Welcome to the Mathrooms!
This is like the Backrooms, but for maths. Once you enter, you start from 1+2=3 but then you continue to explore deeper and deeper levels until before you know it you end up knowing everything you need to know to get a degree in (pure) mathematics, built from first principles.
The Mathrooms is my website for maths which is usable as both a teaching resource and a way to self study Mathematics. Here I always try to explain the why behind stuff. And I also try to explain it at the level where you have to use the stuff, even if this means explanations are long and complicated as I have to develop some theory. I want to eliminate the frustrating experience I've had where I search for the proof of a result applied at A level but all the proofs that exist assume undergraduate background. Also, my video on arithmetic and algebra basics (See level 1) was designed to address many pitfalls and sources of confusion I've seen from people who struggle with maths. In general, the supplementary videos on this website are an informal (compared to the written notes) treatment of topics that still explain the "why".
NOTE: All videos on this website will be replaced with written notes after levels 1-9 are finished and mistakes corrected.
This website also showcases all the maths that I personally know from first principles which is always growing. I was originally making notes to keep my own maths knowledge and proof dependencies organized but eventually noticed that these turned into useful materials that I wish I had when I was an A level student, which is what inspired me to make this. This website is more about building mathematical theory and understanding from first principles and less about how to apply it to the real world or how to do it in a way that will make examiners happy. If you like applied maths or physics this may not be for you, but if you like proofs you should love this.
Sublevels marked bonus are not strictly needed for progression (and if a result from there is needed it will either be moved or explicitly stated) and are optional. These include results that are not part of the core syllabus that I wanted to include.
While rigor is very important in the Mathrooms, this is not a formal logic website, so at some point we do use common sense. I don't insist on proving things that have been proven IF they are sufficiently obvious. For example, I don’t think proof is needed to convince you of the intermediate value theorem, which says that a graph that you draw without taking your pen off the paper that goes from an altitude of 0 meters to 2 meters has a point on it at an altitude of 1 meter (although in that particular case we do formalize it in the level 7 analysis course since I think it's interesting to think about "what does it mean for something to be continuous" and think about how to make that more precise than "you can draw it without lifting your pen"). Note however that anything less obvious than this assertion is probably something we justify here, since intuition can fail..
Some proofs are deferred to later levels (usually to levels 4 and 6 and in other rare cases when I really think it's best) to cater to people who just want to pass in school and not necessarily understand all the proofs, and when this happens I point it out explicitly.
You can contact me at mathsorgerorg@gmail.com to suggest better proofs or explanations, correct errors, or just to chat!
About the creator
I'm currently a maths undergraduate at the university of Cambridge and I've been passionate about Maths for my entire life. I also know pi to 10000 decimal places which is cool and actually was surprisingly fun to do.
Summary of the whole curriculum of this site
- Level 1 is an introduction. It consists of a 3 hour video where I explain how to do basic arithmetic and algebra. It is specifically designed for the general public; the only assumed prerequesite is knowing how to count. I have made an effort to address various pitfalls and struggles that I have seen when talking to struggling mathematicians in this video.
- Levels 2 and 3 transition into high school algebra, the equivalent of GCSE and A level maths. In level 2, we talk about solving equations, graphs, and solving geometrical problems and an introduction to trigonometry. In level 3, calculus is introduced including the derivative and the integral. We also go into more detail on trigonometric identities and functions like the exponential and logarithm. Level 3.3 serves as an introduction to statistics, using Dream's manhunt cheating scandal as a motivating example.
- In Level 4, we pivot the focus from procedures to proofs. I approach introducing this by building on content from earleir levels - specifically it is designed to fill in gaps from levels 2 and 3, i.e. why certain rules that I used actually work. I usually prove stuff as I go but I want GCSE and A level students to be able to use this site as revision and not be forced to deal with proofs they don't need. Due to the nature of this level, it is admittedly not the most organized. We introduce a little bit of real analysis for the purposes of proving the generalized binomial theorem and the fact that the limits of integration work. We also introduce proof techniques such as induction to prove statements previously assumed such as why prime factorization is unique. Level 4 also contains 2 supplementary videos.
- In Level 5, which is supposed to be like A level further maths, we build on the work we did in level 3. In level 5.1 show more methods for doing basic proofs and solving differential equations and ingerals, and in level 5.2 I build on the work I did in statistics in level 3.3 by introducing more hypothesis tests and distributions. Level 5.1 also introduces matrices and does more work on complex numbers - a concept introduced in level 4 to prove trigonometric identities. Level 5 contains a bonus sublevel that covers a proof that pi is irrational and a proof of the Basel problem.
