Learn ALL THE MATH IN THE WORLD from START to FINISH
What this covers
I took all of mathematics and broke it down into 8 core areas. In this video I will show you those 8 areas and the subjects that live inside each of those areas. Below I have suggested some books in case you want to learn more. The books below have been carefully selected. All my books: https://www.amazon.com/stores/The-Math-Sorcerer/author/B0DRSVPVMW (The links below are my affiliate links) My Books Real Superpowers That Will Change Your Life: https://amzn.to/40cTxas Superpowered Calculus: Limits, Derivatives, and Integrals: https://amzn.to/4hmqerE Calculus Alpha: Limits for Math Super Nerds: https://amzn.to/403WCrU Calculus Beta: Derivatives for Math Super Nerds: https://amzn.to/3El7NVQ Precalculus for Legends: https://amzn.to/4gk9GPT Superpowered College Algebra 1: https://amzn.to/3BMHxmx Superpowered College Algebra 2: https://amzn.to/3W40oAB
The Bonus Book: https://amzn.to/3ByNCTA
Foundations of Mathematics Logic and Proof Writing: https://amzn.to/3VKKoDs Number Theory: https://amzn.to/4ftX8Fg Intermediate Algebra https://amzn.to/3ZZX0ch College Algebra: https://amzn.to/4glQiDd Precalculus & Trigonometry: https://amzn.to/4gHFqzm
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Geometry and Topology Geometry: https://amzn.to/4gkQKBC Topology: https://amzn.to/41Hn6lx Algebraic Topology: https://amzn.to/4iMaM9R Differential Geometry https://amzn.to/4gG7tix
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Analysis and Calculus Calculus https://amzn.to/4iFaAsT Advanced Calculus: https://amzn.to/49KH2G8 Real and Complex Analysis: https://amzn.to/3ZL2oPa Functional Analysis: https://amzn.to/3DlbavE Complex Variables: https://amzn.to/4gjHZrj Differential Equations: https://amzn.to/3DiQcO6 Partial Differential Equations: https://amzn.to/3DzlqjO Numerical Analysis: https://amzn.to/49H5QPn Fourier Series https://amzn.to/3P2J3Uw
Applied Mathematics and Modeling Applied Math https://amzn.to/49N7zTq Cryptography https://amzn.to/41HWcdg Physics: https://amzn.to/40aMmzD and https://amzn.to/49LQ1H3 Electrical Engineering: https://amzn.to/3P3w7xN
Probability and Statistics For Beginners: https://amzn.to/4iLn1mP More Advanced: https://amzn.to/4foQVdY
Advanced Topics and Frontiers Nothing to see here:)
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************Udemy Courses(Please Use These Links If You Sign Up!)************* Abstract Algebra Course https://www.udemy.com/course/abstract-algebra-group-theory-with-the-math-sorcerer/?referralCode=B04607DA7A7D0E29272A
Advanced Calculus Course https://www.udemy.com/course/advanced-calculusreal-analysis-with-the-math-sorcerer/?referralCode=0ABDD66D061D976EE232
Calculus 1 Course https://www.udemy.com/course/calculus-1-with-the-math-sorcerer/?referralCode=E853B70ED36571CA9768
Calculus 2 Course https://www.udemy.com/course/calculus-2-with-the-math-sorcerer/?referralCode=BAA5520B32FEA9827D54
Calculus 3 Course https://www.udemy.com/course/calculus-3-with-the-math-sorcerer/?referralCode=296462D1897904C4BEB3
Calculus 1 Lectures with Assignments and a Final Exam https://www.udemy.com/course/calculus-1-lectures-with-the-math-sorcerer/?referralCode=5D425153B63A8F506E30
Calculus Integration Insanity https://www.udemy.com/course/calculus-integration-insanity-with-the-math-sorcerer/?referralCode=D533EEE31F90EDDAFF93
Differential Equations Course https://www.udemy.com/course/differential-equations-with-the-math-sorcerer/?referralCode=4F0D91B41F7DACF4EC28
Differential Equations Lectures Course (Includes Assignments + Final Exam) https://www.udemy.com/course/differential-equations-lectures-with-the-math-sorcerer/?referralCode=89E6CCC307A7A8627CD6
College Algebra Course https://www.udemy.com/course/college-algebra-with-the-math-sorcerer/?referralCode=B2929EE97EF68DB9B69F
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Mathematics can be organized into eight distinct categories—Foundations, Algebra and Structures, Geometry and Topology, Discrete Mathematics and Combinatorics, Analysis and Calculus, Applied Mathematics and Modeling, Probability and Statistics, and Advanced Topics and Frontiers—allowing any mathematical subject to fit into one of these frameworks and helping learners understand the landscape and prerequisites of the discipline.
- Eight-category taxonomy is more practical and intuitive than pure/applied divisions and better reflects how mathematics is actually studied in colleges
- Each category contains identifiable core subjects with clear prerequisites and relationships (e.g., proof writing enables topology, calculus enables differential equations)
- This organizational structure reveals that mathematical literacy requires building foundational skills before progressing to advanced topics
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Linear algebra textbooks exist on a spectrum from computational (focused on applications and procedures) to proof-based (emphasizing rigorous mathematical justification), with proof-based texts typically used for second courses and being more difficult.
“you dealing with matrices linear Maps systems of equations uh also this book here is computational based so it's going to have some proofs but it's not like a proof-based linear algebra book so usually usually you have two types of linear algebra books you've got like the computational ones and then you've got the proof-based ones but it's all inine your algebra and it falls under algebra and structures”
Proof writing is a foundational mathematical skill that enables progression to advanced subjects including abstract algebra, topology, real analysis, and set theory.
“proof writing is based on logic”
Mathematical ability increases when learners recognize that mathematics is inherently difficult and that different types of mathematics exist, because people tend to like and excel at what they understand to be learnable and within their capability.
“a lot of people when they first see math they're frustrated by it because it is challenging right math is hard and when things are hard well it's hard right we want to give up but it's okay once you realize that math is hard and that there's different types of math well that kind of makes you like it more and people tend to like what they're good at so if you like math and you're really interested in it then chances are you can become very very good at it”
Probability and statistics is an underappreciated mathematical field that should receive more attention from mathematics majors than it currently does.
“here's an area that I feel doesn't get much love from people it's probability and statistics I feel like most math Majors I've known aren't really into this and the people who are into it are very much into it”
Mathematics can be systematically organized into eight categories: Foundations of Mathematics, Algebra and Structures, Geometry and Topology, Discrete Mathematics and Combinatorics, Analysis and Calculus, Applied Mathematics and Modeling, Probability and Statistics, and Advanced Topics and Frontiers.
“in this video we are going to go over every single area of mathematics so what I have done is I have created eight categories and you can take any math subject in the world and you can fit them into one of these categories”
The video's pedagogical goal is to enable learners to understand that mathematics is organized into distinct subject areas with identifiable prerequisites and relationships, reducing frustration and increasing engagement.
“and that's my goal of making this video a lot of people when they first see math they're frustrated by it because it is challenging right math is hard and when things are hard well it's hard right we want to give up but it's okay once you realize that math is hard and that there's different types of math well that kind of makes you like it more”
Physics is correctly classified as applied mathematics because physics fundamentally relies on mathematical methods and can be understood as an application of mathematics to physical systems.
“I thought physics let me throw this in here because it's applied math I mean people might say well it's physics it's not math you use a lot of math and it's applied so I decided to throw in physics”
Spherical trigonometry is an obsolete mathematical subject that was widely taught and used during the World War II era but has not been taught in modern colleges, despite its historical importance.
“here's a lost subject you've probably never seen plain and spherical trigonometry right that's pretty cool so spherical trigonometry is no longer taught so I have this Theory um I think it's it was used a lot and it was taught a lot during the World War II era and I know that during the war um I think they they used a lot of this stuff so it was taught a lot back then like what's the copyright on this let's see yeah see 1950 so that was after World War II all the ones I have are older I don't have any newer books on spherical trigonometry it's like they don't exist”
'All the Math You Missed But Need to Know for Graduate School' by Thomas Garrity is an exceptional reference that compresses numerous advanced mathematics topics into one volume by focusing on key objects and basic maps rather than exhaustive exposition.
