18.01 (Calculus I) or equivalent. No prior proof experience required.
: Fields, vector spaces, and permutations. Analysis Concepts : Real number sequences and infinite sets.
The transition to proofs can be daunting. 18.090 offers several advantages:
18.090 is a critical "gateway" course. It provides the crucial necessary for success in demanding upper-level classes. The Pure Mathematics Option explicitly recommends students gain proof experience in 18.090 before tackling 18.100 (Real Analysis) or 18.701 (Algebra I). It is also a Restricted Elective in Science and Technology (REST) , allowing students to fulfill a General Institute Requirement while building this essential skill. 18.090 introduction to mathematical reasoning mit
Writing a proof is exactly like writing code. If there is a single logical flaw or unexamined assumption, the entire proof crashes. 18.090 trains your brain to think with compilation-level precision.
Instructors report that novices struggle most with:
At elite institutions like the Massachusetts Institute of Technology (MIT), mathematics undergoes a radical shift. It transforms from a tool for calculation into a formal language of logic, abstraction, and rigorous proof. Analysis Concepts : Real number sequences and infinite sets
Not everyone at MIT takes 18.090. Some arrive with AP credit in BC Calculus and a strong background in math competitions (IMO, USAMO). For those students, 18.090 might be redundant. However, for the following archetypes, 18.090 is non-negotiable:
Prove that for any integer ( n ), if ( n^2 ) is even, then ( n ) is even.
While 18.100A is often seen as the first proof-based class, 18.090 serves as a gentler, more foundational introduction to mathematical reasoning itself catalog.mit.edu. It provides the crucial necessary for success in
The course famously insists that students write proofs in full, grammatical English sentences—never a chain of mathematical symbols. A proof for 18.090 looks like a paragraph in a detective novel, not lines of code.
This is the heart of the course. Students learn several distinct strategies to prove mathematical theorems: