Start your journey into the Universe’s most beautiful art form…

- Lesson 1: Introduction

- Lesson 2: Sets
- Lesson 3: Subsets
- Lesson 4: Sets of Sets
- Lesson 5: A Paradox
- Lesson 6: Functions
- Lesson 7: Injective, Surjective, Bijective
- Interactive Interlude
- Lesson 8: Bijectivity and Counting
- Lesson 9: Counting Infinity
- Lesson 10: Infinity times 2
- Very Quick Note on Lessons 11-15
- Lesson 11: Infinity times Infinity
- Lesson 12: Infinity times Infinity = Infinity
- Lesson 13: Infinity type 2
- Lesson 14: Infinity Infinities
- Lesson 15: Proving the Pattern
- Lesson 16: Unions and Intersections (New from Old)
- Lesson 17: More Constructions pt. 1
- Lesson 18: More Constructions pt. 2
- Interactive Interlude
- Lesson 19: If I’ve done it once I’ve done it a thousand times
- Lesson 20: Reconsidering High School Algebra
- Lesson 21: The Necessity of Irrationality
- Quick Recap of Notation
- Lesson 22: Giving Life to Sets
- Lesson 23: More fun with groups
- Interactive Interlude
- Lesson 24: Cool Groups pt. 1—Permutations
- Lesson 25: Cool Groups pt. 2—Get Your Hair Did
- Lesson 26: Cool Groups pt. 3—Dihedral groups
- Lesson 27: Group Constructions: Subgroups
- Lesson 28: Group Constructions: Subgroups Part Deuce
- Lesson 29: Group Constructions: Subgroups Finale
- Lesson 30: Homomorphisms: Moving Between Groups
- Lesson 31: New Functions From Old pt. 1
- Lesson 32: New Functions From Old pt. 2
- Lesson 33: Homomorphisms pt. 2: Into Abstraction
- Lesson 34: Like Clockwork
- Lesson 35: Like Riding a Bicycle
- Lesson 36: The Generators
- Lesson 37: New Groups From Old: Product Groups
- Lesson 38: Equivalence Relations
- Lesson 39: Conjugation
- Lesson 40: Group Constructions: Normal subgroups
- Lesson 41: Quotient Groups

Lots and lots more lessons to come!

Check out the book The True Beauty Of Math, Volume 1, The Foundations, which includes bonus material, more detail, extra exercises and full solutions!

To download the book “The Language Of Nature,” which discusses the ideas introduced here as well as much, much more, check out this page.

This is awesome. You are awesome. Thank you!

Haha thanks! Glad you like it 🙂

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These lessons have really opened the door to mathematics for me. Thanks.

That’s fantastic! It’s my pleasure to help open these doors 🙂

Great effort…the entire approach of teaching math through set theory is very unique, However I do feel that this approach doesn’t build problem solving as well and early on as an approach which starts with geometry or combinatorics.

Thanks for the input, and I agree with you completely. However, as I tried to preface on the front page, my goal here isn’t to create trained mathematicians, but rather simply to expose the reader to a different (and likely unfamiliar) side of math, and to hopefully inspire him/her to pursue it further outside of the site. Therefore problem solving skills and/or a solid foundation in these subjects is not going to be acquired simply be reading this site, but hopefully a deeper understanding of what (lots of) math is all about and the thought processes behind it will be.

Thanks again for you input, and please don’t hesitate to continue to reply with your thoughts.

Thanks for revealing such a beauty to the world man ….. great effort 😀 !

My pleasure, glad you like it!

I want to be a mathematician when i grow up

I’m glad to hear it! Hopefully this site serves as the tiniest step in that journey…It should be a fun ride!

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How long did it take you to write “The Language of Nature”?

Several months, but only part time (i.e., a few nights a week for a few hours at a time). Are you enjoying/did you enjoy it?

I haven’t read it yet. I only just discovered it after looking ahead in the “Lessons” page. I only just discovered your blog yesterday and so far I’ve read to Lesson 21. But the textbook looks very good. I’ll tell you what I think of your blog and your textbook after I’ve finished reading them. From what I’ve seen so far, Great Work!

Also, have you ever considered becoming a teacher?

Thanks! Glad you like it, and I hope you continue to do so 🙂 And yes I’ve very much considered it—a teacher and a researcher in my own right. I’m able to do both now and I just hope to continue to do so! It’s late here in the UK so my next response (if necessary) might not be as timely, but I hope you continue to read/enjoy and I’ll respond asap! Cheers

I like riding bikes. Can you please post lesson 32?

haha will do soon GRAbino, the last few months HAve been crazy for Me but expect new lessons appearing BY octobeR. A.

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Holy crap these are amazing. You are really revolutionary in your teaching in a way!

Glad you like it! There will be lots more to come!

wow thank you very much good kind sir

My pleasure! 🙂

Can you please work up from set theory to analysis, advanced algebra (linear, geometric) and calculus of a single or multiple variables! I really want to see you teach those concepts in your intuitive and graceful style! (Also- maybe this is too much to ask, well I’ve asked too much already- but can you keep your discussion centralized so that you do not arbitrarily separate mathematics into overlapping topics?).

I do indeed plan on keeping this thing going for linear algebra and analysis (and therefore calculus), and even onto things like algebraic geometry and categories, but it will take a LONG time to get to all of these topics while keeping the discussion intuitive and friendly. Since I want to maintain intuitiveness, friendliness, as well as rigor, it will take a long, LONG time to get to these more advanced topics, but rest assured, we’ll eventually get there. I’m glad you seem to be enjoying the site so far though! 🙂

And I do indeed plan on keeping the discussion centralized and emphasizing — wherever possible — the fact that mathematics is a beautiful unified and not a collection of distinct classes and textbooks.

Ever since my Analytic Geometry Prof. made us write an overnight quiz – which was hard coz it’s pretty tedious work, but very satisfying – I’ve come to really appreciate Mathematics, not to mention our world including our galaxy and the universe! So I hope I can catch up with everyone else, using your blog! 🙂

That sounds awesome, and I’m glad you like the site! 🙂

Don’t stop it! It

I really liked the way you mentioned the easy to understand structure of lessons and solutions. Keep up the good work.