# Mathematics

*Chair: *Patrick W. Keef

Ibrahim Abdelrazeq

Barry Balof

James Cotts

Robert Fontenot

Russell A. Gordon

David Guichard

Douglas Hundley

Kelly McConville *(on Sabbatical, Fall 2014)*

Albert W. Schueller

Allen Tucker

**Mathematics Department Website »**

Mathematics courses provide an opportunity to study mathematics for its own sake and as a tool for use in the physical, social, and life sciences.

All or part of the calculus sequence is required or recommended by several majors at Whitman and calculus is the most common mathematics course taken by students. However, the department offers other courses (Mathematics 108, 128, 167) that are intended for students who wish to take mathematics but are not interested in or not prepared for calculus. Courses in programming, or with an emphasis on computing, are 167, 235, 270, 350, 467.

P-D-F policy: The department places no restrictions on the use of the P-D-F option for mathematics courses for majors or nonmajors, except that students choosing the mathematics major must take Mathematics 260 for a grade. The department strongly recommends that students majoring in mathematics or completing a joint major with mathematics not use the P-D-F option in mathematics courses.

The senior assessment in mathematics consists of a comprehensive examination in two parts: a four-hour written examination and a one-hour oral examination. The written examination covers three semesters of calculus and one of linear algebra — roughly the first two years of the program. The oral examination covers general and advanced topics.

**Distribution: **Courses completed in mathematics apply to the quantitative analysis distribution area.

**Learning Goals:** Upon graduation, a student will be able to:

**Major-Specific Areas of Knowledge**- Solve mathematical problems. Learn new mathematics independently. Evaluate mathematical arguments. Have depth of study in at least one area of mathematics. Have a basic understanding of several branches of mathematics.

**Communication**- Communicate mathematical ideas effectively both orally and in writing.

**Quantitative Skills**- See other goals.

**The Mathematics major:** A total of 35 credits, to include Mathematics 225, 235, 240, 260; any two of 385, 386, 455, 456; 497 or 498; 12 additional credits in mathematics courses numbered above 200. A grade of B- (2.7) or better in Mathematics 260 is required. Grades of B (3.0) or better in Mathematics 225 and 260 are strongly recommended for any student considering mathematics as a major, and both courses ought to be completed by the end of the sophomore year. Students planning graduate study should take Mathematics 386 and 456 and should acquire a reading knowledge of either French, German, or Russian.

A student who enters Whitman College without a good working knowledge of the material in Mathematics 125 and 126 will have to complete 41 mathematics credits to fulfill the requirements for the mathematics major (including six credits for Mathematics 125, 126).

Twenty-six mathematics credits are required for the mathematics-physics major, 29 mathematics credits for the economics-mathematics major, and 29 mathematics credits for the 3-2 mathematics-computer science major.

**The Mathematics minor:** Fifteen credits or more in mathematics courses numbered 200 or above.

**The Economics-Mathematics combined major:** Mathematics 167, 225, 235, 240, 244, 247, 349, and three additional credits chosen from mathematics courses numbered above 200. Economics 101 or 177, 102, 307, 308, 327, 428, plus one additional course in economics. For economics-mathematics majors, Economics 101 or 177, 102 and Mathematics 247 are the prerequisites for Economics 327, and Economics 227 does not apply toward the minimum major requirements. Economics 493, 494, and other economics courses taken P-D-F may not be used to meet the 28-credit requirement. The senior assessment consists of the written exam in mathematics, the Major Field Test (MFT) in economics, and a combined oral exam scheduled by the economics department.

**The Mathematics-Physics combined major:** Mathematics 225, 235, 240, 244, and nine additional credits in mathematics courses numbered above 200; Physics 155 or 165, 156 or 166, 245, 246, 255, 256, 325, 339, and one additional physics course numbered from 300-480, or BBMB 324. Senior assessment consists of the written exam in mathematics, the written exam in physics, and a combined oral exam scheduled by the physics department.

**Majoring in Computer Science.** Please see the *Combined Plans* section of this catalog.

**Choosing a Calculus Course **Students who wish to take calculus should note the following: Students with a strong background in high school mathematics not including calculus start with Mathematics 125. Students who have taken a high school course in calculus, but who have not taken the BC calculus Advanced Placement Test (see the statement below regarding college credit for the Advanced Placement Test) should take the Advisory Calculus Placement exam offered by the department of mathematics.

Students should note that several programs require the calculus lab, Mathematics 235, in addition to Mathematics 225. Because the lab course teaches skills that are useful in other mathematics and science courses, it is strongly recommended that students take Mathematics 235 as early as possible in their programs. Programs that require the calculus labs are the mathematics major, the economics-mathematics major, the mathematics-physics major, the physics major, the 3-2 engineering program, and the 3-2 mathematics-computer science major.

