A Gentle Introduction To Learning Calculus

Posted May 2, 2008, under Calculus, Guides, Math
Tags:
Related Posts:

A BetterExplained Guide To Calculus

Explanations for everyone.

A Calculus Analogy: Integrals as Multiplication

Learning Calculus: Overcoming Our Artificial Need for Precision

Prehistoric Calculus: Discovering Pi

Like it? Share on:

• 
• 
• 
• 

RSS or Email Subscription |  Print This Post

Get the Math, BetterExplained ebook

Like the site? Take it with you. Develop your math sense using insights, not memorization.

131 Comments »

Trackbacks & Pingbacks

1. Pingback by Mainfram Reality » links for 2008-05-03 — May 2, 2008 @ 11:34 pm

2. Pingback by nulleando.com.ar » Blog Archive » Replanteo … — May 3, 2008 @ 5:16 pm

3. Pingback by The Case For Teaching Calculus Early - Standard Deviations — May 3, 2008 @ 5:56 pm

4. Pingback by Adam @ Dusk » Math Class — May 10, 2008 @ 3:57 pm

5. Pingback by Spy story « The Braden Files — May 12, 2008 @ 8:09 pm

6. Pingback by Prehistoric Calculus: Discovering Pi | BetterExplained — May 29, 2008 @ 9:16 am

7. Pingback by Intuitive Guide to Angles, Degrees and Radians | BetterExplained — July 9, 2008 @ 7:43 am

8. Pingback by Learning Calculus: Overcoming Our Artificial Need for Precision | BetterExplained — August 21, 2008 @ 2:30 pm

9. Pingback by Better Explained « Xavier Seton’s Blog — May 7, 2009 @ 12:42 am

10. Pingback by A Calculus Analogy: Integrals as Multiplication | BetterExplained — July 15, 2009 @ 11:43 am

Comments

1. Sign me up! I did all that crazy “area under the curve” stuff at school, but never understood how it REALLY worked. y=2x^2 => dy/dx = 4x…sure, but what the heck is going on. They lost me when the sines and cosines joined the calculus party.

   I’ve nevertheless remained interested in maths over the 20 years since, and here’s the crack: your article is brilliant. I can absolutely get what you’re talking about. Your circle example is dynamite, and I also found the idea that calculus “finds patterns between equations” very intuitive.

   Now let me remember, my little equation is differentiation. That’s like taking pi.r^2 back to 2pi.r So what you showed was integration; which makes sense to me as you did take the area under a curve. So, to differentiate pi.r^2 I don’t ask for dy/dx, but rather something/dr I don’t see any other letter, an pi is just a number, so maybe the use of y is just convention? So…dy/dr?

   Keep up the good work.

   Paul — May 2, 2008 @ 12:43 pm

2. Beautiful commentary. I’m currently taking Calculus III, and have already finished Differential Equations. For my degree, these would be the final mathematics courses I would need. However, recently I’ve felt that it’s all starting to make sense and come together, and I’ve found that mathematics is quite elegant. After a certain point, I don’t feel awed by its complexity, but rather it’s simplicity. How an incredibly seemingly complex relationship can be broken down into a symbolically simple idea is truly beautiful. I’ve decided to continue taking some mathematics courses in my next semester, and see how far I want to go in that direction.

   It really is a shame that the way mathematics is presented creates a negative impression from grade school on. Conceptually, it is beautiful and elegant and explanatory and all-encompassing. If I had been introduced to mathematics in that form when I was younger, I would have probably been hooked then.

   My favorite moments in horrible math classes in high school and junior high would be when the teacher would digress and just talk about the nature of zero or infinity or other interesting concepts. Of course, the teacher would usually end with something like, “Well, anyway, to get back on topic…” and resume with some cumbersome proof.

   I’m not saying that a conceptual presentation of mathematics should precede basic grade school necessities like arithmetic, but it should definitely have its place. By misrepresenting the elegant nature of mathematics, we are restricting students who would otherwise begin to take interest.

   Again, great article!

   Mike — May 2, 2008 @ 12:50 pm

3. I like these sorts of examples for people who have never seen calculus before because, honestly, the subject is not that hard. Give me an above-average student and I can teach them the basics of calculus in less than a week.

   But it’s rarely the basics that get people. These methods, after all, were how calculus developed up into the mid-19th century — nary a delta or an epsilon in sight.

   Euler was the master of these types of proofs. It wasn’t until mathematicians like Weierstraß started getting counter-intuitive results with these so-called “intuitive” methods that they decided an absolutely rigorous foundation for calculus (and all of mathematics) was necessary.

