I read these in 5th grade or so and learned most of middle school and high school math just from those.
They're written as a fantasy novel in which the characters discover the relevant math concepts as part of the story. I know, you're thinking that's a silly gimmick and it can't possibly be any good, but it was actually done quite well. At least until the later parts of Calculus where the situations have to get pretty contrived.
Highly recommended, at least for precocious kids who like math and fantasy novels.
When I was in HS, I wish my school had used books like Serge Lang's or even Gelfand instead the terrible textbooks we were forced to use. I ended up needing to use Saxon Math to review, which was better because it always builds on previous skills instead of moving on to the next topic and never revisiting what you've already learned again.
I really hoped the trig book would help me get over my mental block, but it still didn’t work. There’s something about Trigonometry that just befuddles me.
Have you seen the animation about how sin/cos/tangent are derived? I was absolutely befuddled by trig although I was able to do it by learning rules, until I saw that thing.
Speaking of books about calculus, I like Robert Strichartz's The Way of Analysis. Robert develops intuition and motivation of key concepts in an amazingly detailed way. He often starts with an intuitive but flawed idea, and then gradually revises the idea to a mature definition or proof. The whole book focuses on "why": why real numbers are defined that way, why limit uses the delta-epsilon notation, why we need to topology, etc and etc. Reading the book is more like an adventure than following a set path. The author explains his approach eloquently: "Mathematics is more than a collection of theorems, definitions, problems and techniques; it is a way of thought....My goal in writing this book is to communicate the mathematical ideas of the subject to the reader. I have tried to be generous with explanations....I believe that what you learn through a process of struggle is more likely to stick with you than what you learn without effort....Understanding mathematics is a complex process. It involves not only following the details of an argument and verifying its correctness, but seeing the overall strategy of the argument, the role played by every hypothesis, and understanding how different theorems and definitions fit together to create the whole. It is a long-term process; in a sense, you cannot appreciate the significance of the first theorem until you have learned the last theorem.”
"Why" is something that was always a noticable omission from my math classes in school. I got by simply by learning the mechanical motions of whatever topic we were learning but there was never any cohesion to the concepts. I've been working through this book (and necessarily brushing up on my algebra which has atrophied for the same reason) and I've been absolutely loving it.
I was a math tutor in college, and many of the tutees would come into the tutor lab completely downtrodden, dreading their homework. I found the most helpful thing I could do with them was to focus more on getting them to understand the motivation behind what they had gone over in class that day, and in particular saying things like "now why would we do this" as soon as I could see in their face that it was clicking. Getting them to understand the why (and getting them to feel like they understood the why) was incredibly effective for helping them feel less insecure when going to class.
When I was taking calculus classes, and struggling, a tutor helped me immensely in the exact same way; explaining both practical applications and often the historical context around the method he was teaching me.
When learning Calc, I always appreciated anyone who could teach the main points without slipping into formal textbook descriptions and terminology. The more advanced one gets in Math, (eg, your teacher or professor with an advanced Math degree), the more they tend to do it, even of they try not to.
I like this text - reminds me of "the complete idiot's guide to calculus" which I used as a companion to my Calc textbook (Stewart) when learning. Not sure I would have faired as well without it.
It gives a great and sometimes humorous understanding, which formed a nice framework when I moved on to Vector Calc, Differential Equations, etc later on. At that point in your math career, you can't (or shouldn't try to) rely on mindless algorithms/processes to solve problems like you may have in high school. You've got to understand what you're doing.
Ps, I doubt many on HN really needed to hear that, but it may be a useful note for somebody.
This is a very important point. Because of the "curse of knowledge", at a certain point you are unable to communicate the subject you're expert in to beginners. That's why one of the best ways to study is with your peers: the one who gets it first has the best chances to transmit their aha moment to their colleagues.
It's difficult because to get to the level of a beginner you need to force yourselves to make many oversimplifications, moreover, you have to deliberately be imprecise otherwise people won't understand you. But some people do have this talent: even though they are experts, they still can transmit their knowledge at various levels of difficulty. Keith Devlin is one. Feynman of course.
