Mathematics
Professors R. Howell, J. Leech, C. R. Rosentrater
Associate Professors D. Hunter, P. Hunter
Description of the Major. Mathematics is a language capable of clear and precise expression and an analytic tool that can solve complex problems. It is important because of its many applications, but many mathematicians view the subject as a creative art in which human reason finds its purest expression. The attention to precise reasoning in mathematics as well as its emphasis on abstraction and creativity identify it as a discipline central to the liberal arts and sciences. Students will find that this perspective permeates the teaching of mathematics at Westmont.
Distinctive Features. The program in mathematics provides solid preparation for graduate study; it also facilitates interaction, both academic and social, with faculty and peers. Among the educational advantages the program offers are opportunities for students to participate in various research projects, problem-solving groups, or work as teaching assistants. Westmont students also help prepare and run an annual high school mathematics contest that the College hosts. This popular event has helped place Westmont as a leader in mathematics education, and its graduates who choose to enter this field are highly regarded. Details regarding recommended strategies for teacher preparation can be found by consulting the department’s web site.
Career Choices. By choosing the appropriate courses, students can prepare for: graduate study in mathematics; a career in secondary education; opportunities in computer science and operations research; or study in disciplines akin to mathematics in methodology (e.g., linguistics) or which rely heavily on mathematics (e.g., engineering, actuarial science, statistics, economics).
Requirements for a Mathematics Major (B.S. Degree): 53 units
Lower-Division Courses: 24 units
- MA 9 Elementary Calculus I (4)
- MA 10 Elementary Calculus II (4)
- MA 15 Discrete Mathematics (4) or MA 19 Multivariable Calculus (4)
- MA 20 Linear Algebra (4)
- One of the following applied course sequences: (8-9)
- CHM 5, 6 General Chemistry I, II (4,4)
- CS 10, 30 Introduction to Computer Science I, II (4,4)
- PHY 21, 23 General Physics I, II (4,4)
Foundation Courses: 8 units
- MA 108 Mathematical Analysis (4)
- MA 110 Modern Algebra (4)
In-Depth Study (Choose one of the following): 4 units
- MA 109 Advanced Mathematical Analysis (4)
- MA 111 Applied Modern Algebra (4)
Interdisciplinary Study: 1 unit
MA 90 Seminar (1)
Capstone Course (Choose one of the following): 4 units
- MA 136 Geometry (4)
- MA 140 Complex Analysis (4)
- MA 155 History of Mathematics (4)
Breadth: 12 units
(Choose any 12 additional units chosen from upper-division mathematics courses or CS 135)
Requirements for a Mathematics Major (B.A. Degree): 45 units
Lower-Division Courses: 20 units
- MA 9, 10 Elementary Calculus I, II (4,4)
- MA 15 Discrete Mathematics (4) or MA 19 Multivariable Calculus (4)
- MA 20 Linear Algebra (4)
- One of the following applied courses: (4)
- CS 5 Fundamentals of Computing (4)
- CS 10 Introduction to Computer Science I (4)
- CHM 5 General Chemistry I (4)
- PHY 21 General Physics I (4)
Foundational Courses: 8 units
- MA 108 Mathematical Analysis (4)
- MA 110 Modern Algebra (4)
Interdisciplinary Study: 1 unit
MA 90 Seminar (1)
Capstone Course (Choose one of the following): 4 units
- MA 136 Geometry (4)
- MA 140 Complex Analysis (4)
- MA 155 History of Mathematics (4)
Breadth: 12 units
(Choose any 12 additional units chosen from upper-division mathematics courses or CS 135)
Requirements for a Mathematics Minor: 24 units
- MA 9, 10 Elementary Calculus I, II (4,4)
- MA 15 Discrete Mathematics (4) or MA 19 Multivariable Calculus (4)
- MA 20 Linear Algebra (4)
- One of the following: (4)
- MA 110 Modern Algebra (4)
- MA 123 Number Theory (4)
- MA 136 Geometry (4)
- MA 155 History of Mathematics (4)
- One of the following: (4)
- MA 108 Mathematical Analysis (4)
- MA 121 Introduction to Numerical Analysis (4)
- MA 130 Probability and Statistics (4)
- MA 140 Complex Analysis (4)
Admissions Math Requirement
The admissions math requirement is a prerequisite for all mathematics courses, unless otherwise noted. The requirement is as follows: Three years of high school math, including Algebra II, or a math SAT I score of 550 or ACT math score of 22. For further information, see p. 232.
Lower-Division Course Descriptions
MA 4 Mathematics in Western Culture (4) Prerequisite: Admissions math requirement. A survey of some of the great ideas and questions in mathematics in the context of their historical/cultural formulation. Emphasis on conceptual rather than computational skills.
MA 5 Introduction to Statistics (4) Prerequisite: Admissions math requirement. Exploratory data analysis, correlation and regression. Distributions: normal, binomial, Student’s t, chi-square, F. Inferential statistics: parametric and non-parametric tests for population parameters; tests for goodness-of-fit and independence; t-tests; one- and two-way analysis of variance. Extensive use of spreadsheets.
MA 7 Finite Mathematics (4) Prerequisite: Admissions math requirement. Discrete mathematics: probability, linear programming, game theory, matrices, Markov chains.