   So, the only caveat is that while these methods might be intuitive and help people just learning calculus, there are limits at which this type of reasoning breaks down and we simply can’t reconcile what is true with what our intuition says is true.

   Jesse Farmer — May 2, 2008 @ 12:51 pm

4. Dude, you rock!

   Being an Engineer, I understand the pain a naive student goes through when he is burdened with truck load of Calculus books having tons of theorems, proofs and unimaginable number of weird questions that have absolutely no relevance to the real world!

   I scored well in my engineering mathematic subjects but I never really understood the point of learning that stuff. Heck, I don’t even remember half of it now.

   I wish we had someone like you who could paint such a wonderful picture and make the subject more relevant to students.

   I look forward to whatever article you come up with next in the series.

   God Bless You!

   (BTW, where are you from? I wud love to meet a genius like you sometime!)

   Prateek Sharma — May 2, 2008 @ 12:57 pm

5. I just wanted to say I’ve been reading your blog for some time now, but I just had to let you know every article is great and very informative, I just wish you wrote more often =) (j/k I know it must be a lot of time to put together these articles, but thanks again!)

   Justin — May 2, 2008 @ 1:30 pm

6. This was just great. Now can someone out there with the requisite skills (I don’t have them) *please* make the circle into triangle thing into a video and post a link to youtube?

   James — May 2, 2008 @ 2:16 pm

7. You said: “Instead, let’s share the core insights of calculus. Equations aren’t enough — I want the “aha!” moments that make everything click.” Amen! Those “aha!” moments make live worth living (or math worth learning )

   Rodrigo — May 2, 2008 @ 2:16 pm

8. “[...] they decided an absolutely rigorous foundation for calculus (and all of mathematics) was necessary”

   Well “they” may have decided that, but they failed. No mathematical system is absolutely perfect. There are always holes to poke. This is the essence of Gödel’s work. Your system will never be rigorous enough to always be right, but it might be rigorous enough to work for the problems you care about.

   x — May 2, 2008 @ 2:51 pm

9. Wow, thanks for the comments guys!

   @Paul: You got it — we were essentially integrating the equation for circumference. But if you call it that from the outset, and define it rigorously, people’s eyes will glaze over .

   And as you said, the use of x (input) and y (output) are conventions. So the regular way would be to say the equation is really 2 * pi * x, where x is the radius (never mind that we always learned it as 2 * pi * r). dy/dr is a perfectly fine way of saying it too.

   One interesting thing about integration is seeing how something that doesn’t “look” like a curve (a bunch of rings) can be twisted into a format that does.

   @Mike: Thanks for the awesome comment! You really nailed it, there are such beautiful ideas buried in math, which could really encourage people, but don’t have a chance because we jump into the details.

   Conceptual discussions & drills have their place. It may be like listening to fun music (rock, rap, etc.) and being inspired to play. Then you start learning an instrument and memorize scales (doing drills). Drills are much more manageable when you have an appreciation for why you’re doing them.

   Those side discussions you mention can be awesome — it highlights the discovery side of math. For every equation, there was someone seeing it for the first time and saying “whoa”.

   @Jesse: That’s a very good point. I see it similar to teaching Physics: we start with Newtonian mechanics, which are “intuitive” to a degree. Then, as people advance, we teach them about the exceptions: strange things happen at the speed of light (relativity) and when you get really small (quantum mechanics).

   But if we started off with relativity and quantum we’d lose everyone along the way.

   @Prateek: Thanks for the kind words! Just a curious learner here. I know what you mean — I’ve taken many math classes, but the formulas just seemed to stay there, and didn’t really change how I viewed the world.

   I’m usually in the Boston or Seattle area, and if you’re around feel free to drop me an email (kalid@instacalc.com).

   @Justin: Thank you for the kind words, that really means a lot. Yeah, I wish I posted more frequently too .

   The articles can be time consuming (10-15 hours) but I think my brain is the bottleneck — procrastination, perfectionism, and sometimes it’s a struggle to have a “good enough” insight (I don’t want to rewrite what’s already on wikipedia). Maybe I can find a way to trick myself into writing more .

   @James: That would be awesome. Unfortunately I don’t have any animation skills either.

   @Rodrigo: I agree — math would be a boring place if it was only about pushing numbers around .

   Kalid — May 2, 2008 @ 3:01 pm

10. @x: You hit the nail on the head. Math, at its core, depends on unprovable axioms and assumptions — at some point you have to say “this seems to work, it’s good enough, let’s run with it”.

    Unfortunately the quest to make calculus rigorous turned it into something which isn’t as easily understood for beginners.