> appreciated anyone who could teach the main points without slipping into formal textbook descriptions and terminology
This is why I can't impart knowledge at all. I get too technical. When I'm in a mentorship position, I need to be pair programming, and I need to be hands off the keyboard. Getting me to explain anything is a fruitless exercise.
And that's fine: not everyone has to be destined to be a teacher. It is important to know that you are not (if you are not), and to thoroughly appreciate those who are.
> not everyone has to be destined to be a teacher. It is important to know that you are not (if you are not), and to thoroughly appreciate those who are.
Well, this should be understood, but it's not. Especially in the current academic setup the professors - experts in the field - should have the skill to transmit their knowledge to students. Unfortunately, many spectacularly fail. They'd better focus on research and leave the job of teaching to more able individuals.
My pet peeve is linear algebra. You can present it in an extremely boring way, and this is what most professors do. It's like they wanted to make this fascinating subject as repulsive as possible. Maybe you could get away with it 50 years ago, but now we have all the tools and areas of application that make teaching linear algebra in the old way a crime.
Linear Algebra was another one I had a tough time with until I found the right "teacher."
Gilbert Strang's MIT Open Courseware series to the rescue. He wrote the textbook I was using and was also pretty entertaining. My college professor made the concepts sound terribly complicated, but Strange made them highly approachable and interesting.
> I am convinced by the experience of others: It is good for students to work in small groups. There are so many reports about the success of this idea that it has to be accepted as valuable. It will be implemented in different ways, and the comments from Ithaca about group projects are representative:
> "The approach changed the students' attitudes toward mathematics. The projects engage the curiosity of the good students and challenge them, but this does not come at the expense of average and weaker students. In fact, cooperative work with good, motivated students bolsters the others.'
> "A common fear about groups is that one student may do nothing but still get the same grade as the members who did all the work. This has not been a great problem . . . (others say the same). Students experience cooperative learning. They talk to each other about mathematical ideas and they form friendships with other mathematics students."
> I personally believe that we too often lose sight of the human part of learning mathematics.
Most calculus books in India never have any information on the history of calculus.
I remember being taught limits, differentials, etc purely as mathematical structures.
No context what-so-ever.
In engineer, I read a book called "Advanced Engineering Mathematics" by Erwin Kreyszig. This was a wonderful book that also had some footnote references to calculus books.
It was pulling on that thread and reading quite a bit of history and other good books (MIT Books) that I got a hand of the inside out of calculus.
How I passed Calculus exams in High school, only god knows.
I am an engineer now and will be ever grateful to the likes of George B Thomas, James Stewart, N Piskunov for teaching me calculus and making me wonder why such a beautiful concept was twisted and mangled by official curriculum books in India.
Learning in India is a misnomer. You don't learn, you cram up. I had an economics teacher who explained the concepts of economics, then she went on pregnancy leave. The replacement teacher came to the class everyday and literally dictated the stuff that we were supposed to note down. Exam time - kids got better marks with the new teacher. Guess who got to brag about being a good teacher.
I am the fool who is mortally afraid of calculus. I am not kidding when I say that I still get nightmares of sitting in a Calculus exam. I am 37 years old
The first chapter turned on a light bulb in my head when it said
d which merely means “a little bit of.”
∫ which is merely a long S, and may be called (if you like) “the sum of.”
I am gonna bookmark it and read it to the end so that when my son grows up, I can teach him.
It's so funny how much of an impression bad experiences in school make on you. I occasionally have dreams where it's finals week and I realize that I haven't attended any sessions or completed any of the homework for some class. (Sadly all too realistic, I really struggled to have my shit together in college.)
I've often thought that if I ever became wealthy enough to not have to work, I'd like to go back to school and do a better job of it. Looking back, I kick myself for wasting so much of a period of time where I could have been filling my brain with useful knowledge.
Oh well, though - I got enough education to land my first programming job, so it wasn't a total waste. And a lot of fun was had along the way. Hindsight can be a bit illusory.
from my perspective, integrals are continuous forms of summations.
so Σ(6;n=1) f(n) becomes ∫(6;x=1) f(x) or ∫(6;1) f(x) dx
The dx term always confused me, especially when it started being used in unfamiliar ways e.g g = (dt+1)/dx - but if you think of it as the step variable/component of a summation term I think it helps.
