Mathematics are the result of mysterious powers that nobody understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs, a mathematician chooses one pattern for beauty's sake and brings it down to earth.

1. Boolean algebra 2. Numeral systems 3. Floating points 4. Logarithms 5. Set Theory 6. Combinatorics 7. Graph Theory 8. Complexity Theory 9. Statistics 10. Linear Algebra Interestingly i learned all of these in college but barely remember anything.

Small correction: the Big O notation doesn't specify how fast something is. A constant time complexity of O(1) doesn't mean that the operation/algorithm is fast, but that the speed is constant as the input scales. As a consequence, an O(n^2) algo can be faster than an O(n) algo, for some range of small inputs.

Dude, your examples are so much better than anything we ever got at school. I kinda wanna become a teacher just to be as good as you one day.

As a Asian Student, I can confirm you that we have learned all those math without knowing any use of it. 🙂

In my opinion the most important math class for programmers is discrete mathematics. They helped me understand recursion and stuff like that mathematically

More math concepts for programmers: - Modular arithmetic (RSA encryption) - Differential geometry (smooth surfaces and curves) - Lambda calculus (functional programming) - Fourier transformation (image/audio compression & filtering) - Information theory (data compression) - Game theory (artifical intelligence) - Quaternions (computer graphics)

I hated math in school, I avoided learning complex equations and algorithms when first learning to program, but once I saw a few of Sebastian Lagues videos where he does stuff in Unity, and started messing around with equations and learning to translate them from paper to code, I literally began to feel like I'm casting magic. Its so amazing what just a few dependent variables can change

Please do more such videos on computer science fundamentals. Like how compilers work, how interrupts work, internal working of CPU, network fundamentals, OS fundamentals, etc.

In an era of AI related content on this channel, I'm so relaxed to see a normal non-AI video recently!

I paused the video in the middle just to say, that was the greatest explanation of logarithmic functions I've ever seen in my entire life, thank you for gracing me with that analogy and visual

I'm currently studying Graph Theory for a test in the 2nd year of uni. And looking at my future classes everything matches up to what you said in this video! Amazing.

4:10 In math it's more common to use a different base (ln(x)), but it's easy to change by multiplying the result by a factor, so it's not really important if you're working in base 2, 10, or Euler's number.

Functional programming is still very closely attached to its mathematical roots and it makes programming experience beautiful.

Very interesting and informative video. Very rarely does fireship ever miss, but some videos feel like they fall into a more repetitive format of namedropping a bunch of languages and frameworks loosely tied to a relevant topic. You're very good at explaining topics that can be hard to visualize, I hope you do more videos on general concepts

I feel like if teachers introduced some concepts as you introduced them here ppl would understand some things way faster, you've done some really nice visual aids, good job!

I've taken some combinatorics classes as a part of my math minor while I finish up my computer science degree. Taking combinatorics has changed the way I think about programming problems. The part I like about it the most is how different it is than other higher level math classes. You don't need to know a lot of any other math to jump in and learn it!

I love this! You did a really good job making these subjects relatable and show how important they are in computer science and software engineering.

As a first year undergrad studying Applied Math who has much interest in Computers, I enjoyed this video a lot! I'm currently studying Abstract Algebra and boy that thing is so tough to grasp.

This was one of the most important and interesting videos of my (fresh) coding career and even my whole education. I wish I had teachers that could share their knowledge in this way because I could rarely connect to them and had to learn the things on my own(the hard way). Thanks again, I wish I could subscribe more times as my behind is too poor to give any other kind of support.

Your videos always feel like a freaking mind reader. Scans mind. Sees things that are a bit shaky. Explains them in easy to understand way that is also entertaining. Huge props! 😂

As a person who skipped linear algebra and trigonometry and jumped directly into gamedev, I'd suggest you to focus on your maths if you truly wanna build something breathtaking.

that log cut in half analogy was so practical and good. m never forgetting log graph in my life

As someone who started off in PL/SQL with Procedural querying using Cursors, newer gen people don't realise how beautiful Set Based operations are in regards to modern SQL, Hashsets, Functional Programming, GroupJoins etc; and its beautiful how it immediately clicks in our brains when we commit to these. E.g imagine a baby given those shape to socket toys, the baby doesn't go through it iteratively, but rather already computes in sets(shapes and sockets) and performs map or fold in one go.

I've studied all of this while doing my UG and PG never found some one that summarize the overall concept of these topics in short video like this. Hat off bro.

I'm a university student studying some math courses and currently studying statistics This video is amazing at revealing how my studies are linked to computer science

You explained logarithmic functions in a way that no upper level or university class ever was able to. Congratulations, you did what a professor could not.

Love your videos! I am a software angineer who almost finished a degree in Mathematics with researchs in advanced boolean algebra before going back to CS in university, and it made my passion for programming only grow more and more! Now I'm finishing my graduation's final paper on Functional Programming and Category theory!

Dude this video is super cool because you were able to explain these concepts in a much simpler and easier to digest way than most of my professors did. It also makes me feel better as I see that while I'm not an expert in every concept, I'm familiar with all of them making me feel way less stupid 😅

Except for calculus, this video was my entire math journey through my first 2 years of my CS degree.

I completely agree with this. Math is crucial in programming even when nowadays some people want to ignore it. It is not hard if you enjoy it. What is wrong is that many math-lover programmers lie about how they hate math and that you don't need to learn math if you wanna be a good programmer like in this video (but he already confirmed but actually by using sarcasm). The reality is: if you have any problem with math you are gonna face REALLY HARD life problems on your job, and projects.

How you explained these concepts with this speed and I still understood you is sick. You are the greatest teacher who walked on this earth. A sensei for all the sensei's. The ring doesn't control you, you control the ring. Amazing!

When you explain base 10 to begin explaining how bases in general work I think it might be better to think of it as each place's decimal value is equal to the number in that place which is less than the base value(10 for decimal) multiplied by the base to the power of the index(with this index starting at 0). This explanation should be easy to understand for programmers since any programmer should understand indexes that start at 0. To express this briefly you can say the decimal value 'D' is the place value 'P' multiplied by the base value 'B' to the power of the index 'I' or D = P*B^I. This is how I like to think about this and it works for any base. EX: for 1011 we have a base value of 2 and index 0, 1, 2, 3 from right to left. to get the decimal value we have D=((1*2^0)+(1*2^1)+(0*2^2)+(1*2^3)) = (1+2+0+8) = 11. Hopefully this helps someone and makes sense.

This is the first time I understood logarithms thanks alot, I think this will 'stick' with me for some time before I forget it

5yrs in programming, thanks for best explanation of big O complexity :D Video is nicely done, progressing from basics to advanced. I believe you will keep your job of educator for noobs even with GPT-10 :D

This video does wonders... been procrastinating a few tasks today but the instant I clicked on this video I instantly jumped and started working on them

After dismissing it all for years when I couldn't appreciate it in high school... "Who would ever need the Pythagorean Theorem?". Now I'm learning about Fibonacci, the Russell paradox and the Mandelbrot set and I wish I would have known (more) about all this earlier and wasn't just taught enough to pass an exam in school. Could I have appreciated it more at an earlier age? I don't know. We're in a complex system that we don't have the tools to understand yet or maybe ever. Just like in AI development they have now found that language is the key to everything human, maths is the key to the universe.

4:56 Set Theory is one of the foundations that enable defining the continuous structures of geometry, topology, and calculus. Zermelo-Fraenkel Set Theory with the Axiom of Choice (normally called just ZFC) can serve to specify the way a set interacts with an equipped topology, or how the set of real numbers contains numbers that cannot be reasoned about due to how uncountably transfinite it is.

Vector vs Scalar quantities are pretty excellent to know when thinking about useful analytics. I'm so happy you mentioned them!

omgggg actually I am at this point in my life where I basically need to understand those topics to progress further in my programming career. You really did a good job explaining those topics short and sweet, even for someone like me who always sucked at math and now has to work with linear algebra etc. for ML in my company. LOL

funny how I was JUST stressing about how much math I'm gonna have to learn for programming (and I'm beyond horrible with math). I feel a bit more confident now since I can grasp the concepts laid out here pretty easily (the excellent explanation makes it easy to understand, so props!) it's just the actual numbers part that gets me lol

I wonder how much time did it take you to compile this information... I understand all the concepts but to summarize them is not easy. Well done!

As a long-time fireship watcher, I can say that your examples are very funny. They help us to remember better and hence, to understand the concepts well.

Wow, you will be VERY familiar with all of these concepts if you got a CS degree, I got one and now and only now I understand why we get to know those concepts, thanks.