    Kalid — May 2, 2008 @ 3:02 pm

11. This is something I’ve learned from my quite limited independent study of calculus, which is my personal way of looking at it: calculus is all about how things change. The derivative is one tiny change, and the integral is the sum of many tiny changes. That explanation works quite well, to me, for setting up equations that use calculus. It also makes the fundamental theorem of calculus very simple to understand.

    I have to agree about math education; I’m reminded every day that there are people intelligent enough to understand math who don’t get it because it’s not explained in a way that makes sense intuitively. It wasn’t even until about a year or two ago that I started to really understand math and not just use the equations I was given.

    I’m sick of the way the education system teaches math, so much that I’ve considered writing a textbook in the style I think math should be taught. To me, it’s simple: learn the way that it was originally discovered. It was discovered through intuition, and that’s the best way to learn it.

    I’ll cut short my rambling here. I’ve given you too much to read as it is.

    Zac — May 2, 2008 @ 7:19 pm

12. Hi Zac, thanks for the comment. Yep, seeing the derivative and integral that way (in terms of changes) can really give an intuitive feel — and the fundamental theorem becomes that much clearer.

    I agree with you about math education — I think many people are capable of learning the subject, but it’s not presented in the best way. We tend to show the final result without all the steps along the way — and those steps are what build intuition. It surprises me that people don’t often write about their own insights (vs. formulas), so just trying to take a stab at it.

    Always appreciate an interesting discussion!

    Kalid — May 2, 2008 @ 9:37 pm

13. Another good explanation. Thanks Kalid.

    Viru — May 2, 2008 @ 11:11 pm

14. You’re welcome Viru, glad you enjoyed it.

    Kalid — May 2, 2008 @ 11:28 pm

15. Hi,

    Wow! You have communicated a beautiful simplicity. I have several books on calculus (Calculus for Dummys, Math for the Millions, etc. etc.–never was able to read them) but your explanation is what I have needed all these years. Congratulations, and thanks.

    Doug Hogg
    Former Prinicpal of Pinewood Academy

    P.S. Since it only communicates to people who know calculus, I think you could leave this line out:
    “I’d feel I cheated if I called calculus “the study of limits, derivatives, integrals, and infinite series”.

    Doug Hogg — May 3, 2008 @ 1:51 am

16. “You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity).”

    Sounds like just as strong an argument for Creation if you ask me! Sugar and fat are provided to aid survival, and our bodies are designed to make use of them in an optimum way. Sweet fruits encourage consumption and hence spreading of seeds for survival; sounds like a good “plan” to me.

    I enjoy your articles, but weakly weaving religion into an article on math is unnecessary and, frankly, I didn’t think it was your style.

    Mark — May 3, 2008 @ 6:12 am

17. I have always, ALWAYS hated math. I’m actually pretty decent at it when I understand it, but it is such a painful process to get to where I understand it that by the time I do, I’m sick of it and don’t want to do it anymore. I would be so much better at it if I bothered to practice it, but I hate it so much that I don’t WANT to practice it. I’m in my first year of college, and the placement test put me in trigonometry (I don’t know how, because I only made it through a year and a half of high school algebra before I gave up), but I only have to take college algebra to transfer, so that’s what I’m going to do next fall because it stands a chance of not making me crazy.

    But reading this post…well, it kind of made me want to learn how to like math. It made me CURIOUS about numbers, which has honestly never happened before. The rings-into-triangle thing was the biggest “AHA!” moment I’ve ever had regarding math. It made sense, so I liked it. (I like things when I understand them, see. Like, solving gigantic equations is ridiculously fun, because I know how to do it.)

    Anyway. I am rambling. But thank you, thank you! I feel like there’s a glimmer of hope that I might be able to get a handle on math if I just look at it differently. I never thought of it being ideas; it was just brain-numbing formula memorisation until now. And I hate it when I’m unable to do something, so I really would love to be able to do math and not excuse myself by saying it isn’t my subject. Your definition of calculus made so much more sense than the ones I’ve heard.

    Kat — May 3, 2008 @ 6:36 am

18. @Mark

    2nd Paragraph:
    “Calculus relates topics in an elegant, brain-bending manner. My closest analogy is Darwin’s Theory of Evolution: once understood, you start seeing Nature in terms of survival. You understand why drugs create stronger germs (survival of the fittest). You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity). It all fits together.”

    I guess I don’t see where Kalid is “weakly weaving religion into an article on math.” Or was your comment meant to be taken sarcastically?