One of the confusing things with integration is the difference between definite and indefinite integrals. What you're describing is a definite integral which is indeed a continuous sum. Indefinite integrals are maybe better called antiderivatives and it's less clear what it's all about. Most people just remember a set of rules for calculating derivatives and antiderivates (like integration by parts, the chain rule etc.). I find this really throws people off as it's like two separate concepts both, confusingly, called integration.
They're not totally separate concepts though, no? The definite integral is the difference of two values of the indefinite integral (antiderivative).
I suppose if you're talking about solving them computationally, you can calculate/approximate the definite integral by taking the limit of the sum as you reduce the step size, whereas performing the indefinite integral requires a different method (chain rule etc.). But when doing them by hand usually the first step in calculating a definite integral is finding the antiderivative.
That's the thing. What tends to stick in people's heads is the bit that hurts. You can tell someone that integration is "just a continuous sum" and you might see a ligthbulb go on. But then the bad news: you still need to find this magical function F(x) and to do that requires intuition and guesswork. Understanding it's "just a sum" doesn't help at all with finding the antiderivative.
if definite integrals can be represented as a sum of a finite series, I can't see why an Indefinite integral isn't the sum of an infinite series.
You can just take any definite integral and replace it's interval with variables. Now it's just algebraic, much like if you replaced numbers in an equation with variables.
> Indefinite integrals are maybe better called antiderivatives
but why? I can't see why definite integrals aren't equally better called antiderivatives over an interval. Concepts wrt calculus, and method of solving them are separate things to me - you can understand how gravity works in general, without the specific means of solving orbital equations.
The Internet Archive has scans of the second edition of Calculus Made Easy, from 1914 [1]. The book is also available as PDF and TeX at Project Gutenberg [2].
The calculusmadeeasy.org website does include at least one useful footnote that the other versions don’t: “The term billion here means 10^12 in old British English, ie, trillion in modern use.” [3]
I've no idea whether this is an effective way of teaching calculus, but it's not how I think about the subject. I think of differentiation as the slope of the curve, and integration as the area under the curve. Possibly I am more visually-minded than most people.
Yeah, that's pretty much the standard approach to teaching calculus and I would guess that the majority of people with any understanding of calculus understand it roughly that way. It probably is the most intuitive and easiest way to get students from zero to being able to solve problems and use the calculus that they might encounter in many other fields (basic physics, statistics, etc).
It's not really a "wrong" interpretation of calculus, but it has limitations. Eventually the geometric intuition breaks down. You can use it pretty well as long as you are working in standard euclidian spaces with up to three dimensions. Past that, you really can't visualize things and your intuition will lead you to incorrect conclusions or at least just get in the way.
A course in Analysis will tear down that intuition and (hopefully) replace it with a more rigorous one based on limits of infinite series. It's a longer and harder road, but leads to an approach to calculus that carries over to N dimensions almost trivially (vector and tensor analysis). It also extends to limits of series of things other than just Real numbers, eg, calculus on complex numbers, limits of series of functions (harmonic/Fourier analysis), stochastic processes, or variations in functions (calculus of variations). Not being inherently tied to our normal ideas of euclidian spaces even allows for some really weird things like being able to integrate the Dirichlet function (a function where rational numbers have the value 1 and irrational numbers have the value 0). Many of these turn out to be very important and useful in advanced physics and other fields.
> It's not really a "wrong" interpretation of calculus, but it has limitations. Eventually the geometric intuition breaks down.
As long as you realise it's an analogy, it's not the actual thing, I don't see the problem. So for example gradient descent in neural networks is partial differentiation along 100s or 1000s of dimensions, and I wouldn't try to visualise that directly; I might use an analogy of a marble rolling down a 3-dimensional landscape to find a local minimum, but that's clearly an analogy.
