The academic catalog is currently being updated for the 2019-20 year. View the Catalog Archive to access the 2018-19 catalog as well as catalogs from previous years.

**Eric Westlund** (department head)

The department offers two majors: mathematics and mathematics/statistics.

Mathematics is the study of numbers, measurements, patterns, shapes, equations, relations, functions, change, symmetry, structure, sets and operations; the modeling of physical phenomena to better understand and predict nature; the development of theorems from accepted axioms through logical proof. It is abstract and applied, theoretical and experimental. Mathematics is perhaps the oldest academic discipline, yet mathematics is the primary language and theoretical foundation of modern technology. It is an extremely versatile major. Mathematics majors are encouraged to explore applications of mathematics in other disciplines, and it is a popular second major for students pursuing advanced degrees.

Statistics is the science of reasoning from uncertain empirical data. Statisticians build mathematical models to solve problems in business, the natural sciences and the social sciences. The intent of the mathematics/statistics major is to provide adequate preparation to attend graduate school in statistics or to pursue a career such as actuarial science.

**Requirements for majors:**

**Mathematics major.** MATH 220, 240, and 253; CS 150 or 160; MATH 215, 322, or 327; with a minimum of eight courses (32 credits) in mathematics numbered 200 or above, including at least three courses (12 credits) in mathematics numbered 300 or above. Writing requirement completed with MATH 220 or 240. (No more than two of MATH 215, 271, 322, 327, 328 can count toward the mathematics major.)

**Mathematics/Statistics major.** MATH 220, 240, and 253; CS 150 or 160; MATH 271, 322, 327, and 328; MATH 454 or DS 320. Writing requirement completed with MATH 220 or 240. (A student may not major in both mathematics and mathematics/statistics.)

**Mathematics minor.** At least five courses (20 credits) in mathematics, including MATH 220, 240 and three additional courses, of which two are numbered 200 or above.

Suggested electives for majors planning careers in the following areas:

- Graduate study in mathematics: MATH 351, 454, 456, 471, 472.
- Graduate study in statistics: A mathematics/statistics major with both MATH 454 and DS 320. Consider additional courses in computer science, data science, or a science discipline.
- Secondary education: MATH 360, 365, 471. See education department for secondary education minor requirements.
- Actuarial science: A mathematics/statistics major, ECON 247, ECON 248, MGT 353, MGT 365, and more courses in computer science or data science.
- Computer or data science: MATH 327, 328, 471.
- Management (accounting, economics): MATH 327, 454.
- Science: MATH 327, 328, 351, 452, 453, 454, 456.
- Applied statistics: A mathematics/statistics major with DS 320 and other courses in computer science and data science.

NOTE: Students earning a C or below in MATH 220 or 240 are advised not to take 300+ level courses.

The mathematics department placement procedure uses high school records, scores on ACT or SAT tests, and a placement test in mathematics as a basis for a recommendation. MATH 110 and MATH 115 are designed for students who will not be taking calculus. MATH 123 is only for students who major in elementary education. Students with good algebra and trigonometry skills should begin with the traditional Calculus I course, MATH 151. Students who need calculus for their major but also need a review of algebra and trigonometry should take the MATH 140 and 141 sequence. Students whose math placement suggests they require more in-depth review of algebra should consider completing MATH 100 before registering for MATH 140. Students who have completed a year of calculus in high school and perform well on the Advanced Placement A/B Exam or the calculus portion of the mathematics placement test should start in Calculus II, MATH 152. Students who perform well on the Advanced Placement B/C Exam should start in MATH 220 or 240.

- A student will receive credit for MATH 151 with a score of 4 or 5 on the AP Calculus A/B Exam. A student will receive credit for MATH 151 and MATH 152 with a score of 4 or 5 on the AP Calculus B/C Exam. A student will receive credit for MATH 151 with an IB Mathematics HL score of 4-7.
- A student who places into MATH 152 will receive, upon completion of the course with a grade of C or better, Luther College credit for the preceding calculus course, MATH 151 (if credit has not already been awarded through AP or transfer credit).
- A student who places into MATH 240 may petition the head of the mathematics department, upon completion of the course with a grade of C or better, to receive Luther College credit for the preceding calculus course(s) MATH 151 and/or MATH 152 (if credit has not already been awarded through AP or transfer credit). Approval of the petition will depend upon whether the student has previously covered the content of MATH 151 and/or MATH 152.
- A student who achieves a score of 4 or 5 on the AP Statistics Exam will receive credit for MATH 115.

NOTE: AP credit for MATH 115, 151, or 152 satisfies the all-college requirement for quantitative perspective (QUANT).

- 2 hours

This course is focused on strengthening algebraic and quantitative skills required for success in science, economics, or business majors. By preparing students for the first semester of Calculus, this course is appropriate for those desiring an entry level college mathematics course before completing Math 140 in the following semester. Topics include simplifying mathematical expressions, functions and graphs, solving polynomial/rational equations in one variable, exponents, quantitative reasoning and mathematical models.