- Level 6 shifts the focus back to proofs. Once you get here, you're in undergraduate territory and conceptually it is the hardest level yet, with about 100 pages as well as 2 supplementary videos. This is where if you can manage it, you can definitely cope with higher level math. Like level 4, this level is designed to fill in gaps from earlier levels. This gets very involved, with level 6.1 being a crash course on real analysis and linear algebra used mostly to prove things from level 5.1, and level 6.2 is a crash course on measure theory as well as other analysis results, forming the backbone for level 6.3 where I fill in gaps from levels 3.3 and 5.2 - the statistics levels. As well as several smaller theorems, the 2 big theorems I prove in level 6.3 are the central limit theorem, which requires the measure theoretic framework from level 6.2, and (infamously) chi squared tests, where I have my own non-standard proof that is about 15 pages of algebra (the thing that makes this differ from standard proofs is that I prove a local limit theorem directly for the relevant case instead of doing it via a general theorem like the multivariate CLT), since proofs not requiring background are hard to find. Level 6.4 is shorter, it consists of a few analysis theorems used to justify implicit differentiation.
- It may not be entirely true to say level 6 is a difficulty spike, since later levels aren't really much easier. However, an alternative progression path is stated on the level 6 page that may be easier for some readers. The progression path is 6.1, 6.2 except the measure theory chapter, chapters 1-7 of 6.3 which prove all stats results except CLT and chi squared and stuff building on those, then 7.1, 7.2 and 7.3 to gain some mathematical maturity, then to go back and do the harder parts of level 6 and then do level 7.4 and continue.
- In Level 7, we finally reach my notes for the Cambridge Maths Tripos. Roughly speaking, levels 7 and 8 cover the first year / part IA, levels 9 and 10 cover the second year / part IB, and levels 11-14 cover the third year / Part II. However, this is not a hard rule as I do the courses slightly out of order in places.
- Level 7 serves as an introduction to the language and techniques of undergraduate maths, particularly in Numbers and Sets. The Groups course also introduces the concept of a Group, a mathematical object we study more later that ultimately will help us study numbers and shapes, and the Vectors and Matrices course is designed to connect algebraic and geometric ideas. We also have a course on Real Analysis which will cover material that will be very important for later courses. Level 7 contains a bonus sublevel that covers a few fun results including some analysis constructions and a fun number theory proof.
- With the general language and techniques of undergraduate maths in hand, you as the reader have reached the point of no return, as if you got this far you are probably very good at maths and passionate. Level 8 teaches foundational topics that are useful for undergraduate maths - specifically it has Linear algebra, an introduction to Topology, Complex analysis, Probability and Differential equations (DEs includes a supplementary video). Level 8 has a bonus sublevel that currently covers a proof of unsolvability of the quintic using topological ideas.
- With the foundations of undergraduate maths in hand, levels 9 and 10 build on this with more theory and applications. Theoretical courses include courses on Number theory, a course on Groups rings and modules, and a course on Logic and set theory. We also have applicable courses such as Optimization, Statistics, Markov chains, Numerical analysis and Vector calculus, with Vector calculus being in level 9.4 with level 9.5 existing to fill in gaps that were not proven in Vector calculus, which is focused more on computation and Heuristic arguments.
- Beyond this point we get into the kinds of Maths where most people won't know what half of the words even mean. Level 11 is where we expand on the measure theory and analysis we did in levels 6 and 7 and build the framework of functional analysis, which we do for our ultimate goal of applying it to understand cool mathematical techniques like Fourier series, the Fourier transform, the Laplace transform, differential equations and the Calculus of variations.
- In levels 12-15, this is where most of the Part II tripos courses will live. While details have not been finalized yet, there are Algebraic courses, Geometric/Topological courses, Applicable courses, and courses that may fit into more than one of those categories. They are Geometry and groups (no longer a part II course), Analytic number theory (not a part II course), Galois theory, Algebraic number theory (Corresponds to Number fields tripos course), Representation theory, Algebraic geometry, Algebraic topology, Differential geometry (the old IB geometry course may be merged into this), Topics in analysis (may be merged into other sublevels since most of the content overlaps), Graph theory, Further complex methods, Dynamical systems, Principles of statistics, Automata and formal languages, Coding and cryptography and Applied probability. Also, Asymptotic methods and Numerical analysis are 2 courses that I'm not fully sure whether I want to do.