**Advanced Placement **The policy for advanced standing and credit for the College Board Advanced Placement program is as follows:

- Students with a 4 or 5 on the BC calculus test are considered to have completed the equivalent of Mathematics 125 and 126 and receive six credits in mathematics.
- Students with a 4 or 5 on the AB calculus test (or on the AB subtest of the BC test) are considered to have completed the equivalent of Mathematics 125 and receive three credits in mathematics. These students should take the placement test offered by the department of mathematics to determine whether they should enroll in Mathematics 126 or Mathematics 225.
- Students with a 4 or 5 on the computer science (A) test are considered to have completed the equivalent of Mathematics 167 and receive three credits in mathematics.
- Students with a 4 or 5 on the statistics test are considered to have completed the equivalent of Mathematics 128 and receive three credits in mathematics. Students should consider taking Mathematics 247 if they have also completed the equivalent of Mathematics 125.

A student has the option of repeating a course for which AP credit has been granted, but with a commensurate reduction in advanced placement credit.

#### 108 Introduction to Mathematical Thinking

3; not offered 2014-15

This course will introduce students to mathematical thinking by studying a variety of mathematical topics. Topics may include problem-solving strategies, recreational mathematics and puzzles, mathematics of finance, voting power, and game theory. This course is not designed to prepare students for calculus and is intended primarily for the nonmath major. *Prerequisite**:* two years of high school mathematics.

#### 119 Programming with Robots

3; not offered 2014-15

An introduction to programming techniques applicable to most languages using personal robotics kits (Lego Mindstorm NXT’s provided). The programming language used is most similar to the C programming language. Frequent programming projects are required in both independent and group settings. Traditional computer science topics like logic and algorithms, simple networking, event loops, and threading also will be explored.

#### 125 Calculus I

3, x Keef, Cotts

A brief review of some precalculus topics followed by limits, continuity, a discussion of derivatives, and applications of the derivative. *Prerequisites**:* two years of high school algebra; one year of plane geometry; and knowledge of trigonometry and exponential/logarithmic functions or consent of instructor.

#### 126 Calculus II

3, 3 Fall: Balof, R. Gordon; Spring: Hundley

A continuation of Mathematics 125, covering integration, techniques for computing antiderivatives, the fundamental theorem of calculus, applications of the definite integral, and infinite series.

#### 128 Elementary Statistics

3, x Staff

Probability and statistics including methods for exploring data and relationships in data, methods for producing data, an introduction to probability and distributions, confidence intervals, and hypothesis testing. *Prerequisite**:* two years of high school mathematics.

#### 167 Introduction to Computer Programming

3, 3 Schueller

An introduction to programming techniques applicable to most high-level programming languages. Covers core programming topics including logic, loops, functions, and objects. Uses an object-oriented programming language like C++ or Java. Frequent programming projects are required.

#### 203, 204 Special Topics in Introductory Level Mathematics

1-3

On occasion, the mathematics department will offer courses on introductory topics in mathematics that are not generally covered in other introductory courses. Possible topics include Introduction to Number Theory, Chaos and Applied Discrete Probability. Any current offerings follow.

#### 204 ST: Philosophy and Foundations of Mathematics

x, 3 Doyle

An introduction to the study of the philosophy and foundations of mathematics. Topics may include: philosophical and mathematical background of the early 20th century crisis in the foundations of mathematics; survey of 20th century accounts of mathematical truth and knowledge advocated by both mathematicians and philosophers, including Formalism, Logicism, Intuitionism, and Structuralism; late 20th century work by philosophers and mathematicians that aims to reconcile philosophical/foundational theory with the social and technological aspects of mathematical practice. The course will assume familiarity with calculus and mathematical proof. *Pre-* or *corequisite:* Mathematics 260 or consent of instructor. Distribution area: quantitative analysis.

#### 225 Calculus III

4, 4 Fall: Schueller; Spring: Balof

Topics include partial derivatives, gradients, extreme value theory for functions of more than one variable, multiple integration, line integrals, and various topics in vector analysis.

#### 235 Calculus Laboratory

x, 1 Hundley

A laboratory to investigate ways in which the computer can help in understanding the calculus and in dealing with problems whose solutions involve calculus. No programming required; a variety of existing programs will be used. *Pre-* or *corequisite:* Mathematics 225.