- 4 hours
- Fulfills: Quantitative

Quantitative literacy plays an important role in an increasing number of professional fields, as well as in the daily decision-making of informed citizens in our changing society. This course is designed to improve students' quantitative reasoning and problem-solving skills by acquainting them with various real-world applications of mathematical reasoning, such as fair division, voting and apportionment, graph theory, probability, statistics, the mathematics of finance, check digits and coding, and geometry. This course is recommended for students who wish to take a non-calculus-based mathematics class as they prepare for their lives as informed members of a larger world. Prerequisite: high school algebra.

- 4 hours
- Fulfills: Quantitative

The course uses data sets from the social and natural sciences to help students understand and interpret statistical information. Computer software is used to study data from graphical and numerical perspectives. Topics covered include descriptive statistics, correlation, linear regression, contingency tables, probablity distributions, sampling methods, confidence intervals, and tests of hypotheses. This class does not count towards the mathematics major or minor or the mathematics/statistics major. Students who earn credit for BIO 256, MGT 150, PSYC 350, or SOC 350 may not earn credit for MATH 115. Prerequisite: high school algebra.

- 4 hours
- Fulfills: Quantitative

This course provides pre-service K-8 teachers a strong foundation in the mathematics content areas as described in NCTM's Principles and Standards for School Mathematics. The content standards include: Number and Operations, Algebra, Geometry, and Measurement. This course will engage students in standards-based mathematics learning to prepare them for the pedagogical practices they will learn in EDUC 325. Prerequisite: one year of high school algebra, one year of high school geometry and admission into the teacher education program. Co-requisite: EDUC 325 and 326.

- 4 hours
- Fulfills: Quantitative

MATH 140 and 141 cover all the material in MATH 151; Calculus I, while concurrently reviewing precalculus material. Algebraic and graphical representations of functions including: polynomial, rational, exponential, and logarithmic; techniques of solving equations and inequalities; modeling with various functions. An introduction to calculus concepts such as instantaneous rates of change: limits; derivatives; continuity; applications of derivatives. Graphing calculator use may be required. Prerequisite: a suggested placement.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 140

Continuation of topics of MATH 140: trigonometric functions, derivatives, chain rule, the mean value theorem, Riemann sum approximation for integrals, definite integrals, antiderivatives, and applications. Graphing calculator use may be required. (Students who earn credit for MATH 141 may not earn credit for MATH 151.)

- 4 hours
- Fulfills: Quantitative

Topics related to instantaneous rates of change: functions, limits, continuity, derivatives, mean value theorem and applications; antiderivatives and definite integrals. Graphing calculator use may be required. (Students who earn credit for MATH 151 may not earn credit for MATH 140 or 141). Prerequisite: three years of high school mathematics including algebra, trigonometry, and geometry, and a suggested placement.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 141 or 151 or suggested placement

Applications of the definite integral, techniques of integration, differential equations, power series, Taylor series, and an introduction to computer algebra systems.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 152

This course introduces students to logic, set theory, and methods of proof. The emphasis will be on learning to write rigorous mathematical proofs. Topics will include induction, functions, equivalence relations, cardinality, combinatorics, and recurrence relations.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 152

Theory, computation, abstraction, and application are blended in this course, giving students a sense of what being a mathematics major is all about. Assignments will include computations to practice new techniques and proofs to deepen conceptual understanding. This course starts by solving systems of linear equations, views matrices as linear transformations between Euclidean spaces of various dimensions, makes connections between algebra and geometry, and then extends the theory to more general vector spaces. Topics include matrix algebra, vector spaces and subspaces, linear independence, determinants, bases, eigenvalues, eigenvectors, orthognality, and inner product spaces.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 240

The tools of calculus are developed for real-valued functions of several variables: partial derivatives, tangent planes to surfaces, directional derivatives, gradient, maxima and minima, double and triple integrals, and change of variables. Vector-valued functions are also studied: tangent and normal vectors to curves in space, arc length, vector fields, divergence and curl. The fundamental theorem of calculus is extended to line and surface integrals, resulting in the theorems of Green, Stokes, and Gauss, which have applications to heat conduction, gravity, electricity and magnetism.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 240

Why is it so difficult to make accurate predictions about seemingly chaotic physical systems like weather? This course explores the behavior of nonlinear dynamical systems described by iterated functions. A variety of mathematical methods, including computer modeling, is used to show how small changes in initial conditions can drastically change the future behavior of the system. Topics will include periodic orbits, phase portraits, bifurcations, chaos, symbolic dynamics, fractals, Julia sets, and the Madelbrot set. Offered alternate years.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 220 or 240