    Zack — May 3, 2008 @ 11:18 am

19. @Doug: Thanks for the note, I’m glad you found it helpful! Good point on the note — I changed the wording a bit. It makes me chuckle when I see complex subjects (calculus) explained in terms of other complex subjects (limits, integrals, etc.), without at least _some_ plain-english explanation. How is a beginner looking up what calculus means supposed to have an idea of what it does?

    @Mark: I’m not sure I understand the connection to creation — the goal was to use evolution as an example of a simple, unifying theory that can explain a lot of natural behavior.

    Animals that hated sugar, fat and other high-calorie foods probably starved when times were tough. But their siblings with a sweet tooth probably survived, which selected for that trait. Evolutionary pressure gives an explanation of why sugar would seem sweet to us today (I’m not a biologist, there may be other reasons too).

    Anyway, the point is that calculus finds similar connections/underlying themes between math — there are nice (simple) reasons why the formulas are linked.

    Without calculus, the similarity in the equations just looks like a happy coincidence, much like “sugar is sweet and spoiled food tastes bad” might seem like a lucky coincidence without the theory of evolution. Hope this helps clarify what I meant.

    @Kat: That’s awesome! I love getting those “aha” moments and I’m happy you were able to get excited about calculus ideas (it’s a rare thing in this day and age).

    You definitely can get a handle on math — I really believe it’s a skill like writing. Once upon a time, everyone thought reading & writing were “hard” and only for scribes; today everyone does it.

    The hardest part about math can be staying interested and keeping your motivation, so hang in there! Seeing it as just another way to talk about an idea can help get the big picture. And you’re right, when you get it, even solving gigantic equations can be fun .

    Kalid — May 3, 2008 @ 5:37 pm

20. @Kalid:

    Your implication appears to be that evolution is THE theory that provides the “aha” level of understanding the natural world. Yet the example you provided is just as easily explained by creation. It came off a bit preachy to me and detracted from an otherwise well-written article.

    Mark — May 3, 2008 @ 6:38 pm

21. I always wanted to learn this calculus stuff. Tho I seemed to have survived the last 40 years of electronics and computer theory without it, I’ve always had a curiosity about just what all those squiggly lines were on the old chalk boards. I think you have succeeded in clearing up some of the fog. (so far so good anyway) Please keep up the good work you have been doing on this web site. I really have enjoyed all of your articles.

    Paul — May 3, 2008 @ 9:03 pm

22. @Mark: Point taken, and happy for the discussion. I think the key point behind it all is that the sweetness of sugar serves a purpose (to help us survive) — but if we don’t notice this underlying theme then we miss many of connections that exist in the real world.

    @Paul: Thanks for dropping in, and for your comment! Glad to make things clearer as I can — the funny thing is that despite using the squiggly lines many times, they tended to stay in the realm of abstract symbols without much inherent meaning. So I’m trying to go back and relearn the stuff with the viewpoint of “it has to mean something!”. I’ll keep writing as best I can .

    Kalid — May 4, 2008 @ 11:45 am

23. Absolutely magnificent. One of the best things I’ve ever stumbled upon. The analogy with finger painting only after learning chemistry/physics/anatomy is so very accurate.

    Keep it up!!

    Grey — May 4, 2008 @ 5:43 pm

24. Thanks Grey, I’m thrilled you enjoyed it so much! Yes, not letting people fingerpaint (with the absence of tests & grades) can destroy a child’s interest in a subject. “Drill & kill”, I’ve heard it been called.

    Kalid — May 4, 2008 @ 8:45 pm

25. Many Thanks for Sharing, such a valuable information.

    Best Regards
    Team
    Web Hosting Sri Lanka

    Derive Host — May 4, 2008 @ 10:28 pm

26. Hi Kalid,

    I’ve been waiting for this article/series about calculus for few months since started reading your blog. I tried to learn calculus myself few times. I’ve learned something I knew how to compute some simple examples but I’ve been missing that “Aha” moments so much. So I’ve decided that I’ll wait until you start to write about the calculus, since you explained so well every area you wrote about so far (exponential functions, natural log, complex numbers, …) and in the meantime I’ll spend my math time in other areas. I’m very lucky that I did such choice. Your article is, as always, so enlightening and clear. I’m very happy to have such a great math teacher! Thanks you so much for such material.

    Also, I would like to mention the book by Keith Devlin, “The Language of Mathematics: Making the Invisible Visible”, which actually brings me to the interest in math, one or two years ago. Without that book I would probably not read this blog and would not believe in my bright math days So for others asking “Why Math?” or searching for a lot of “Aha!” moments, the Keith’s book is great reading during waiting for next Kalid’s article

    Martin — May 5, 2008 @ 12:35 am

27. @Martin: Thank you for the wonderful comment — I’m glad you’re finding the articles helpful! I’ll try to keep them that way .

