Mathematics

Eric Westlund (department head)

The department offers two majors: mathematics and mathematics/statistics. For purposes of advising students, advanced courses may be classified as follows: MATH 253, 351, 452, and 462 are called applied; MATH 365, 454, 456, 459, 471, and 472 are called foundational; and MATH 321, 322, 327, and 328 are called statistical.

Mathematics is an extremely versatile major. Mathematics is both the primary language and the theoretical foundation of modern technology, yet its roots in Western culture are as deep as those of any classical discipline.

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 cover the undergraduate mathematics necessary for an understanding of statistics.

Requirements for majors:

Math: Plan I (Theory). MATH 220 and 240; either MATH 454 or 471; CS 150 or 160; and a minimum of three additional courses in the math department numbered 200 or above, including at least one additional from MATH 365, 454, 456, 459, 471, or 472. Writing requirement completed with MATH 220, MATH 240, and MATH 454 or 471.

Math: Plan II (Applications). MATH 220 and MATH 240; either MATH 454 or 471; CS 150 or 160; and four additional courses in the math department numbered 200 or above. Writing requirement completed with MATH 220, MATH 240, and MATH 454 or 471. Only two of MATH 321, 322, 327, and 328 may count towards a mathematics Plan II major.

Math: Plan III (Teaching). A Plan I mathematics major that includes MATH 321, 365, and 471. Complete one of MATH 253, 322, or 351. Writing requirement completed with MATH 220, MATH 240, and MATH 454 or 471. Credit hours of required and elective mathematics courses must total a minimum of 30 hours. See education department for secondary education minor requirements.

Mathematics/Statistics: MATH 220, 240, 253, 321, 322, 327, 328, and CS 150 or 160. MATH 454 is recommended. Writing requirement completed with MATH 220, MATH 240, and MATH 327 or 328. (A student may not major in both mathematics and mathematics/statistics.)

Required for a mathematics minor: At least five courses in mathematics, including MATH 240 and two additional courses numbered 200 or above.

Required for a second teaching area: At least 24 credit hours, including MATH 151, 152, 220, 240, 321, and 365. CS 150 or 160, and EDUC 352 are required. The second teaching area license is offered only in the state of Iowa.

Suggested patterns for majors planning careers in the following areas:

  • Graduate study in mathematics: Plan I including MATH 253, 321, 351, 454, 456, 459, 471, 472.
  • Statistics: A mathematics/statistics major.
  • Actuarial science: A mathematics/statistics major plus ECON 247 and ECON 248, MGT 353 and MGT 365, and courses in computer science.
  • Computing/data science: MATH 220, 321, 322, 327, 328, 462, 471.
  • Management (accounting, economics): MATH 253, 321, 322, 327, 454.
  • Science: MATH 253, 321, 322, 327, 328, 351, 452, 454, 456, 462, 471.

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

First-Year Placement

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 who need calculus for their major but also need a review of algebra and trigonometry should take the MATH 140 and 141 sequence. Students with good algebra and trigonometry skills should begin with the traditional Calculus I course, MATH 151. Students who have completed a year of calculus in high school and perform well on the Advanced Placement A/B Exam or the Calculus I portion of the mathematics placement test should start in Calculus II, MATH 152. Students who perform well on the Advanced Placement B/C Exam or the Calculus II portion of the mathematics placement test should start in MATH 220 or 240.

Advanced Placement Credit

  • 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 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).

Mathematics Courses

MATH 110 Mathematics in Our World

  • 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.

MATH 115 Introduction to Statistics

  • 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.

MATH 123 Mathematics for Elementary School Teachers

  • 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.

MATH 140 Calculus I with Review, Part A

  • 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.

MATH 141 Calculus I with Review, Part B

  • 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.)

MATH 151 Calculus I

  • 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.

MATH 152 Calculus II

  • 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.

MATH 220 Mathematical Reasoning and Writing

  • 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.

MATH 240 Linear Algebra

  • 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.

MATH 253 Vector Calculus

  • 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.

MATH 258 Chaotic Dynamical Systems

  • 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.

MATH 260 Elementary Number Theory

  • 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.

MATH 321 Probability and Statistics I

  • 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.

MATH 322 Probability and Statistics II

  • 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.

MATH 327 Applied Statistics I

  • 4 hours
  • Fulfills: Quantitative
  • Prerequisites: MATH 321, 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.

MATH 328 Applied Statistics II

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

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

MATH 351 Ordinary Differential Equations

  • 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.

MATH 365 Geometry

  • 4 hours
  • Fulfills: Quantitative
  • Prerequisites: MATH 220 and MATH 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.

MATH 452 Partial Differential Equations

  • 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.

MATH 454 Principles of Real Analysis

  • 4 hours
  • Fulfills: Quantitative
  • Prerequisites: MATH 220 and MATH 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.

MATH 456 Functions of a Complex Variable

  • 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.

MATH 459 Topology

  • 4 hours
  • Fulfills: Quantitative
  • Prerequisites: MATH 240 and MATH 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.

MATH 462 Numerical Analysis

  • 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.

MATH 471 Abstract Algebra I

  • 4 hours
  • Fulfills: Quantitative
  • Prerequisites: MATH 220 AND MATH 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.

MATH 472 Abstract Algebra II

  • 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.

MATH 490 Senior Project

  • 1, 2, or 4 hours