# How to self study mathematics ## Background I have been self studying mathematics for around 2 years. I started out with a very basic and vague highschool math understanding and worked my way towards understanding multivariable calculus and linear algebra. This hasn't come easy and there were many strugles along the way. My goal here is to show you the things that helped me and to share the best resources I've found along the way. ## Introduction As a self taught software engineer I sometimes feel like I'm missing out on a proper education and that I'm lacking the proper theory for certain things. I always knew that in the early days computer systems were designed and build by mathematicians and that mathematical concepts are spread all over computer science. So for me this was reason enough to really start learning the actual mathematics underlining the systems I use on a day to day basis. Only to discover that Mathematics is so much more. > **Word of Warning** > You're going to need a whole lot of self dicipline and dedication to get this done properly. This means that if you don't have the drive to really persevere you're probrably going to give up. However if you make it through the countless hours of struggling it will be more than worth it. You have to stay curious and geniuly interested. After all the foundation of modern world is mathematics. ## Starting from highschool algebra I started out with a very basic understanding of algebra and the minimal understanding of Calculus. However even this I hadn't used in a long time so I needed a refresher. I used [Brilliant](https://brilliant.org/home/) in order to do this but [Khan Academy](https://www.khanacademy.org/) is also a great resource however for me it was a little bit too slow and I didn't really enjoy the exercises and explainer videos. The downside of Brilliant is that sometimes it was going a little bit too fast it focuses a lot on building intuition which is good but to really remember and get a feel for it needs more exercises. So I did the following Brilliant courses for algebra [Algebra I](https://brilliant.org/courses/algebra-extensions/), [Algebra II](https://brilliant.org/courses/graphing-and-modeling/) and [Complex Algebra](https://brilliant.org/courses/complex-algebra/) although you will only use Complex Algebra later on. ## Calculus Calculus is all about dealing with numbers that change over time. How things have changed and how they are going to change. Calculus has many applications, you can for example use it for building a game engine, where you want to predict how object move over time and how they should react to certain forces. I've done the following courses for the fundamental understanding of calculus [Calculus in a Nutshell](https://brilliant.org/courses/calculus-nutshell/), [Pre Calculus](https://brilliant.org/courses/pre-calculus/), [Trigonometry](https://brilliant.org/courses/trigonometry/), [Calculus done right](https://brilliant.org/courses/calculus-fundamentals/) and [Integral Calculus](https://brilliant.org/courses/calculus-ii/). I would also highly recommend the videos from 3Blue1Brown [Essence of calculus](https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr). For more intuition and history behind Calculus you can read Infinite Powers by Steven Strogatz. It is easy to read and explains where calculus came from and why it makes sense. After this you should get ready to get into Multivariable Calculus. This is a big step up from the calculus you've done before. I've done the course on [MitOpenCourseware](https://ocw.mit.edu/courses/18-02sc-multivariable-calculus-fall-2010/pages/syllabus/) and I can't understate how good this resource is. You get everything from lectures to assignments to exams. I really like the focus on practical applications, which is really motivating to learn more as you can directly see why it is useful. If you ever need extra explanation videos I would recommended the [Multivariable Calculus playlist by khanacademy](https://www.youtube.com/playlist?list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7). Funny enough a lot of the videos there are made by Grant Sanderson, the same guy behind 3Blue1Brown. I recommend taking each exam and exercises seriously and to keep repeating exercises until you understand it. ## Linear Algebra If you want to know about the relationships between a big mess of numbers then Linear Algebra is for you. The concepts of linear algebra are extremely useful in physics, economics, computer science and engineering. Everything with lots of data uses Linear Algebra. Maybe you've heard of Machine Learning or AI, yes all Linear Algebra. I've done the legendary [course of prof. Gilbert Strang on MitOpenCourseware](https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/pages/syllabus/). I would recommend watching the lectures x1.25 or x1.5, it can be a little slow at times. I have watched the lectures and did the recommended exercises. If I didn't understand the topic well enough I would read the chapters from the book he wrote [Introduction to Linear Algebra, Fifth Edition](https://dokumen.pub/qdownload/introduction-to-linear-algebra-fifth-edition-5nbsped-0980232775-9780980232776.html) and make some or all exercises in that chapter. For more intuition about the topic I would recommend checking [Essence of linear algebra](https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab) by 3Blue1Brown. ## Future work ## When you get stuck So you're definitely going to get stuck or have trouble understanding some topic. It is very important NOT to ignore this topic and to move on. This will most likely haunt you in the future when you're working on a more advanced topic build on something that you might have ignored (Math is all about building on previously taught topics). When you get stuck it can be because either you don't know where some solution came from and which steps were made or you don't understand what is asked or in general what is going on. In the former case you can use [Wolfram Alpha](https://www.wolframalpha.com/) and if you have the Pro subscription you can look at how it calculated the solutions, this has saved me countless times, although try to avoid it whenever you can calculating things by hand can really strengthen intiution. In the latter case you can watch videos on Youtube about the subject or use things like [ChatGPT](https://chat.openai.com/) to ask for specific questions and nudge you in the right direction. Ask it about the things you don't understand in a question and piece by piece desect everything you need to know to answer something you're working on. Have patience! ## All Resources *Calculus* Infinite Powers by Steven Strogatz