MATH 75A and B; no credit if taken after MATH 108. Application of statistical procedures to examples from biology, engineering, and social science; one- and two-sample normal theory methods; chi-square, analysis of variance, and regression; nonparametric methods. Computerized statistical packages are used.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 107. Introduction to Probability and Statistics

Prerequisite: MATH 77 (may be taken concurrently). Basic concepts required for applications of probability theory; standard discrete and continuous models; random variables; conditional distributions; limit theorems.

Units: 3

Course Typically Offered: Fall

###### MATH 108. Statistics

Prerequisite: MATH 107. Criteria used for selecting particular procedures of data analysis; derivation of commonly used procedures; topics from sampling, normal theory, nonparametrics, elementary decision theory.

Units: 3

Course Typically Offered: Spring

###### MATH 109. Applied Probability

Prerequisite: MATH 107. Introduction to stochastic processes and their applications in science and industry. Markov chains, queues, stationary time series.

Units: 3

Course Typically Offered: Spring

###### MATH 110. Symbolic Logic

(Similar to PHIL 145; consult department.) Prerequisite: MATH 75 or MATH 75A and B. An informal treatment of the theory of logical inference, statement calculus, truth-tables, predicate calculus, interpretations applications.

Units: 3

Course Typically Offered: Spring

###### MATH 111. Transition to Advanced Mathematics

Prerequisite: MATH 76. Introduction to the language and problems of mathematics. Topics include set theory, symbolic logic, types of proofs, and mathematical induction. Special emphasis is given to improving the student's ability to construct, explain, and justify mathematical arguments.

Units: 3

###### MATH 114. Discrete Structures

Prerequisite: MATH 111. Counting techniques, matrix algebra, graphs, trees and networks, recurrence relations and generating functions, applied modern algebra.

Units: 3

Course Typically Offered: Fall

###### MATH 116. Theory of Numbers

Prerequisite: MATH 111. Divisibility theory in the integers, primes and their distribution, congruence theory, Diophantine equations, number theoretic functions, primitive roots, indices, the quadratic reciprocity law.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 118. Graph Theory

Prerequisite: MATH 111. Trees, connectivity, Euler and Hamilton paths, matchings, chromatic problems, planar graphs, independence, directed graphs, networks.

Units: 3

Course Typically Offered: Spring - even

###### MATH 121. Numerical Analysis I

Prerequisites: MATH 77 and CSCI 40. Zeros of nonlinear equations, interpolation, quadrature, systems of equations, numerical ordinary differential equations, and eigenvalues. Use of numerical software libraries.

Units: 3

Course Typically Offered: Spring

###### MATH 123. Topics in Applied Mathematics

Prerequisite: MATH 77. Vector spaces and linear transformations, eigenvalues and eigenfunctions. Special types of linear and nonlinear differential equations; solution by series. Fourier transforms. Special functions, including gamma, hypergeometric, Legendre, Bessel, Laguerre, and Hermite functions. Introduction to partial differential equations.

Units: 3

Course Typically Offered: Spring - odd

###### MATH 128. Applied Complex Analysis

Prerequisite: MATH 77. Analytic functions of a complex variable, contour integration, series, singularities of analytic functions, the residue theorems, conformal mappings; emphasis on engineering and physics applications.

Units: 3

Course Typically Offered: Fall

###### MATH 133. Number Theory for Liberal Studies

Prerequisite: MATH 10B or permission of instructor. The historical development of the concept of number and arithmetic algorithms. The magnitude of numbers. Basic number theory. Special numbers and sequences. Number patterns. Modular arithmetic.

Units: 3

Course Typically Offered: Fall

###### MATH 134. Geometry for Liberal Studies

Prerequisite: MATH 10B or permission of instructor. The use of computer technology to study and explore concepts in Euclidean geometry. Topics include, but are not restricted to, properties of polygons, tilings, and polyhedra.

Units: 3

Course Typically Offered: Spring

###### MATH 137. Exploring Statistics

Prerequisite: MATH 10B or permission of instructor. Descriptive and inferential statistics with a focus on applications to mathematics education. Use of technology and activities for student discovery and understanding of data organization, collection, analysis and inference.

Units: 3

Course Typically Offered: Fall

###### MATH 138. Exploring Algebra

Prerequisite: MATH 10B or permission of instructor. Designed for prospective school teachers who wish to develop a deeper conceptual understanding of algebraic themes and ideas needed to become competent and effective mathematics teachers.

Units: 3

Course Typically Offered: Spring

###### MATH 139. Advanced Algebra for Middle School Teachers

Prerequisite: MATH 6 or MATH 138. Basic structures of modern algebra from a middle school mathematics curriculum perspective. Algebraic structures, polynomial equations, and elementary linear algebra.