Fibonacci numbers, divisibility theory in the integers, prime numbers, Euclidean algorithm, Diophantine equations, congruences, divisbility tests, Euler's theorem, public key cryptography. Usually offered in alternate January terms.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 152

Axioms and laws of probability, conditional probability, combinatorics, counting techniques, independence, discrete and continuous random variables, mathematical expectation, discrete probability distributions, continuous probability distributions, functions of random variables, joint probability distributions and random samples, statistics and their distributions, central limit theorem, distribution of a linear combination of random variables.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 321

Sampling distribution of mean, standard deviation and proportion, theory of estimation, methods of point estimation, hypothesis testing, large and small sample confidence intervals, inferences for means, proportions and variances. Distribution free procedures.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 115, MATH 321, MATH 322, BIO 256, PSYC 350, SOC 350, ECON 242 or MGT 150

Regression Analysis: Least square estimates, simple linear regression, multiple linear regression, hypothesis testing and confidence intervals for linear regression models, prediction intervals, and ANOVA. Model diagnostics including tests of constant variance assumptions, serial correlation, and multicollinearity. Time series: Linear time series, moving average, autoregressive, and ARIMA models. Estimation and forecasting. Forecast errors and confidence intervals.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 327

Design and analysis of experiments; analysis of variance techniques; fixed, random, and mixed models; repeated measures.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 240

Differential equations is an area of theoretical and applied mathematics in which there are a large number of important problems associated with the physical, biological, and social sciences. Analytic (separation, integration factors, and Laplace transforms), qualitative (phase and bifurcation diagrams), and numerical (Runge-Kutta) methods are developed for linear and nonlinear first- and higher-order single equations as well as linear and non-linear systems of first-order equations. Emphasis is given to applications and extensive use of a computer algebra system.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 220 and 240

This course follows the historical development of geometry, indcluding the important question of which parallel postulate to include. This is a proof-oriented course focusing on theorems in plane Euclidean and hyperbolic geometry, with some mention of elliptic geometry. We examine the development of a lean set of axioms, (incidence, betweenness, congruence, continuity) and investigate which theorems about points and lines can be derived using them.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 351

An introduction to initial and boundary value problems associated with certain linear partial differential equations (Laplace, heat and wave equations). Fourier series methods, including the study of best approximation in the mean and convergence, will be a focus. Sturm-Liouville problems and associated eigenfunctions will be included. Numerical methods, such as finite difference, finite element and finite analytic, may be introduced, including the topics of stability and convergence of numerical algorithms. Extensive use of a computer algebra system.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 220 and 240

The course studies functions of a real variable and examines the foundations of calculus, with an emphasis on writing rigorous analytical proofs, and follows the historical development of analysis from Newton to Leibniz through Lagrange, Cauchy, Bolzano, Weierstrass, Cantor, Riemann, and Lebesgue. Topics include the topology of the reals, sequences, series, limits, continuity, pointwise and uniform convergence, differentitaion, Taylor series, and integration.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 253

What happens when calculus is extended to functions of a complex variable? Geometry and analysis combine to produce beautiful theorems and surprising applications. Topics include complex numbers, limits and derivatives of complex functions, Cauchy-Riemann equations, harmonic functions, contour integrals, the Cauchy integral formula, Taylor and Laurent series, residues, and conformal mappings with applications in physical sciences. Offered in alternate years.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 240 and 220

What properties of a space are preserved by a continuous transformation? Point-set topology uses the concept of an "open set" to extend the definition of continuity to many different spaces. Topics may include the order, metric, product, and subspace topologies; limit points, connectedness, compactness, countability axioms, separation axioms, the Urysohn lemma, and the Urysohn metrization theorem. Usually offered in alternate January terms.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 240 and CS 150

Numerical analysis is a blend of computational methods and mathematical analysis of their convergence, accuracy, and stability. Topics include roots of equations and solutions of systems of linear equations, function interpolation and approximation, numerical differentiation and integration, optimization, and numerical solutions of ordinary and partial differential equations. Offered in alternate years.

- 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 220 and 240

Real numbers and integers satisfy many nice properties under addition and multiplication, but other sets behave differently: matrix multiplication and composition of functions are noncommutative operations. Which properties (associativity, commutativity, identity, inverses) are satisfied by operations on sets determine the basic algebraic structure: group, ring, or field. The internal structure (subgroups, cosets, factor groups, ideals), and operation-preserving mappings between sets, (isomorphisms, homomorphisms)are examined. Emphasis is on theory and proof, although important applications in symmetry groups, cryptography, and error-correcting codes may also be covered.

- 1, 2, or 4 hours
- Fulfills: Quantitative
- Prerequisites: MATH 471

Topics may include simple groups, Sylow theorems, divisibility in integral domains, generators and relations, field extensions, splitting fields, solvability by radicals, Galois theory, symmetry and geometric constructions. Offered on demand.

- 1, 2, or 4 hours