MA 9, 10 Elementary Calculus I, II (4,4) Prerequisite for MA 9: Admissions math requirement. Prerequisite for MA 10: MA 9 or equivalent. Functions, graphs, limits, differentiation, integration, sequences, series. Introduction to numerical methods.
MA 10H Honors Calculus II (4) Prerequisite: MA 9 or equivalent and instructor approval. Functions, graphs, limits, differentiation, integration, sequences, series. Emphasis on theoretical aspects of the calculus, with extensive computer use to illustrate patterns and perform complex computations.
MA 15 Discrete Mathematics (4) Prerequisite: MA 9. The study of ideas of discrete mathematics including sets, permutations, relations, graphs, trees, and finite-state machines. Using these concepts, students will learn mathematical skills such as: methods of proof; problem solving via advanced counting techniques; problem solving through the creation of algorithms.
MA 19 Multivariable Calculus (4) Prerequisite: MA 10 or 10H. Elements of vector analysis. Functions of several variables. Differentiation, partial differentiation, gradient, implicit functions. Integration, multiple integrals, line integrals, Green’s Theorem.
MA 20 Linear Algebra (4) Prerequisite: MA 10 or 10H. Vector spaces, linear transformations, matrices, eigenvalues and eigenvectors; orthogonality; applications to differential equations, and optimization problems.
MA 40 Differential Equations (4) Prerequisite: MA 10 or 10H. First-order equations, linear equations, systems of linear equations. Series solutions, transform methods, numerical methods. Applications. Existence and uniqueness theorems.
MA 90 Seminar (1) Required attendance in the seminars offered by the Natural and Behavioral Science Division during a given semester. Students enrolled will be under the guidance of a faculty mentor and will meet periodically to discuss the wide range of topics presented in the seminars. Seminars usually occur on Friday afternoons.
Upper-Division Course Descriptions
MA 108 Mathematical Analysis (4) Prerequisite: MA 20. Topology of metric spaces, Riemann-Stieltjes integration, differentiation, sequences and series of functions, power series.
MA 109 Advanced Mathematical Analysis (4) Prerequisite: MA 108. Measure and integration theory, space of functions, Fourier series.
MA 110 Modern Algebra (4) Prerequisite: MA 20. Groups including permutation groups, subgroups, factor groups and isomorphism theorems. Rings and ideal theory. Fields and their extensions. Applications to solving polynomial equations and geometry.
MA 111 Advanced Modern Algebra (4) Topics will be selected from among the following: Group actions and Burnside’s Theorem; Sylow Theorems; subnormal subgroup series, the Jordan-Holder Theorem; structure theorems for finitely generated abelian groups. Extension fields and their automorphism groups, Galois Theory; solvability of polynomials by radicals. Unique factorization in integral domains.
MA 121 Introduction to Numerical Analysis (4) Prerequisite: MA 10 or 10H, Recommended: CS 10. Numerical methods in the solution of equations; polynomial approximations; integration, and the solution of differential equations. Use of computer where applicable.
MA 123 Number Theory (4) Prerequisite: MA 19 or MA 20. Prime factorization and the distribution of primes. Congruences and residue class arithmetic; quadratic residues and Gauss reciprocity. Primality testing and pseudoprimes with applications to cryptography. Arithmetic functions. Theorems on sums of squares and other results inspired by Fermat.
MA 130 Probability and Statistics (4) Prerequisite: MA 10 or 10H. Probability spaces, random variables, discrete and absolutely continuous distributions, independence, conditional probability. Normal, binomial, Poisson distributions, joint distributions. Moments. Central Limit Theorem. Hypothesis testing, point estimation.
MA 135 Formal Languages and Automata (4) Prerequisite: CS 30. Regular languages; finite automata. Context-free languages; pushdown automata; Turing machines, halting problem. Computability. (Offered in alternate years, spring semester.)
MA 136 Geometry (4) Prerequisite: MA 20. Axiomatic systems; finite geometries, neutral and hyperbolic geometries, transformations of the Euclidean plane, projective geometry.
MA 140 Complex Analysis (4) Prerequisite: MA 19. Complex numbers, analytic and harmonic functions, integrals, series, residues and poles, conformal maps, Fundamental Theorem of Algebra and the classical theorems obtained in complex analysis. Discussion of some of the great topics in complex analysis such as the Tiemann Hypothesis and Bieberbach Conjecture (now a theorem).
MA 150 Topics (4) Prerequisite: MA 19 or MA 20. Course content will be determined by student interest and need.
MA 155 History of Mathematics (4) Prerequisite: MA 19 or MA 20. Survey of the historical development of mathematics from antiquity through the early twentieth century. Topics included: mathematics in ancient Greece, mathematics in China and India during the medieval period, the mathematics of Islam, the evolution of ideas in such areas as geometry, number theory, calculus, algebra, and set theory. Includes exploration of historiographical questions and of questions about the nature of mathematical discovery and proof. Emphasizes use of primary sources.
MA 160, 165 Fundamentals of Mathematics I, II (4,2) Not for credit toward mathematics major. Logic, sets, numbers, natural numbers, numeration systems, algorithms for arithmetic operations, geometry, probability. (GE Reasoning Abstractly for MA 160; Quantitative and Analytical Reasoning for MA 65)