    Thanks for the book recommendation, I’ll need to check that out. I’m always interested in resources that can help people understand & appreciate math more.

    Kalid — May 5, 2008 @ 11:34 am

28. WOW.

    amazing stuff, when you first told me you were going to write an article how real world calculus I thought it’d be a stretch. this was very impressive, and made it easy to understand.

    I like your approach, I used to learn very complex subjects by picking up the kids editions of things, it gave me the 80% i needed to know to be able to converse in very little time.

    nice job.

    Pham — May 6, 2008 @ 10:06 am

29. It’s Paul from comment #1 again. Thanks for the reply Kalid. Again, the article is brilliant.

    I wonder, does the triangle analogy also work with squares instead of discs? If the side length is x, the perimeter is 4x. I apply your awesome triangle procedure and get (1/2).x.4x which is 2x^2; but I was hoping for x^2.

    Best wishes,
    Paul

    Paul — May 7, 2008 @ 1:44 am

30. @Pham: Thanks man, glad you enjoyed it . Yeah, it’s funny how explaining stuff “for kids” can force you to distill all the mumbo-jumbo into its most basic elements (and therefore making it more clear for everyone).

    @Paul: Thanks for dropping by. That’s a great question — I think using a square should work. The tricky part is that even with “square rings”, we only want to take the radius (x/2).

    Looking at the jagged triangle, you can see how you could bend the sides all the way around to make a circle. Thus, we’re only measuring the “outward” distance from the center, since the perimeter wraps around. Similarly for the square, you can imagine that we’re bending the jagged triangle into 4 corners — we move from the center to the right side, but the height of each line can wrap around the entire square. So we only go from 0 to x/2.

    The equation turns into (1/2)(base)(height) = (1/2)*(x/2)*(4x) = x^2.

    Hope this makes sense, I had to think about it for a bit. I think it’s weird because we aren’t used to talking about the “radius” of a square.

    Kalid — May 7, 2008 @ 1:08 pm

31. Keep up the fantastic maths analysis. Your diagramatic, pictoral explanations should be taught around the world.

    Interested Reader — May 8, 2008 @ 8:21 am

32. Many thanks, glad you enjoyed it!

    Kalid — May 8, 2008 @ 11:15 pm

33. Another great article from a great writer.

    Ferenc — May 11, 2008 @ 2:02 pm

34. Hi Ferenc, thanks for the support!

    Kalid — May 11, 2008 @ 5:38 pm

35. Amazing! Four semesters of mind-numbing calculus in engineering and I was blown away by the circle triangle example. Never really looked at such a basic relation in this light! Can’t wait for more!

    Goldust — May 11, 2008 @ 9:35 pm

36. Awesome, glad it helped you! I know what you mean — sometimes we get stuck in the nitty-gritty of integrals and derivatives that we don’t realize that calculus was buried inside the formulas we learned in middle school .

    Kalid — May 11, 2008 @ 9:46 pm

37. One of the things that I have tried to encourage engineers and mathematicians to do is to tell things to me as though I were 8 years old. You’ve done that here and I am a wee bit wiser for it. Nothing is difficult if the teacher cares enough to make it simple. Congratulations for a [formerly] calculus-shy lawyer.

    Richard Bash — May 12, 2008 @ 8:16 pm

38. Hi Richard, glad you found it useful! The funny thing is that many engineers & mathematicians would prefer the 8-year old version too! Many people end up learning the mechanics but not the insight of the operations. Thanks again for the comment.

    Kalid — May 12, 2008 @ 9:12 pm

39. Kalid, Thanks for such a wonderful article. So far I have never understood maths “the way it needs to be understood”. Your article was enlightening.

    I also appreciate your efforts in replying to each of the comments.

    My wish is that you write a book on Maths in “Simple & easy to understand” way and i would definitely recommend it. Many Thanks Again.

    Vasanth — May 13, 2008 @ 10:30 pm

40. Hi Vasanth, thanks for the message . I think math can be understood a variety of ways (intuitively, mechanically, etc.) and you need them all to have a good grasp. Usually, though, we only focus on the mechanical aspects.

    On the book, I think it would be a great idea. Currently I’m looking into collecting these pages and organizing them into a series. Maybe after I get a few calculus posts under my belt .

    Kalid — May 14, 2008 @ 12:34 am

41. I’ve been reading your blog for months now, and I think everything you write is well thought out, informative, and above all, interesting!

    Calculus was by far my favorite math subject. I had so many “a-ha!” moments that I felt like the world was different after I learned it.