Your intuition can carry over in specific cases like a gradient descent algorithm, but you'll get into trouble if you assume that your 3-dimensional intuitions work for other problems in higher dimensions. Eg, sphere packing is sensible in low dimensions but goes to some weird places as you move up in dimensions.
This is exactly my standard mental model also. I guess the exception is that I sometimes think of integration as accumulation depending on the context. But I have very difficult time thinking about a derivative without the visual of a tangent line popping into view.
Now I knew a little calculus from http://djm.cc/library/Elements_Differential_Integral_Calculu... which I had found in a used bookstore.
But I was frustrated. At the time, the famous book “Thomas” was in two volumes, and I had bought Vol. 1 in the Cornell bookstore after my junior year in HS.
So I sat in my hot, un-air conditioned bedroom in Brooklyn for the entire month of August 1967, and solved EVERY odd numbered problem in that book (the answers were in the back.)
Maybe that’s not universal, but the way the notation was taught to me in school (Germany) was that the finite sum Σf(x)Δx is simply written as ∫f(x)dx in the limit. So dx is really just an infinitesimal Δx, and ∫ is the summation symbol, only written with a large Latin (as opposed to the usual Greek) “S”.
That’s useful to know because several integration tricks become easy if you keep that relationship in mind. If you can transform a sum, you can transform the corresponding integral. (With the usual caveats around degenerate cases.)
Similarly, for differentiation, the limit of Δy/Δx is written as dy/dx. Again, knowing this enables you to derive otherwise complicated looking transformations.
Glad to see something like this. The calculus I took in college was very difficult to learn with a professor who seemed to use extremely outlier/difficult problems on tests that were hard to make sense of when the classes were spent learning the basics but not applying them to any difficult problems.
That, coupled with his inability (or refusal) to actually explain any kind of homework problems when asked- or he would start to, and then never give enough detail for us to make sense of it, made me hate calculus and math became such a sour point for me that I completely changed majors. After 2-3 really bad math class experiences in college, I realized I was spending time and paying money to get nowhere.
Now I find myself wanting to relearn these concepts somewhat out of spite, but more out of just knowing what I missed out on. I ended up going into graphic/web design and development and not getting enough from my classes to even think about employment. By the time I graduated, everything I learned was outdated (CSS tables for layout anyone?)
I wish I had either changed colleges or stayed the course despite the pain and gone into programming. One of my biggest regrets, but it's okay. I'm mostly happy where I am now, doing other stuff that is IT-related. Maybe someday I'll be able to get back into math and programming on a significant scale and do something with it.
The very first page after the prologue of this book (https://calculusmadeeasy.org/1.html) speaks to me more clearly than entire semesters of math classes did back then.
My high school teacher did explain many of the things commenters are complaining they were never taught. Gave bits of history, talked about summing very small pieces, apologies to speed and acceleration, etc, etc.
And calculus was still very difficult to understand and warp your head around. And as a high schooler, the history seemed like a sidetrack.
I guess my point is, this stuff is difficult no matter how you slice it and it needs time, reflection and retelling in order to make sense.
I love how maths books from 100 years ago are perfectly readable even by modern standards, while philosophy texts from the same period are completely inscrutable.
I think most maths books from back then are pretty inscrutable (and I'm a mathematician). In particular, printing pictures was expensive, so they wrote complex descriptions instead, often using terminology that's hardly seen any more. Maths books have in general become much more accessible at school and undergraduate level.
Depends on the exact branch of mathematics I guess. As a CS major, I found Turing’s paper on decidability very readable, for example. Gödel’s original proof of his incompleteness theorem isn’t too bad either.
Well those are research papers, which have a different purpose, and I suspect in some fields research papers are actually more inscrutable now than then. I'm talking about books intended for an audience like the linked one, high school or early undergraduate. I haven't done an exhaustive survey or anything, but most of the ones I've looked at on calculus, trigonometry, etc. just wouldn't work for most students now; with the improved accessibility, much higher numbers of more diverse students are learning this kind of maths.
I very much disagree - what do you consider philosophy, for instance? I'm a bit of a fan of the history of philosophy, and I've very happily read translations of Plato, Aristotle, Schelling, Feuerbach, and Marx - the only one I've had difficulty with is Hegel.