As a COmputer Sciences grad student, I had to take a course on every single subject mentioned (and more like calculus, numeric methods, group theory, differential equations and even some basic physics).

Very concise and excellent content. Shall we make one on Theory of Computation?

I love that you give us as an expert that overall picture of our future path, and releives me a lot, thank u

the reference to Schrodinger's cat at 1:27 was aweoxme. Its always the details in FireShip's videos man.

Thanks for being on of the best tech YT channels!

I was getting there but I see someone else is first, but putting my results here anyways. I only needed first 2 decoded, after searching for them I could find the binary quote Morse quote below: Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth.

PLEASE DO MORE VIDEOS LIKE THESE. Maybe dive deeper in the chapters shown in this video, it will be extremely helpful. For someone who wants to slowly start learning about neural networks, going to 3Blue1Brown and watching their series can get overwhelming due to so much maths on screen, ur graphical and creative methods are much less brain loading.

Excellent video! Really well done, I appreciate the information and memes a lot!

Your explanation of the logarithm is outstanding

Now time for the prerequisites: strong understanding of high school math, calc 1-3, and proofs for the discrete math.

As a math challenged programmer/gamedev, please give us more content like this! 😮🤯

Nice. Thanks man. Please never stop these sorta videos. I am fond of this type of stuff.

There's 4 years of a CS degree in less than 10 minutes! AWESOME as usual!

The base64 you wrote through the video is the quote "Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth", notice that you forgot the dot in the end which would be Lg== in base 64. Which is also funny because Lg is binary logarithms if im not mistaken

i can't believe how great your videos are as physicist we learn some similaire, we basically learn same maths the difference is in the amount of focus on those sub-section just wanted to point one thing which is in linear algebra you skipped one level which is tensors. good luck and keep up the learning and sharing

I can’t believe the amount of useful information you just taught me in a matter of minutes. THANK YOU!!!

The secret is a quote from Bertrand Russell on mathematics! Or at least one in a similar vein. Maybe you asked ChatGPT to make a quote about mathematics and it came up with this one lol. here's the text: "Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs, a mathematician chooses one pattern for beauty's sake and pulls it down to earth."

As an class 10th student looking for career in cs. I can say that I understood nothing 😢😂 but I can guarantee that I will come back to this video after 4 years ..love you ❤❤

Mr. Fireship, forget math. You are magic. I have hated math since I was 7 but you just lit up a flame in me with this video and every other video you make!

2:58, the thing I find wrong about how floating point numbers are handled is the same reason get that .30...4, the exponent is used to represent where the decimal point is when the better way is to represent where the smallest support unit is so that the mantissa can be treated as an integer with N 0s in front/behind it from the known smallest of the containing bytes. For example, let's say we have a piece of paper that can support up to 32 digits (yes I'm referencing a standard float's number of bits), since we know that we can designate 8 digits to represent how far from the smallest digit we are, next we use half the max it can hold to tell us how small the smallest unit should be, so in this case 255 / 2 is 127.5, we'll round that down so that the fastest matching operating in a computer would be a simple right shift by 1, so now our smallest possible unit is 127 digits behind the decimal point & an exponent of 0 means exactly that, an exponent higher than 23 ( our remaining digits when accounting for the + & - prefixes) is definitely bigger than one with an exponent of 0 while any value between 23 & 0 would have to add the difference between the 2 values to the position of the highest digit to compare against the highest digit of the smaller exponent's mantissa. Yes we sacrifice the implied bit but instead we get 100% accurate results beyond the decimal point.. provided they're not longer than 127 digits that is *coughPI*

With content like these, it'll be long before AI replaces you... Kudos.

As a computer science student its nice to refresh all of this concepts that i've been learning all over this years

Diving deeper of the harder topics here would be nice, like a 15 minute linear algebra video or something

I am a math teacher in a high school teaching to students studying informatics curriculum. Thanks a lot for this one!! 🙏🏻

This may be the best maths for programmers video I have seen so far. Surprisingly detailed for how quickly and consistently you go through each concept.