    I think it would be great if in a follow-up article you discuss the relationship between velocity and acceleration. In this modern world there’s so many everyday analogies to be made, and I think determining the rate of change of a rate of change is something that is easily overlooked, yet so elegant once you realize it. It might also be too simple for your blog, but I’d love to see a Kalid explanation for it!

    Another related topic I think would be simple yet interesting is events happening in instantaneous vs discrete time, although personally I can’t think of any good examples for that. I just remember how shocked I was that we could determine an object’s velocity at any given instant and totally remove change in time from the equation, yet it is still inherently dependent on time!

    Kai — May 14, 2008 @ 5:21 pm

42. Hi Kai, thanks for the message! I’m glad you’re enjoying the articles, I really believe that there are interesting nuggets in any subject — sometimes we just have to dig for them .

    I think the relationship of acceleration to velocity is a good one, I’m thinking about how best to present it. Even in a car, you don’t set your *speed* — you push down the gas or brake, which accelerates you, which changes your velocity, which changes your distance. So really, the distance you travel is ultimately a “function” of where your foot is on the pedal. I think it’d be an interesting topic — no subject is too simple .

    The use of instantaneous rates is intriguing as well, I’m trying to figure out the best way to approach the limit concept. It’s essentially a machination from the 1800s to deal with “infinitely small changes”/infinitesimals which had been used intuitively before then. Again, a topic that will need a bit of thinking.

    Thanks again for the comment!

    Kalid — May 14, 2008 @ 10:29 pm

43. “
    Did you know we don’t learn calculus the way Newton and Leibniz discovered it?
    “

    I have often thought about this one.. Thanks!

    Sarnath — May 15, 2008 @ 5:26 am

44. Hi, I’d like to first congratulate on that example. And I’d like to point out that that’s the first time i’ve seen that, or anything like that, witch should be mind bogling since i’m a seniour student in a technical college and my knowledge of math and physics is way above that of the average layman.
    I have my own example of math being tought moronically. I remember once when our high school physics teacher asked us what was the integral(antiderivative) of 1/VdV(the work being done in an isothermal transformation), and no one had a clue, witch was rather odd given that we we’re pretty good at math, and all of us knew the antiderivative for 1/xdx however the antiderivative of 1/VdV, was a whole together diferent story.

    Paul — May 15, 2008 @ 6:05 am

45. @Sarnath: Yes, I consider it ironic that Newton probably wouldn’t recognize calculus as we teach it today .

    @Paul: Thanks for the message. Yep, sometimes we get so deep into the nitty-gritty that we forget how calculus can help us see relationships between “everyday” equations. And sometimes we get sidetracking when a variable is replaced .

    Kalid — May 15, 2008 @ 7:46 pm

46. I “StumbledUpon” this a few days ago, and thoroughly enjoyed it! I had, especially in retrospect, a great calculus teacher in high school. We learned a lot of the intuitive aspects of the subject, but at the time I didn’t realize how unusual and great that was! Unfortunately, it been a long time, and I’ve forgotten a lot. Thanks for publishing this explanation, it makes me want to revisit the subject!

    But the real reason I’m posting is because of what Mark said in the comments above. I wasn’t going to say anything, because religion vs science arguments belong in a different forum, and I felt it was very big of Kalid to say, “@Mark: Point taken, and happy for the discussion.” And then he very deftly sidestepped the whole subject and restated his original point without the “offending” reference to scientific theory. It was definitely a very mature way of handling an immature poster, and I’m hesitant to re-open the subject.

    However, it really bothers me that we tend to treat these pushy religious types as if their “theories” of the origin of life and the origin of the universe deserve respect, or as if they’re harmless. Now people will always believe crazy things, and that’s fine. But a lot of these people are making an organized, concerted effort to undermine human progress, and with our ecosystem in such a delicate position, we as humans can’t afford to let them.

    “I enjoy your articles, but weakly weaving religion into an article on math is unnecessary and, frankly, I didn’t think it was your style.” The first word or concept on the page having anything to do with religion is in Mark’s previous paragraph, when he brings up “Creation.” Apparently, Mark is saying that the theory of evolution is a religion. The simple fact that he believes this shows how ignorant we can be if we hold false beliefs. Ignorance is often very dangerous. Especially widespread ignorance. This also shows how rude and pushy these religious types can be, while making it seem as if we are slighting them, by (in this case) not making allowances for somebody’s random, wacky religious beliefs in a discussion about math.

    “Your implication appears to be that evolution is THE theory . . .” Just as the theory of gravity is THE theory we have to explain the observable fact that gravity exists, the theory of evolution by natural selection is THE theory that explains the observable fact that evolution exists. There are no others. Go ahead, try and name one. But remember, in order to qualify as a theory, it must explain the available evidence, and it must make predictions which are testable. In other words, it must be refutable. Otherwise, it is not a theory.