“it's a lot of subjects in one book he covers all the key theorems or a lot of them right and then you've got some exercises there at the end so I mean pretty incredible”
A differential geometry textbook that includes solutions to every proof in the back is exceptionally valuable for self-study learners despite not making the subject 'easy', as it provides verification and guidance without eliminating difficulty.
“this one is really fun this one has solutions to every proof in the back of the book let me show you check this out this one you can but you might say oh so it's easy I can learn uh I I wouldn't go that far just because you have the answers doesn't mean it's easy so it's still really tough again for topology you you you really you know ideally you want to have some Advanced calculus and Analysis”
Graph theory is sufficiently important as a mathematical subject that it arguably deserves its own dedicated category in mathematics taxonomy rather than being subsumed under discrete mathematics.
“it really does it really does it really does all right let's keep going both of these stacks of books contain analysis and calculus books”
A calculus textbook that does not require knowledge of trigonometry is exceptionally rare and valuable because most calculus textbooks assume prior trigonometric knowledge as a prerequisite.
“this book is wonderful because this book does not require any trigonometry I've used this book to teach a certain class in college and the prct for that class is only college algebra so this is a very special book because there's not many books like this in the world that allow you to learn calculus uh without knowing trig”
The textbook 'A Survey of Modern Algebra' by Birkhoff and MacLane is an exceptionally beginner-friendly abstract algebra textbook that is highly recommended as an entry point to the subject.
“another legendary book a survey of modern algebra by burkoff and mlan my favorite abstract algebra book this is the one I learned abstract algebra algebra with I read this entire book and my only complaint about this book is that it didn't have enough information... but it's a great little book super begner friendly highly recommend getting this one if you're going to get one this is probably the easiest one”
Engineering mathematics textbooks that cover linear algebra, differential equations, calculus, and complex variables are valuable resources despite high costs because they provide rough overviews and are always in high demand.
“an advanced engineering mathematics book so this book actually contains linear algebra differential equation calculus complex variables it has a lot of material it gives you a rough overview of everything um but it's Advanced engineering mathematics I wanted to throw it in with this stack here wonderful book the worst thing about this book is the price it's never cheap because it's always in demand that's true for a lot of books some books are just like always expensive because people want them”
Mathematical statistics (proving all results) is significantly harder than engineering statistics because mathematical statistics textbooks are taught by mathematicians with greater rigor, whereas engineering statistics textbooks may be taught by engineers lacking mathematical training.
“I don't want to get any hate in the video but I just think it's a little bit harder like this is a harder class like if you take if you take mathematical statistics uh in college it's going to be harder than stats for engineer it's going to be this is going to be way harder this is way harder I know took I took I took two semesters of it and I've seen I've seen what's taught in this because I have I've had students uh who have taken this class and it's hard”
"Principles of Mathematical Analysis" by Walter Rudin is a famous analysis textbook known for containing very difficult examples and exercises that test deep understanding; speaker prefers another analysis textbook (by Buck) because it provides more explanatory examples
“a very famous book it's called principles of mathematical analysis was written by Walter Ruden so this is an analysis book or an advanced calculus book but people call it an analysis book it is called mathematical analysis in any case smell it this book has really hard examples and really hard exercises and it's very famous I like the book by Buck better because it has more examples”
The pure mathematics versus applied mathematics categorization is inferior to an eight-category system because the former results in unwieldy book stacks and does not provide learners with accurate insight into different subject areas.
“now there are different categorizations of mathematics typically people do it by saying okay this is pure math and this is applied math I decided not to do that for a couple reasons one the stacks of books would be huge and they would reach the ceiling two I think this gives you a better idea of the different subject areas”
Modern physics is a beautiful subject that includes concepts from Einstein's theories, such as relativity and time dilation, and is appropriate for undergraduate physics education.
