Mathematics
Omnia in mensura et numero et pondere disposuisti.
Thou hast ordered all things in measure, and number, and weight.
But when we begin to look above ourselves again, we find that numbers transcend our minds and remain fixed in the truth itself.
Historically, from Plato’s Academy, to Aristotle’s Lyceum, to the school at Alexandria, to the great medieval universities and those of our own day, mathematics has always been an integral part of a liberal arts education. The application of mathematics to the natural sciences has become instrumental in understanding the natural world and in the many advances in science, engineering, and technology. More importantly, mathematics has been prized since antiquity for its capacity to train the intellect. It is a discipline in which precise definition and clarity of logical demonstration provide a level of certitude that other fields of study do not possess. Ultimately, as one of the speculative sciences, mathematics is worthy of study for its own sake. As such it plays a part in leading the student to contemplation of the good, the true, and the beautiful, and to the end (télos) of education, which, in the words of Saint John Henry Cardinal Newman, is “... the enjoyment of a divine citizenship.”
Mathematics Core
To fulfill the core requirement in mathematics students must take one mathematics course, usually in the freshman year. There are several courses to choose from, depending on the semester, each of which introduces the student to basic concepts and methods of mathematics and its place in man’s attainment of knowledge, wisdom, and the philosophical habit of mind. By honing the powers of reason, the core mathematics courses enhance the student’s critical thinking and problem-solving abilities.
Mathematics Major
The Department offers a mathematics major comparable in both scope and rigor to a standard Bachelor of Arts in Mathematics at institutions across the country. The difference at Christendom is that unlike schools that focus solely on science and technology, our courses are placed within the context of the liberal arts curriculum.
Opportunities for math majors abound. Some students pursue graduate study in mathematics or a teaching career. However, math majors also excel in engineering, law, medicine, economics, finance, data science, technology, and a host of other professions in business, industry, and government.
In the freshman and sophomore years, the requirements for the major include a three-semester calculus sequence, two semesters of physics, and a bridge course in the fundamentals of advanced mathematics. Students may take advantage of Christendom’s Junior Semester in Rome in the fall semester of the junior year, and then take linear algebra and differential equations in the spring. The senior year math courses comprise real analysis, abstract algebra, two electives, a one-hour senior seminar, and the senior thesis, which affords the opportunity to work with a professor and delve into a specific area of mathematics, connecting it to the broader context of the core curriculum.
REQUIREMENTS FOR THE MAJOR AND MINOR
Ten upper-level courses in Mathematics (34 semester hours) are required of majors – eight designated and two electives. The designated courses are:
MATH 201 Calculus I (4 cr.)
MATH 302 Calculus II (4 cr.)
MATH 303 Calculus III
MATH 304 Differential Equations
MATH 351 Fundamentals of Advanced Mathematics
MATH 353 Linear Algebra (4 cr.)
MATH 401 Real Analysis (4 cr.)
MATH 402 Abstract Algebra
MATH 509 Senior Seminar (1 cr.) (Fall)
MATH 512 Senior Thesis (3 cr.) (Spring)
SCIE 204 & SCIE 204L General Physics I & Lab (4 cr.)
SCIE 205 & SCIE 205L General Physics II & Lab (4 cr.)
To complete the mathematics major, students must take at least two additional mathematics elective courses numbered 300 or above.
Note: MATH 201 satisfies the core requirement for mathematics and SCIE 204 satisfies the core
requirement for natural science. A course grade of at least C- is necessary in a math course to
fulfill the department’s requirements for a major or minor, and students must demonstrate competency in physics by receiving a passing grade (D or above) in SCIE 204 and SCIE 205, along with their lab components.
A minor in mathematics may be obtained by completing the calculus sequence (MATH
201, MATH 302, MATH 303) and three additional math courses numbered 300 or above. (A General Physics course with lab may be substituted for one of the latter if a grade of at least C- is earned and it is not used to satisfy the core science requirement. MATH 201 satisfies the core requirement for mathematics).
FOUNDATIONAL CURRICULUM
MATH 101 Introduction to Mathematical Thought This course focuses on our changing conception of the notion of extension leading to the rise of the various branches of mathematics and the application of mathematics to describing the universe.
MATH 102 The Integers Explores the fundamental properties of the integers. Topics include: natural numbers, integers, rational numbers, irrational numbers, prime numbers, the division algorithm, prime factorization, modular arithmetic, and RSA encryption. This course has no prerequisites and satisfies the College core requirement in mathematics. (3 credits)
MATH 103 Euclidean Geometry A study of selected books from Euclid’s Elements. Topics covered include plane geometry, theory of proportions, and classical arithmetic. Students will also investigate the relation between mathematics and more comprehensive philosophical issues.
MATH 150 Introduction to Statistics The purpose of the course is to introduce the student to the ideas and concepts of statistics and the statistical models used for the decision making in different areas of life. Topics covered include description of sets of data, elementary probability, discrete and continuous random variables, the binomial and normal random variables, confidence intervals and hypothesis testing.
MATH 201 Calculus I Basic course in differential calculus with an introduction to integration. Topics covered include limits and continuity, the notion of the derivative, techniques of differentiation, the definite and indefinite integral, and the fundamental theorem of calculus. Prerequisite: Precalculus or permission of the instructor. (4 credit hrs)
ADVANCED COURSES
MATH 302 Calculus II Continuation of MATH 201. Topics include inverse functions, techniques of integration, sequences and series, the conic sections, and the polar coordinate system. Prerequisite: At least a C- in MATH 201 or equivalent. Required of all majors (4 credit hrs)
MATH 303 Calculus III Continuation of MATH 302. Topics include limits and continuity in three dimensions, vectors, vector functions, partial derivatives, multiple integrals, the notions of gradient, divergence, and curl, and the basic theorems of vector calculus. Required of all majors.
MATH 304 Differential Equations This course covers the basic techniques for solution of ordinary differential equations. Topics include first and second order linear equations, non-linear equations, systems of linear equations, the fundamental matrix, series solutions of differential equations, numerical methods and introduction to stability theory. Required of all majors.
MATH 332 Probability and Statistics Introduction to the basic notions of probability and statistics. Topics covered include combinatorial probability, distribution functions, discrete and continuous random variables and distributions, conditional probability, the central limit theorem, and typical applications in reliability, sampling, and estimation theory including Bayesian Inference.
MATH 351 Fundamentals of Advanced Mathematics This course introduces the student to modern mathematical structures that are not present in introductory mathematics courses and aims to develop a student’s skill in composing and writing proofs. Topics include elementary logic, methods of proof, philosophies of mathematics, set theory, functions and relations, cardinality, and elementary number theory. Required of all majors.
MATH 353 Linear Algebra Introduction to the concepts and theory of linear algebra. Topics include vector spaces, bases, matrices, linear mappings, scalar products and orthogonality, determinants, bilinear forms, eigenvalues and eigenvectors, diagonalization, the spectral theorem and the SVD decomposition. Required of all majors (4 credit hrs)
MATH 401 Real Analysis This course is a rigorous introduction to the fundamental theorems of the introductory calculus courses. It aims to develop in the student a sense of the unity of mathematics and further expose him to the importance of rigorous proof in mathematics. Topics include: the real number system, sequences and limits, continuity of functions, the derivative, and the Riemann integral. Required of all majors. (4 credit hrs).
MATH 402 Abstract Algebra This course is an introduction to the ideas of modern algebra which enables one to reinterpret the results of classical algebra, giving them a greater unity and generality. Topics include: equivalence relations, functions, properties of the integers, groups, rings, integral domains, ideals, and fields. Required of all majors.
MATH 403 Foundations of Geometry Exploration of axiomatic systems of geometry, beginning with the neutral geometry of Euclid’s Elements, in contrast to that of spherical and taxicab geometries, which motivates Hilbert’s axiomatization of Euclidean geometry. Further topics include independence, consistency, and completeness of axiomatic systems of geometry, Euclid’s geometry with the parallel postulate, and hyperbolic geometry. Prerequisite: Calculus II and MATH 351.
MATH 409 Number Theory An algebraic and historical approach to the theory of numbers. Topics include: the natural numbers and their properties, the Euclidean Division Algorithm, unique prime factorization, modular arithmetic, RSA Encryption, the Gaussian integers and other quadratic integer rings, and the Law of Quadratic Reciprocity. Prerequisite: MATH 351.
MATH 462 Combinatorics Study of questions concerning enumeration and arrangements of objects. Topics include the Pigeonhole Principle, the Principle of Inclusion/Exclusion, recurrence relations, generating functions, graph theory, and other algebraic counting techniques. Prerequisite: MATH 351
MATH 490-99 Special Topics or Directed Studies in Mathematics A topic chosen according to the interests of the students and the instructor, such as: applied mathematics, game theory, discrete mathematics, number theory, philosophy of mathematics, history of mathematics.
MATH 509 Senior Seminar A seminar on topics in mathematics to be determined by the instructor in consultation with the students. The student gains experience in learning the development of a topic, research methods, organizing and writing a mathematical paper. (1 credit hr)
Math 512 Senior Thesis Direction of the students with his senior thesis, a major scholarly paper on a mathematical topic of his interest. The student receives instruction and individual assistance in development of a topic, research methods, organizing and writing a mathematical paper.