UW-Madison offers several distinct courses that introduce students to writing formal mathematical proofs. These courses are primarily intended for students who plan to take our 500- and 600-level courses, almost all of which are proof-based. In most cases, any one of these courses formally satisfies the introductory proofs prerequisite of the advanced proof-based courses. However, students should check the official requisites in the Guide.
In order to complete the major in mathematics you must take an introduction to proofs class or the Applied Mathematical Analysis sequence (MATH 321-322). The named options explicitly require such a course under the “Intermediate Mathematics” requirement. For the standard major, it is impossible to complete the “Analysis, Topology, Algebra” requirement without first taking either an introduction to proofs class or MATH 322.
The purpose of this page is to describe the essential differences between the four introduction to proofs classes. Note that we do allow students to take more than one of these courses to count toward the total courses/credits required for the math major. However, students must consult the Guide to verify if a particular course will count toward their requirements.
MATH 341 (Linear Algebra)
MATH 341 is a linear algebra course which is also meant to be an introduction to proofs and proof-writing. The linear algebra content of this course is comparable to that in MATH 340, but at a more accelerated pace because of the devotion of time to proofs and writing proofs. Students who complete the course should be well-prepared to move on to any upper-level course, in particular MATH 521, 541, or 551.
It is the recommended linear algebra course for majors interested in moving to advanced undergraduate courses quickly. Students who prefer to separate their introduction to linear algebra from their introduction to proof should consider MATH 340 for linear algebra and MATH 421 or MATH 467 for introduction to proof.
Due to the more intensive proof-writing nature of the course, students will probably find the course more demanding than MATH 340, and for this reason MATH 341 carries the Accelerated Honors (!) label.
Students who complete this course and would also like exposure to differential equations should consider MATH 319.
In summary, MATH 341...
- Is Honors-level;
- Is accepted in both the major and certificate programs;
- Is a good introduction to proofs and proof-writing;
- Covers material in MATH 320 and therefore credit for only one of MATH 320 or 341 can be applied to the math major or certificate;
- Covers material in MATH 340 and therefore credit for only one of MATH 340 or 341 can be applied to the math major or certificate;
- Covers material in MATH 375 and therefore credit for only one of MATH 341 or 375 can be applied to the math major or certificate;
- Will give students access to advanced-level undergraduate math courses.
Suggested further courses are...
- MATH 421 for another exposure to formal mathematical arguments at the introductory-level;
- Any math course above the 500-level (possibly assuming other prerequisites).
MATH 375 (Topics in Multi-Variable Calculus and Linear Algebra)
MATH 375 is an Accelerated Honors (!) course which features the role that linear algebra has in multivariable calculus. It also provides students with an introduction to proofs and proof-writing. In terms of subsequent coursework, MATH 375 generally fulfills any prerequisite that includes MATH 341.
It is assumed that students who complete this course will move on to complete the sequel course, MATH 376. (Students who complete MATH 375 and not MATH 376 are not considered to have completed the content of MATH 234! By enrolling in MATH 375 in the fall, students should be prepared to enroll in MATH 376 in the spring. Otherwise, they will need to enroll in MATH 234 in order to complete multivariate calculus.)
In summary, MATH 375:
- Is Honors-level;
- Enrollment is by permission only;
- Is accepted in both the major and certificate programs;
- Is not a course you can take if you have credit for one or more of MATH 234, 319, 320, 340, or 341;
- Is a good introduction to proofs and proof-writing;
- Covers material in MATH 320 and therefore credit for only one of MATH 320 or 375 can be applied to the math major or certificate;
- Covers material in MATH 340 and therefore credit for only one of MATH 340 or 375 can be applied to the math major or certificate;
- Covers material in MATH 341 and therefore credit for only one of MATH 341 or 375 can be applied to the math major or certificate;
- Will give students access to a number of advanced-level undergraduate math courses, although some may also require MATH 376.
More information on the MATH 375/376 sequence can be found on our Honors calculus page.