#### 240 Linear Algebra

3, 3 Fall: Schueller; Spring: Keef

This course first considers the solution set of a system of linear equations. The ideas generated from systems of equations are then generalized and studied in a more abstract setting, which considers topics such as matrices, determinants, vector spaces, inner products, linear transformations, and eigenvalues. Prerequisite: Mathematics 225.

#### 244 Differential Equations

3, 3 Fall: Hundley; Spring: R. Gordon

This course includes first and second order linear differential equations and applications. Other topics may include systems of differential equations and series solutions of differential equations. *Prerequisite:* Mathematics 225.

#### 247 Statistics with Applications

3, x Staff

An introduction to statistics for students who have taken at least one course in calculus. Focuses on learning statistical concepts and inference through investigations. Topics include, but are not limited to, exploratory graphics, sampling methods, randomization, hypothesis tests, confidence intervals, and probability distributions. A statistical software package will be used. *Prerequisite: *Mathematics 125 or equivalent.

#### 248 Statistical Modeling

x, 3 McConville

This course follows introductory statistics by investigating more complex statistical models and their application to real data. The topics may include simple linear regression, multiple regression, non-parametric methods, and logistic regression. A statistical software package will be used. *Prerequisite: **Mathematics 128, Mathematics 247, Biology 228, Economics 227, or Environmental Studies 207.*

#### 260 An Introduction to Higher Mathematics

x, 3 R. Gordon

An introduction to some of the concepts and methodology of advanced mathematics. Emphasis is on the notions of rigor and proof. This course is intended for students interested in majoring in mathematics; students should plan to complete it not later than the spring semester of the sophomore year. *Prerequisite**:* Mathematics 225.

#### 270 Data Structures with C++

3, x Tucker

We study fundamental methods used to store, access, and manipulate data in computers. Storage structures to be covered include files, lists, tables, graphs, and trees. We will discuss and analyze methods of searching for and sorting data in these structures. *Prerequisite**:* Mathematics 167.

#### 281, 282 Independent Study

1-3, 1-3 Staff

A reading project in an area of mathematics not covered in regular courses or that is a proper subset of an existing course. The topic, selected by the student in consultation with the staff, is deemed to be introductory in nature with a level of difficulty comparable to other mathematics courses at the 200-level. May be repeated for a maximum of six credits. *Prerequisite:* consent of supervising instructor.

#### 287 Independent Study in Geometry

3, 3 R. Gordon

This independent study in geometry will include a review of high school geometry, a few topics in advanced Euclidean geometry, a reading of Books I and II of Euclid's Elements, and an introduction to hyperbolic geometry. The grading for the course will be based on a journal (20%), a two-hour written midterm exam (40%), and a one-hour oral final exam (40%). Since the student will be working independently on the material, a disciplined work ethic is required. *Prerequisite: **Mathematics 225.*

#### 299 Problem-Solving in Mathematics

1, x Balof

Students will meet weekly to discuss problem-solving techniques. Each week a different type of problem will be discussed. Topics covered will include polynomials, combinatorics, geometry, probability, proofs involving induction, parity arguments, and divisibility arguments. The main focus of the course will be to prepare students for the William Lowell Putnam Mathematics Competition, a national examination held the first Saturday in December. Students who place in the top 500 on this exam nationwide have their names listed for consideration to mathematics graduate programs. Graded credit/no credit. May be repeated for a maximum of four credits. *Prerequisite**:* consent of instructor.

#### 337 Geometry

3; not offered 2014-15

Essential for prospective high school mathematics teachers, this course includes a study of Euclidean geometry, a discussion of the flaws in Euclidean geometry as seen from the point of view of modern axiomatics, a consideration of the parallel postulate and attempts to prove it, and a discussion of the discovery of non-Euclidean geometry and its philosophical implications. *Prerequisite**:* Mathematics 126.

#### 339 Operations Research

3; not offered 2014-15

Operations research is a scientific approach to determining how best to operate a system, usually under conditions requiring the allocation of scarce resources. This course will consider deterministic models, including those in linear programming (optimization) and related subfields of operations research. *Prerequisite:* Mathematics 240; 167 or 235.

#### 349 Probability Theory

x, 3 Staff

A formal introduction to probability and randomness. The topics of the course include but are not limited to conditional probability, Bayes’ Theorem, random variables, the Central Limit Theorem, expectation and variance. Both discrete and continuous probability distribution functions and cumulative distribution functions are studied. *Prerequisite**:* Mathematics 225.

#### 350 Mathematical Modeling and Numerical Methods

3, x Hundley

This course explores the process of building, analyzing, and interpreting mathematical descriptions of physical processes. Topics may include feature extraction, partial differential equations, neural networks, statistical models. The course will involve some computer programming, so previous programming experience is helpful, but not required. *Prerequisite:* a first course in Statistics or consent of instructor.