The terminology chonen by the translator to reflect the author's intention is often strange, but it's no different to, say, reading Homer in translation.
Personally, I've found that philosophic words and their meanings don't change too much - and there are good arguments with formalism vs anti-formalism to the point where I think that mathematical and formal logic methods may not reflect the insight of some of this "older" philosophy as a whole. That is to say - the fact of being "completely inscrutable" can be something of an advantage if meaning is what you're seeking.
There is lots of brilliant philosophy from a houndred years ago (or thereabouts). Try reading Bergson, James, Nietzsche, Whitehead, Russel and Husserl and see if that changes your opinion.
I don’t disagree, but my point is about clarity, not relevance. There’s a reason it’s so uncommon for people to read Nietzsche unguided, or without secondary literature.
Clarity in philosophy is, to draw an analogy with Rich Hickey's argument, akin to ease in programming. A dangerous mask for complexity. Are Plato's dialogues really easier to read than Hegel's dialectic? Only superficially.
I got into philosophy with a gnarled copy of Thus Spoke Zarathustra in my bag. I read it on my own, without guides. And though yes there was a huge amount that I missed, it's anyway an inexhaustible work, and I thoroughly enjoyed it and it transformed my way of thinking. I would argue that's especially what that era sought to achieve (especially as a retort to Neo-Kantianism).
Philosophy, like code, should be as hard as the concepts it's trying to express. Not more, and not less.
Integration and differentiation came like the next step in school. You learned addition and subtraction, and its rules, then multiplication and division and its rules, etc. This went on and built on each other until you learned integration and differentiation.
Both had rules that weren't conceptually much more difficult than what came before, but in university it didn't help that I understood all this.
In the end I had to learn some tricks to find out how to modify a formula so I could integrate them. I would walk in circles all the time and end up with the same formula.
The missing intuitive link for me was the limit. I did not really understand calculus as a concept, a thing of distinct abstracted value, until I learned it starting from the idea of the limit, continuity, and how it differed from actual computation of specific values of a function.
I learnt calculus in maths. I was quite good at it. I could differentiate anything (as this is an easy, mechanical task), and I was good at integrating (this takes more cleverness).
But I didn't get what any of it was about until I learnt about distance, velocity and acceleration. Then it all clicked. I really think any lesson about calculus should start with practical examples and everyone these days can understand the relationship between distance, velocity and acceleration so it's an obvious place to start.
I read "No. 4"--Simplest Cases. This is just like how I remember calculus from high school--a bunch of rules that I memorized but had not idea how to apply to real-world scenarios. It's all about x, dx, y, dy, etc.
Maybe it's easy, but it's hard for me to know how its useful.
I think mathematicians who write math instruction love math for maths sake. The rest of us want to know how we can use it to solve problems we face in our everyday lives.
I found math concepts much easier to understand after seeing their application in engineering. Linear algebra and tensor calculus make sense when you study mechanics. Complex numbers make sense when you study electromagnetic fields and circuits. Differential equations are also way easier to understand when you use them in a real instead of an abstract problem.
If you are never going to get into these (or similar) fields, you should not worry.
I'll second Strang. I recently re-read his Introduction to Linear Algebra and was struck with how clearly he explains things compared to most other math textbooks (which I've been reading a lot of lately).
Heed Axler's warning; the compact version is meant as a refresher, not a book to learn from.
> Linear Algebra Abridged is generated from Linear Algebra Done Right (third edition) by excluding all proofs, examples, and exercises, along with most comments. Learning linear algebra without proofs, examples, and exercises is probably impossible. Thus this abridged version should not substitute for the full book. However, this abridged version may be useful to students seeking to review the statements of the main results of linear algebra.
Maybe you previously worked through the full LADR (or some other math-major linear algebra book like the older, classic Halmos), but your memory is fuzzy. That might be a use for it.
Still, I don't know why you'd read through a 150 page refresher rather than the full book. If you can't afford it, the 3rd edition is on libgen like most other textbooks.