“and another physics book I thought physics deserves another another uh another book here so I threw in physics here as well... physics again this is specifically modern physics I actually use this for a course in college called physics 3 really beautiful subject you do stuff you do some of the stuff with Einstein in there like you do like relativity and stuff so time dilation pretty fun”
Set theory requires significant prior mathematical maturity including proof-writing ability to be accessible to learners, making books like Hausdorff's set theory inappropriate for beginners without this foundation
“set theory this is probably the most famous set theory book ever written it it's the one by Felix hausdorf... I have to smell it I'm sorry this is a first edition set the it's a little bit hardcore um the prere for this one by the way um I think you really should know how to write proofs before jumping into this”
Alan Hatcher's 'Algebraic Topology' is an extremely difficult textbook representing the frontier of undergraduate mathematics difficulty.
“very famous book algebraic topology by Alan Hatcher extremely hard and another algebraic topology book again extremely extremely difficult”
Calculus is mathematically defined as 'the mathematics of change' and is based on the concept of limits, which enable precise definition of derivatives (rates of change) and integrals (accumulation).
“what is calculus people will say that calculus is the mathematics of change and that's a pretty true statement typically in calculus you start by learning what's called limits and basically it's the idea of getting infinitely close to something but never actually getting there and a limit kind of kind of you know that's what it defines it allows uh you to get close to something allows functions to get really close to numbers when X values get really close to numbers”
Algebra is a necessary prerequisite for calculus because students must master algebraic concepts before progressing to calculus.
“you do need algebra in order to um you know progress right you have to learn algebra so that you can take calculus”
Numerical analysis is a distinct subject within analysis that focuses on approximation methods and computational approaches to solving mathematical problems.
“numerical analysis here we go a different subject so numerical analysis is like analysis basically but you're doing approximation so um you can take a whole course on this stuff you do a little bit of numerical analysis and differential equations but you do a lot more when you take an entire course”
Abstract algebra is a subject covering groups, rings, fields, and modules (algebraic structures), and students studying abstract algebra benefit from having access to comprehensive reference texts even if those texts are challenging.
“abstract algebra this is a more advanced book this is an Incredible Book because it contains so much content I bought this as a reference my biggest regret is buying it used because look my copy is is is not is not in good condition... so very useful you can use it to learn abstract algebra the prere for this is knowing how to write proofs”
Functional analysis is a graduate-level course that requires prerequisite knowledge of analysis, algebra, and topology, and represents advanced mathematical study.
“functional analysis this is graduate level graduate level course here um you need analysis for this you need algebra you need topology you need to know a lot of mathematics before you do functional but it's really really fun”
Statistics is a mathematical field that enables inferences about populations by analyzing samples (subsets of populations), including techniques like hypothesis testing and estimation of population parameters.
“in statistics you basically you go out right and you're you're trying to say something about a population this is just one example and what you do is you take a sample which is a subset of that population and then you can do things with that sample right you can estimate the uh population mean or you can ask questions about the population mean you know is the average age of people in this area uh greater than 40 and you can do like a hypothesis test so there's different ways of applying statistics to the real world”
Applied mathematics and modeling category could theoretically encompass an entire room of books because applied mathematics permeates numerous fields including physics, engineering, and cryptography.
“this this could really fill an entire room of books because there's so much applied math out there and there's so much modeling”
College algebra courses teach foundational algebraic techniques including quadratic equations, graphing parabolas, polynomial operations, and systems of equations.
“this is mainly like um you know quadratic equations uh graphing parabas you know you've got some stuff with polinomial um so systems of equations”
Partial differential equations involve partial derivatives (rate of change with respect to one variable in a multivariable function) and require prerequisite knowledge of both calculus and ordinary differential equations.
“partial differential equations it's all out of order so this is basically partial differential equations in order to learn this you have to know ordinary differential equations and in order to learn that you have to know calculus so the order is calculus differential equations and then partial differential equations they're called partial because they involve partial derivatives remember earlier I said derivatives describe the rate of change of a function so partial derivatives are very similar but you just look at the rate of change with respect to one of the variables in a multivariable fun function”
Complex analysis applies calculus to complex numbers, with most derivative rules and series concepts remaining identical to real calculus, but introducing new concepts like harmonic functions and the Cauchy integral theorem.