MATH 421 (The Theory of Single Variable Calculus)
This course is designed to introduce students to formal mathematical proof in the context of single variable calculus. In MATH 221-222, our introductory single variable calculus courses, there typically is not much focus on proofs. Here, however, students learn how to prove statements about familiar ideas from single variable calculus. Exact calculus topics will vary from semester-to-semester, as the focus is on proof writing, but students might expect to learn how to give delta-epsilon proofs about limits, prove differentiation rules, understand the definite integral more deeply through proofs, or prove facts about series convergence.
Students in MATH 421 have typically taken or concurrently take an introductory linear algebra course, such as MATH 320 or MATH 340. Students who have taken MATH 341 and are confident in their proof skills are generally recommended to try a 500-level course, but taking MATH 421 as a second transition to proof experience for students who wish to get more practice with writing proofs before moving on to 500-level courses is not uncommon. Taking MATH 421 before linear algebra makes MATH 341 a great linear algebra course, as students will already be familiar with writing proofs and can refine those skills while learning the fundamentals of linear algebra in MATH 421.
In summary, MATH 421…
- Is a suitable introduction to proofs for 500- and 600-level mathematics courses;
- Is accepted in both the major (all named options included) and certificate programs;
- Introduces proof writing in a context with which students are already familiar; and
- Will give students access to advanced-level undergraduate math courses.
Suggested further courses are…
- MATH 341 (only if another linear algebra course has not already been taken);
- MATH 467 for additional introductory-level exposure to proofs and proof-writing; or
- Any math course above the 500-level (possibly assuming other prerequisites).
MATH 467 (Introduction to Number Theory)
This course is designed to introduce students to formal mathematical proof in the context of number theory, which is the area of mathematics that studies the properties of prime numbers. Compared to MATH 341, this course introduces fewer complicated mathematical structures that will be new to students, which gives a bit more time for students to focus on learning about writing proofs. However, compared to MATH 421, which is based almost entirely on mathematical concepts students are already familiar with, there are still a number of new concepts related to prime numbers that students will encounter in MATH 467.
This is the only of our four introduction to proofs courses that can be taken before a student has completed MATH 234, as a student may enroll in MATH 467 after taking MATH 222 and any of MATH 240, 320, or 340. Because of this, many times a student wishing to use MATH 467 as the introductory proofs requisite for advanced courses will also need to have completed MATH 234 before enrolling in the advanced course. There are also some 500- and 600-level proof-based courses for which MATH 467 is not accepted as the introduction to proofs requirement. To enroll in those courses, students may either take another course at this level (typically MATH 421) or first complete MATH 521, which can be used as the requisite for many other advanced courses. Students should check the requirements for mathematics named option programs of interest to them, as MATH 467 does not always satisfy the intermediate mathematics requirement.
In summary, MATH 467…
- Is a suitable introduction to proofs for most 500- and 600-level mathematics courses;
- Is accepted in the standard major and certificate programs, but is not accepted by some named option programs of the mathematics major;
- Introduces proof writing in a context with which students have some familiarity but also introduces a number of new mathematical ideas; and
- Will give students access to math advanced-level undergraduate math courses.
Suggested further courses are…
- MATH 341 (only if another linear algebra course has not already been taken);
- MATH 421 for additional introductory-level exposure to proofs and proof-writing; or
- Any math course above the 500-level (possibly assuming other prerequisites).
MATH 321-322 (Applied Mathematical Analysis)
Students who complete the sequence MATH 321-322 also satisfy the requisite for MATH 521 and some other proof-based mathematics courses. The sequence also satisfies the intermediate mathematics requirement in all the named option plans of the mathematics major other than Mathematics for Secondary Education. For students interested in applications of mathematics in the physical sciences and engineering, this sequence may be of particular interest. Because of the focus on applications, these courses do less to introduce students to writing formal mathematical proofs than the other courses discussed on this page. Because MATH 322 is only a requisite for some proof-based math courses, students wishing to use this course to prepare for proof-based courses should consult with a math major advisor.