Welcome to maths.orger.org!

The aim of this site: To start from 1+2=3 and end with everything you need to know to get a degree in (pure) mathematics, built from first principles.

This 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.

The misc results are not strictly needed as dependencies for levels (and if they are they will be clearly cited) but to get the full experience you should read them once you understand all levels content up to the named appropriate level for that result. Levels are designed to be studied sequentially or skipped if you scan them carefully and are satisfied that you know everything in that level.

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.

Be mindful that while I have gone through great lengths, and I think done a good job, to make these as accessible as possible, some of the proofs are very quick while others are long and technical and may require perseverance to follow, but don’t give up! This is the tradeoff to do this “Used at that level ⇒ proved at that level, no higher background assumed” philosophy that I hold. For example, the central limit theorem is used at A level but takes about 15 (give or take) pages of algebra to prove rigorously from previous ideas at an A level background. However the good part about this is we learn some important undergraduate maths in the process!

Note that 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 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 8 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. However, while rigor is important I do not believe in being pedantic about it so at some point we do use common sense.

Some proofs are deferred to levels 4 and 6 to cater to people who just want to pass in school and not necessarily understand all the proofs, however at undergraduate level I feel that anyone who can understand the content can understand the proofs, so everything used in levels 6 or more is proved at that level.

You can contact me at mathsorgerorg@gmail.com to suggest better proofs or explanations, correct errors, or just to chat! If an error affects a complicated proof, please suggest an alternative self‑contained route.

About me

I'm a maths student starting at University of Cambridge (Trinity college) in 2025 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.