Units: 4

Course Typically Offered: Fall

###### MATH 143. History of Mathematics

Prerequisite: MATH 75 or MATH 75A and 75B. History of the development of mathematical concepts in algebra, geometry, number theory, analytical geometry, and calculus from ancient times through modern times. Theorems with historical significance will be studied as they relate to the development of modern mathematics.

Units: 4

Course Typically Offered: Spring

###### MATH 145. Problem Solving

Prerequisite: MATH 111; EHD 50 (may be enrolled concurrently). A study of formulation of problems into mathematical form; analysis of methods of attack such as specialization, generalization, analogy, induction, recursion, etc. applied to a variety of non-routine problems. Topics will be handled through student presentation.

Units: 3

Course Typically Offered: Fall

###### MATH 149. Capstone Mathematics for Teachers

Prerequisites: MATH 151, MATH 161, and MATH 171 (MATH 161 and MATH 171 may be taken concurrently). Secondary school mathematics from an advanced viewpoint. This course builds on students' work in upper division mathematics to deepen their understanding of the mathematics taught in secondary school. Students will actively explore topics in number theory, algebra, analysis, geometry.

Units: 4

###### MATH 151. Principles of Algebra

Prerequisite: MATH 111. Equivalence relations; groups, cyclic groups, normal sub-groups, and factor groups; rings, ideals, and factor rings; integral domains and polynomial rings; fields and field extensions.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 152. Linear Algebra

Prerequisite: MATH 77. Vector spaces, linear transformations, matrices, determinants, eigenvalues and eigenvectors, linear functions, inner-product spaces, bilinear forms, quadratic forms, orthogonal and unitary transformations, selected applications.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 161. Principles of Geometry

Prerequisite: MATH 111. The classical elliptic, parabolic, and hyperbolic geometries developed on a framework of incidence, order and separation, congruence; coordinatization. Theory of parallels for parabolic and hyperbolic geometries. Selected topics of modern Euclidean geometry.

Units: 3

Course Typically Offered: Spring

###### MATH 165. Differential Geometry

Prerequisite: MATH 77 and MATH 111. Study of geometry in Euclidean space by means of calculus, including theory of curves and surfaces, curvature, theory of surfaces, and intrinsic geometry on a surface.

Units: 3

Course Typically Offered: Fall

###### MATH 171. Intermediate Mathematical Analysis I

Prerequisite: MATH 111. Natural and rational numbers, real numbers as a complete ordered field, its usual topology, sequences and series of real numbers, functions of a real variable, limits, continuity, uniform continuity, differentiability, generalized mean value theorem, Riemann integrals, and power series.

Units: 4

Course Typically Offered: Fall, Spring

###### MATH 172. Intermediate Mathematical Analysis II

Prerequisite: MATH 77 and MATH 171. Pointwise and uniform convergence of sequences and series of functions, convergence of sequences in higher dimensions, continuity and differentiability of functions of several variables. The inverse and implicit function theorems; topics in integration theory in higher dimensions.

Units: 4

Course Typically Offered: Spring

###### MATH 181. Differential Equations

Prerequisite: MATH 81 or MATH 123. Definition and classification of differential equations; general, particular, and singular solutions; existence theorems; theory and technique of solving certain differential equations: phase plane analysis, elementary stability theory; applications.

Units: 3

Course Typically Offered: Fall

###### MATH 182. Partial Differential Equations

Prerequisites: MATH 81 or MATH 123. Classical methods for solving partial differential equations including separation of variables, Green's functions, the Riemann-Volterra method and Cauchy's problem for elliptic, parabolic, and hyperbolic equations; applications to theoretical physics.

Units: 3

Course Typically Offered: Spring

###### MATH 190. Independent Study

See Academic Placement. Approved for RP grading.

Units: 1-3, Repeatable up to 6 units

Course Typically Offered: Fall, Spring

###### MATH 191T. Proseminar

Prerequisites: Permission of instructor. Presentation of advanced topics in mathemathics in the field of the student's interest.

Units: 1, Repeatable up to 9 units

###### MATH 192. Undergraduate Mathematics Seminar

Prerequisite: MATH 76 or consent of the instructor. Presentations on various topics in mathematics. The course is intended for STEM students with a strong interest in mathematics. It is an upper division elective course.

Units: 1, Repeatable up to 3 units

Course Typically Offered: Fall, Spring

###### MATH 198. Senior Project

Prerequisites: Senior standing or permission of instructor; MATH 151, MATH 171, and MATH 152. Independent investigation and presentation of an advanced topic in mathematics. Satisfies the senior major requirement for the B.A. in Mathematics.

Units: 3

###### MATH 202. Fundamental Concepts of Mathematics

Prerequisites: MATH 151, MATH 161 and MATH 171. Fundamental notions regarding number theory, number systems, algebra of number fields; functions.