    “It came off a bit preachy to me.” This is so ridiculous I just had to include it. Sorry. It would make me laugh if it didn’t make me lose so much hope for our future.

    I just think that the most important way to combat the kind of ignorance that leads to the election of incompetent public officials is to combat ignorance whenever we encounter it in our daily lives. Sort of a “think globally, act locally” plan.

    Now this post really is preachy! Sorry, Kalid. Now I’ve had my say, I’ll leave it alone. I Promise.

    Jeff — May 16, 2008 @ 10:35 pm

47. I have to disagree with you on your Shakespeare example. Sure, it gets at the idea “be yourself,” but there’s a reason Shakespeare didn’t just say “be yourself.” The Shakespeare quote is beautiful, and to figure out why, we can diagram the sentence, figure out the meter, look at word choice — in short, figure out *why* it is beautiful. All of those components contribute to the way the sentence functions, so it’s important to look at them. There’s a reason we don’t read simplified, abridged, plain-language versions of Hamlet in English class!

    Gabe Murchison — May 17, 2008 @ 2:51 pm

48. This is great!

    I flunked math all through high school and ended up doing basic math and algebra in a community college. I found a great teacher there who could turn the lessons around 180º and explain it so I could finally understand it. I found out I was a visual learner, but when I got to calculus class, it all fell apart again. I could understand the concepts – I couldn’t put together the equations.

    Funny thing is, I love numbers and thanks to Mechanical Universe, I like physics. I found a book called Physics Without Calculus and truly enjoyed it because I could “see” the problems. As soon as calculus was added, the pictures “disappeared” again.

    Kelly — May 18, 2008 @ 1:47 pm

49. Hi,

    it’s probably not your favorite topic, but since you mention evolution, couldn’t you write an article about it? So many people walk around and think they can argue against it, while the effects are so plain obvious.

    darwin — May 19, 2008 @ 10:02 am

50. excellent job on this one

    wlid — May 19, 2008 @ 5:37 pm

51. @Jeff: Thanks for the comment, glad you enjoyed the post! Yeah, mixing “religion & science” can be a touchy subject. To me, science is about knowledge and religion is about ethics, and you shouldn’t use one to determine the other. I usually don’t address it in posts since it’s unlikely for either person to change their opinion based on a few paragraphs on a website. In this particular example, I’m not as interested in anyone’s reasons why something is so, as much as the effects (sugar/fat serve a higher purpose of encouraging survival). But great points .

    @Gabe: Good point. I think the key is being able to appreciate the intricate structure _and_ the meaning (not just one or the other). Unfortunately, math education tends to focus on the former.

    @Kelly: Thanks, glad you enjoyed it! Yes, calculus can be visual and intuitive, but often it’s buried underneath a pile of equations. The equations are useful, but can be really dense without any kind of intuitive grasp (I had a similar experience with vector calculus — it didn’t really start making sense until I visualized it). Appreciate the book suggestion, I’ll have to add this to the reading list.

    @darwin: It would be an interesting article, though I’m currently not that well versed in the biological particulars . But yep, it would be cool as it’s a very powerful and far-reaching observation.

    @wlid: Thanks, glad you liked it.

    Kalid — May 19, 2008 @ 10:48 pm

52. To start, forgive my english, its my third language.

    Ive always enjoyed math classes, I guess that Ive been lucky enough to have good teachers although they did use the conventional teaching system. The things is that the only way I could study math and remember formulas was to make sense of them. So, while others tried to memorize what formula and when to aply it I was trying to understand why that formula and why use it there. After a while I realized that studying math was quite enjoyable and easy if you went and understood what were you doing with the numbers. That was back in 9th grade.
    Its been more or less 7 years since that revelation was made and Im still studying math related degree, economics. Coming across this article has been interesting since it describes exactly what I felt back in that day, and even though its elemental math for me, its reminded me of the beauty that math has and I had forgotten with subjects like Econometrics and advanced statistics, cause if you dont go to class its very hard to internalize and understand huge formulas. LOL. But anyways.

    Very well written but most of all, I admire someone whos making this effort just for the sake of math and the bad name it has among the young, and I hope that many of them come across this to learn to see math with different eyes.

    Luck and thanks!

    Manu

    Spain

    manu — May 21, 2008 @ 4:21 am

53. Hi Manu, thanks for the message! Yes, I’ve always enjoyed knowing the “why” not just the “how” — unfortunately, for many students it’s not obvious that this understanding is important unless they stumble upon it themselves or their teacher encourages this line of thinking.