“when you study calculus you deal with real numbers right you have functions that have real numbers so here it's very similar except all of a sudden you have complex numbers but a lot of the same rules apply right like all your derivative rules are the same all the series stuff is the same there's some really cool new stuff like harmonic functions and Koshi integral theorem”
Michael Spivak's 'Calculus' textbook has a cult following because it provides beautiful explanations, though it contains less material than modern comprehensive calculus textbooks.
“this is a calculus book called calculus and was written by a man named Michael spivac now this book is very famous it has a cult following because it has beautiful explanations it does not have as much material though as some of the modern books”
Computer science majors are typically required to take discrete mathematics courses because discrete math is essential for computational problem-solving.
“this is the kind of thing you would see like as a computer science major um computer science Majors typically have to take discrete math”
James Stewart's calculus textbook is the most popular calculus textbook in the United States and Canada because it is widely adopted by schools for Calculus 1, 2, and 3 courses.
“this is probably my favorite calculus by James Stewart definitely the most popular calculus book in the world because it's used I mean definitely in the US and Canada this is the most popular one for sure um many schools use this book”
The speaker is a serious book collector who maintains extensive archives of mathematics textbooks and has preserved books from the personal collection of Augustus Prince, the first Black radar man in the US Navy and a nuclear physicist at Brookhaven National Laboratory.
“I have uh I call I collect books I'm a collector right so I have a lot of math books and so like you know the history of the book matters and so I was able to pick up some books from his collection uh he passed away several years ago so really sad yeah really really sad”
Cryptography textbook prerequisites are minimal (basic proof-writing ability helps but is not strictly required) because the subject begins with accessible classical cryptographic systems before advancing to complex concepts.
“the prere for this is smells great I guess it's nothing thing I mean it helps if you know how to write per and you had some other math in your life like you know you have some foundational math but you can see it starts with classical crypto systems and it goes through it just gives you like some simple examples”
Geometry is not commonly taught as a college course in the United States; geometry courses are typically taken by mathematics education majors as a prerequisite for their degree program.
“you don't you don't do a lot of geometry in um in college like there a geometry course is not a course that uh many people take in college for some reason um typically people who take geometry courses are math education Majors that's usually a prere for the math education degree”
The speaker has created online courses covering trigonometry, college algebra, Calculus 1, 2, and 3, differential equations, and abstract algebra available on Udemy.
“I have courses right I do have courses they're on you to me but if you get them use my links they're in the description of this video or they're on my website maths sourcer docomomo price I've got courses on trig uh college algebra Cal 1 Cal 2 calc 3 differential equations abstract algebra I have a course on analysis uh I have some courses that have some proofs and stuff”
A cryptography textbook written before Bitcoin's creation does not cover Bitcoin as digital currency but does cover pre-Bitcoin digital cash concepts.
“it does talk about digital currencies but this book was written before the existence of Bitcoin so you get a pre- Bitcoin introduction to digital currency because Bitcoin didn't exist right people don't realize how big of a deal it was when it was created so um so this book was created before Bitcoin so it does not cover that but it does have um a chapter on digital cash”
Logic books vary significantly from one another in their approach and content, unlike more standardized subjects, making the choice of which logic textbook to use particularly important.
“there aren't that many books on logic compared to like calculus books uh and logic books vary from one book to the other so one logic book can be very different from another logic book it just depends uh on the book”
Analysis and calculus is an exceptionally large category of mathematics encompassing many subjects (limits, derivatives, integrals, complex analysis, real analysis, differential equations, Fourier series, partial differential equations, Hilbert spaces, and functional analysis), and the number of textbooks in this category far exceeds other mathematics categories
“both of these stacks of books contain analysis and calculus books so that's a lot of books right and the reason there's so many books and I could have added so many more is that there's a lot of different subjects that live inside this analysis and calculus category”
Bernoulli differential equations are an advanced topic within differential equations that the speaker has created video content about; they represent a specific class of differential equations with particular solution methods
“dyd X is the derivative of y and so basically you have to solve this for y... yeah I have some videos uh here on my channel of beri differential equations”