Units: 3

###### MATH 216T. Topics in Number Theory

Prerequisite: MATH 116. An investigation of topics having either historical or current research interest in the field of number theory. (Formerly MATH 216)

Units: 3, Repeatable up to 6 units

###### MATH 220. Coding Theory

Prerequisites: MATH 151 and MATH 152. Basic concepts in coding theory, properties of linear and on-linear codes, standard decoding algorithms, cyclic codes, BCH-codes.

Units: 3

###### MATH 223. Applied Operator Theory

Prerequisite: graduate standing or permission of instructor. Fundamentals of abstract spaces and spectral theory of operators with applications. Resolvent set and spectrum of a linear operator. Bounded and unbounded linear operators. Compact operators and the Fredholm alternative. Symmetry and self-adjointness.

Units: 3

###### MATH 228. Functions of a Complex Variable

Prerequisite: MATH 128. Representation theorems of Weierstrass and Mittag-Leffler, normal families, conformal mapping and Riemann mapping theorem, analytic continuation, Dirichlet problem.

Units: 3

###### MATH 232. Mathematical Models with Technology

Prerequisite: graduate standing in mathematics or permission of instructor. A technology-assisted study of the mathematics used to model phenomena in statistics, natural science, and engineering.

Units: 3

###### MATH 250. Perspectives in Algebra

Prerequisite: graduate standing in mathematics or permission of instructor. Study of advanced topics in algebra, providing a higher perspective to concepts in the high school curriculum. Topics selected from, but not limited to, groups, rings, fields, and vector spaces.

Units: 3

###### MATH 251. Abstract Algebra I

Prerequisite: MATH 151. Direct and semi-direct products of groups; quotient/factor groups; isomorphism theorems. Group actions; Sylow theorems; classification of groups; finitely generated Abelian groups. Domains (ED, PID, UFD); polynomial rings. Quotient/factor rings; field extensions; automorphisms of fields.

Units: 3

###### MATH 252. Abstract Algebra II

Prerequisite: MATH 251. Rings and ideals, modules, linear and multilinear algebras, representations.

Units: 3

###### MATH 260. Perspectives in Geometry

Prerequisite: graduate standing in mathematics or permission of instructor. Geometry from a transformations point of view. Euclidean and noneuclidean geometries in two and three dimensions. Problem solving and proofs using transformations. Topics chosen to be relevant to geometrical concepts in the high school curriculum.

Units: 3

###### MATH 263. Point Set Topology

Prerequisite: MATH 172. Basic concepts of point set topology, set theory, topological spaces, continuous functions; connectivity, compactness and separation properties of spaces. Topics selected from function spaces, metrization, dimension theory.

Units: 3

###### MATH 270. Perspectives in Analysis

Prerequisite: graduate standing in mathematics or permission of instructor. An overview of the development of mathematical analysis, both real and complex. Emphasizes interrelation of the various areas of study , the use of technology, and relevance to the high school mathematics curriculum.

Units: 3

###### MATH 271. Real Analysis

Prerequisite: MATH 172. Lebesgue's measure and integration theory on the real line. Limit theorems and types of convergence. Lp spaces. Differentiation and integration.

Units: 3

###### MATH 290. Independent Study

See Academic Placement -- [-LINK-]. Approved for RP grading.

Units: 1-3, Repeatable up to 6 units

###### MATH 291T. Seminar

Prerequisite: graduate standing. Presentation of current mathematical research in field of student's interest.

Units: 1-3, Repeatable up to 6 units

###### MATH 298. Research Project in Mathematics

Prerequisite: graduate standing. Independent investigation of advanced character as the culminating requirement for the master's degree. Approved for RP grading.

Units: 3

###### MATH 298C. Project Continuation

Pre-requisite: Project MATH 298. For continuous enrollment while completing the project. May enroll twice with department approval. Additional enrollments must be approved by the Dean of Graduate Studies.

Units: 0

###### MATH 299. Thesis in Mathematics

Prerequisite: See Criteria for Thesis and Project. Preparation, completion, and submission of an acceptable thesis for the master's degree. Approved for RP grading.

Units: 3

# Requirements

## Mathematics Minor Requirements

The minor requires 20 units in mathematics, MATH 75 (or MATH 75B) or above, excluding MATH 75A, 100, 133, 134, 137, 138, 139, and 149.

**Minor Advising Note**

Courses in the minor may be taken CR/NC, subject to university regulations. The Mathematics Minor also requires a 2.0 GPA and 6 upper-division units in residence.

# Faculty

For faculty phone numbers and e-mail, see the campus directory.

For more on the faculty, see the faculty pages.

The faculty pages are updated by the department or program.