All what was shown her i learned in a humanist class in high school and at first year at the University. But i'm born and raised in Poland, so for plenty of us it's very common. Now i know why polish programmers are very often by deep data science, ai (including chatgpt) or security. Nice video :)

Matrices are definitely very important to learn as they have many uses and are very simple and easy to understand.

joins are NOT set unions/intersections. In particular, the way you displayed left/right joins would mean they just return the original tables, which is obviously not the case. SQL *does* have UNION and INTERSECT operators for queries. But an INNER JOIN e.g. is the graph of the relation defined after the ON, and the other ones are unions (!) of the latter with elements of A x {null} or {null} x B

I used basic differential equations all the way through my PhD. But if I’m not using particular maths a lot then the core of the concept may not stick. So I love watching other people explain math concepts and you made byte binary make concrete and intuitive sense. Loved the idea of thinking of it as each position is a yes or no at a power of 2 and adding the result.

A little tip: exponents, roots and logarithms are interchangeable. If 3^2=9, then sqrt(9)=3 and log3(9)=2. Any exponentiation has three parts (base, exponent and result) and so there is a nth root that will give its base as a result and a logarithm that will give its exponent.

This is so good, it's like a math teacher is actually trying to teach you something instead of flexing his math muscle :D

Chat GPT can decrypt the message instantly. It also explains that it is encrypted in Base64 which is a common encoding scheme used to represent binary data in ASCII format. It even corrects misspellings in the encryption.

I am amazed by how programmers are still avoiding math.

Literally explained the most confusing part of binary to me in 10 seconds. Finally made the connection lol. Thank you!!!

This is the first time I really see (and so finally understand) what "left join" is and why it is necessary to have "left join" and other types of "join" (because as shown there is obvious distinction between them)

Using logs to explain logarithm, brilliant ❤

Small nitpick: linear regression isn't just for lines. Fitting a polynomial or some linear combination of functions makes it linear in the linear algebra sense.

This video is very interesting...I wanted to revise most of these concepts but i have been procrastinating to do so....This video is a really good motivation for me. Thank you! :)

Thanks for the video. What I missed here would be an example for each math concept in real world. For example how Boolean algebra can simplify otherwise complex conditions etc. while I knew these concepts from university beforehand, for most of them I am not sure where and how to use them.

Fireship actually have ability to convince me that I need certain subject or tool for my career

Linear algebra for RSA. I think linear algebra is more related to AES.

Mathematicians will unironically say something like "The volume of a superegg can be derived via squigonometry, a generalization of trigonometry to squircles. It is related to the gamma function" and expect people to understand.

We believe teachers are the core of our childrens education and therefor their future. My math teacher in primary school always told me how stupid I was etc and that I would never be able to properly function in society. She was partly right and partly wrong. I love being the odd one out and enjoy my personal space to the extend it may appear peculiar, this she nailed. What may offend her endlessly is to know that once I realised I could in fact do math it changed my universe and so many things around me came to blossom. Now I can program stuff she never even had the imagination to comprehend even if it was forced upon her. May her soul rest in peace, I have found mine. I mean this in all humbleness because talent is a gift and not an entitlement. Did I also mention my family were not rich. And I forgot to say she was an Apartheid teacher, very judgemental alike Apartheid religion. She could not teach me math, but she taught me to be strong. Thank you ms? Dreyer.

7:52 JIAN YAAAAAANG

Great video, im gonna learn all of these concepts in 10 minutes with notes each using chatgpt and put it on my resume, if i get a job where i feel ill need it, at least ill know the concepts at least.

Really good video, I loved it, very well explained like always. I just missed numerical methods and will be 10/10

Thanks for the amazing video, Jeff! Your content is pure gold!

I definitely need boolean algebra. Trying to learn to write drivers for my mcu peripherals in C, and being able to do that with ternary operators would make easier in the end.

3:23-3:26 "Now that we know how numbers work..." My brain is fried and I am in tears

In linear regression, the input/output variables don't have to be linear wrt eachother. The linearity comes from the coefficients (\betas) being linear, such that we can have a vector representation of them.

Small nitpick: RSA doesn't involve linear algebra at all. I think you might have meant to say AES instead, which represents the data to be encrypted as matrices and involves matrix multiplication in the MixColumns step

Fireship , please make a full computer science course in your style of teaching. i would definitely buy such a thing

This video takes me back to the good ol' days and when i wrote sprite routines using algebra and 6502 machine code.

I wish I knew more about all this years ago.

1:07 this is a sensitive scene >:(

The first ever video I have watched at velocity x0.75. And going to watch again. Bravo!