#### 358 Combinatorics and Graph Theory

3; not offered 2014-15

Topics in elementary combinatorics, including: permutations, combinations, generating functions, the inclusion-exclusion principle, and other counting techniques; graph theory; and recurrence relations. *Prerequisites**:* Mathematics 260 or consent of instructor.

#### 367 Engineering Mathematics

x, 3 Schueller

An introduction to mathematics commonly used in engineering and physics applications. Topics may include: vector analysis and applications; matrices, eigenvalues, and eigenfunctions; boundary value problems and spectral representations; Fourier series and Fourier integrals; solution of partial differential equations of mathematical physics; differentiation and integration of complex functions, residue calculus, conformal mapping. *Prerequisite:* Mathematics 244.

#### 368 Complex Variables

x, 3 Schueller

Complex analysis is the study of functions defined on the set of complex numbers. This introductory course covers limits and continuity, analytic functions, the Cauchy-Riemann equations, Taylor and Laurent series, contour integration and integration theorems, and residue theory. *Prerequisite**:* Mathematics 225.

#### 385, 386 Abstract Algebra

4, 4 Fall: Balof, Guichard; Spring: Guichard

The first semester is an introduction to groups and rings, including subgroups and quotient groups, homomorphisms and isomorphisms, subrings and ideals. Topics for the second semester may include fields, simple groups, Sylow theorems, Galois theory, and modules. *Prerequisite**:* Mathematics 260.

#### 438 Statistical Theory

4; not offered 2014-15

This course studies the mathematical theory of statistics with a focus on the theory of estimation and hypothesis tests. Topics may include properties of estimators, maximum likelihood estimation, convergence in probability, the central limit theorem, order statistics, moment generating functions, and likelihood ratio tests. A statistical software package will be used. *Prerequisites:* Mathematics 349 and one of Mathematics 128, Mathematics 247, Biology 228, Economics 227, or Environmental Studies 207.

#### 455, 456 Real Analysis

x, 4 (455 offered spring; 456 not offered 2014-15) Schueller

First semester: a rigorous study of the basic concepts of real analysis, with emphasis on real-valued functions defined on intervals of real numbers. Topics include sequences, continuity, differentiation, integration, infinite series, and series of functions. Second semester: content varies from instructor to instructor but includes topics from metric spaces, the calculus of vector-valued functions, and more advanced integration theory. *Prerequisite**:* Mathematics 260.

#### 467 Numerical Analysis

3; not offered 2014-15

An introduction to numerical approximation of algebraic and analytic processes. Topics include numerical methods of solution of equations, systems of equations and differential equations, and error analysis of approximations. *Prerequisite**:* Mathematics 167. *Pre-* or *corequisite**:* Mathematics 240.

#### 471, 472 Special Topics

1-3

On occasion, the mathematics department will offer courses on advanced topics in mathematics that are not found in other course offerings. Possible topics include topology, number theory, and problem-solving. Any current offerings follow.

#### 471 ST: Number Theory

3, x Keef

An introduction to the theory of numbers. Topics to be considered will include congruences, sums of powers of integers, quadratic reciprocity and quadratic forms, Farey and continued fractions, arithmetic functions and the Mobius inversion formula. *Prerequisite:* Math 260 or consent of instructor. Distribution area: quantitative analysis.

#### 472 ST: Introduction to Point-set Topology

x, 3 Guichard

Standard topics in point-set topology, including open sets, closed sets, continuous functions, connectedness, compactness, metric spaces. *Prerequisite:* Math 260. Distribution area: quantitative analysis.

#### 481, 482 Independent Study

1-3, 1-3 Staff

A reading or research project in an area of mathematics not covered in regular courses. The topic is to be selected by the student in consultation with the staff. Maximum of six credits. *Prerequisite*: consent of supervising instructor.

#### 497 Senior Project

x, 4 Schueller

Preparation of the senior project required of all graduating mathematics majors. Each student will be matched with a faculty member from the mathematics department who will help supervise the project. Course objectives include developing students’ abilities to independently read, develop, organize, and communicate mathematical ideas, both orally and in writing. A final written and oral report on the project is completed.

#### 498 Honors Thesis

4, 4 Staff

Preparation of an honors thesis. Required of and limited to senior honors candidates in mathematics. Students will be a part of the Mathematics 497 *Senior Project* class (described above), but their work will be held to a higher standard. P*rerequisite: *admission to honors candidacy.