The book is not easy. It's more like a traditional textbook, albeit a great one. In my experience, the book is great at showing how seemingly hard concepts can be systematically developed, layer by layer, under the framework of vector space and linear transformation. When reading the book, I often marveled like this: that's it? Wow! A few references to the previously listed theorems and definitions and this is proved?
Axler's approach puts the cart behind the horse where it belongs. Learning to do row reduction by hand is a really pointless exercise that misses the forest for the trees. But if want want the nitty gritty on computation, Golub and Van Loans "Matrix Computations" is pretty classic.
Only if you want to be a real mathematician. Most people don't and are better served by learning to calculate, since that teaches you how the techniques of linear algebra are applied.
You actually do need to learn to do row reduction by hand because it teaches you what is going on. Same reason you need to learn to integrate by hand even though you'll never do it again after your coursework.
BTW, I learned linear algebra from Axler first, so I have some basis for comparison. There is a reason Axler is not considered an introductory text even though it is not very hard.
The YouTube channel 3Blue1Brown has an "Essence of Linear Algebra" series that I found helpful (along with an Essence of Calculus series and videos on other tricky subjects such as differential equations and Fourier transforms). I found that the visual approach of the videos helped make the concepts more intuitive.
There's Serge Lang's Basic Mathematics which discusses algebra and geometry, including a bit of linear algebra. Introduction to Linear Algebra, by the same author is a bit deeper. Those are the textbooks I would pair with Calculus Made Easy.
For slightly less mature students, there's Algebra by Gelfand.
I really waned to like the Gefland books, and Algebra starts off so well, but they are actually intended as barebones exercise books to go along with lectures which introduce the problems. Or, they could for advanced students who are interested in revisiting the basics. As books, the experience is maddeningly incomplete, and they do not teach mathematics to the uninitiated. I wouldn’t recommend them to anyone who either wants to relearn high school math or learn from ground zero. During self study, I hit a wall about 25% through, and after checking solutions online it turned out that the problem was extremely advanced and also had an error of some kind which renders the solution impossible. I learned some behind the scenes aspects of the book such as the course and lecture which are missing, and I gave up at that point
I think Lang, though tough, actually would be a better book to learn from. It actually seems comprehensive and designed for motivated people who want a very solid understanding of math, but are actually possibly new to the subject.
I liked this. It’s a different approach so take a look at the free pages. Not for everyone but worth a look to see if it fits you. Bonus: the author is responsive.
it's mostly just a few key concepts such as the eigenvalue, eigenvector, the determinant, the inverse, and some others. Even the wiki does a good job describing it + youtube videos
Finally, let me explain the key to understanding calculus. If you get this, calculus is forever just algebra. (Of course, if you cannot, or are unwilling to DO the algebra, you are lost.)
“The derivative of the area under the curve of a function is the function itself.”
I made the mistake of buying Thomas' Calculus after watching a lecture by Knuth where he lauded having Thomas as his teacher. My oh my is Thomas' Calculus not a book to teach yourself calculus with without lectures and seminars to guide you.
I’d also recommend “Quick Calculus,” by Kleppner. It has helped a lot of people. I would search for an early, used addition, as textbooks tend to get more elaborate and worse pedagogically as they are “improved.”
First time I heard about this book I was young and thought I was good at maths and (naturally) dismissive of it because of its title. Surprised it is quite good! Never used it for serious study though.
Algebra the Easy Way - https://www.goodreads.com/book/show/6781.Algebra_the_Easy_Wa...
Trigonometry the Easy Way - https://www.goodreads.com/book/show/2077099.Trigonometry_the...
Calculus the Easy Way - https://www.goodreads.com/book/show/3074478-calculus-the-eas...
I read these in 5th grade or so and learned most of middle school and high school math just from those.
They're written as a fantasy novel in which the characters discover the relevant math concepts as part of the story. I know, you're thinking that's a silly gimmick and it can't possibly be any good, but it was actually done quite well. At least until the later parts of Calculus where the situations have to get pretty contrived.
Highly recommended, at least for precocious kids who like math and fantasy novels.