    Glad you’ve enjoyed the article .

    Kalid — May 21, 2008 @ 7:44 am

54. I can’t imagine how I did pass all my math subjects back in college. My professor presented the subject just the way you presented your circle-triangle area presentation.

    This is awesome. I love mathematics!

    tiny — May 21, 2008 @ 9:37 pm

55. Awesome, glad you’re finding it useful .

    Kalid — May 22, 2008 @ 5:06 pm

56. Hi,

    I am an engineer by profession. I scored well in math during my school days and college days too(That is the beauty of current math education you can score well without understanding anything about solution). For nearly 12 years right from my higher secondary school to college, I have asked all my teachers why we should learn calculus and where we should apply it (The best ones gave the example of speed,velocity and accelaration nothing more than that). Others gave a list of formulas to memorize.

    I went through this article in your site and atlast found the answer. We are really blessed to have this internet,stumble and blog. God bless you. I have been searching through lots of books and lots of sites regarding calculus. But never had that A-ha moment. I had it when I read your article. I dont have words to explain my happiness. you have unwound a knot that was tied 12 years before. Thanks for all your effort keep up the good work.

    With lots of love and respect,
    Ferose Khan J

    Ferose Khan J — June 2, 2008 @ 2:44 am

57. Hi Ferose, thank you for such a wonderful message! It means a lot that the article was able to help understand this subject — I know what you mean about the memorization vs. understanding, I had plenty of “cram and forget” sessions in school. It just wasn’t satisfying to manipulate the equations without knowing what they were really for.

    Again, I’m really happy the article was able to help, I’m planning on doing a series on calculus so I hope they are useful for you too!

    Kalid — June 2, 2008 @ 11:35 pm

58. I very much enjoyed your article Kalid. As many posters have, I applaud your effort in sharing knowledge, and excitement about it, with others.

    But the real reason I’m posting… in a nutshell (I love those books) “the greatest measure of intellect and knowledge is recognizing how much you don’t know.” Jeff’s premise that ignorance=belief in creation and education=belief in the theory of evolution is belied by the fact that many of the most educated and intellectual people (including many, many scientists) do not believe in that theory. Einstein professed belief in intelligent design. And no matter what you may say when it comes to (as science likes to call it) “first cause” your mouths open and close as you stutter to say something but you have no coherence.

    I do appreciate your respect for other people’s opinions Kalid. I wish more people would realize how much claimed “knowledge” really is just opinion and more respect should be accorded. I also wish I didn’t over-react to those that don’t.

    Again, excellent article and I look forward to reading more from you.

    Corey — June 5, 2008 @ 12:31 am

59. Hi Corey, thanks for the comment, I’m happy you’re enjoying the article.

    The question about ignorance and knowledge is a good one. In fact, I think an admission of ignorance is a prerequisite for understanding because everyone needs to accept the possibility that their current position may be flawed.

    Otherwise, we’d still believe in a geocentric universe, chariots pulling the Sun, etc — you cannot teach someone who thinks they already know.

    The primary difference, to me, is what constitutes the gap when we don’t understand something, like the origin of the universe. Is the gap filled by something fundamentally unknowable (God, Nature, etc.), or is it just an idea we haven’t discovered yet, like gravity moving the planets?

    That’s more of a personal/philosophical question that isn’t along the lines of what I discuss today, and unfortunately can lead to counter productive discussions (it can become a heated topic, and I don’t know many people who said “I drastically changed my opinion based on a comment I read online” ).

    So, I’m primarily interested in explaining what we currently understand, knowing it may not be complete (Newtonian Physics to Einstein’s Physics to whatever comes next). However, we’ve got to start somewhere: All models are flawed, but some are useful.

    Kalid — June 5, 2008 @ 11:35 am

60. Great thx for pictures

    Sbs Matematik — June 6, 2008 @ 3:52 pm

61. lol i already love math, and this article just made me love it even more ..

    Anonymous — June 9, 2008 @ 7:47 pm

62. @Sbs: Glad you liked it.

    @Anonymous: That’s great! Yes, sometimes math can be really, really painful or really really fun — I’m trying to find ways to turn the former into the latter .

    Kalid — June 9, 2008 @ 7:51 pm

63. I just finished reading both this article and the prehistoric calculus one on discovering pi, and I just have to say wow.

    I’m recently finished with the 9th grade, and I do have to agree with what you’ve said on the majority of the math programs being taught today. I love math (yep, I’m a math geek) but that’s only because I always focus on the intuition of it. I absolutely hate having to memorize numbers of formulas, so instead I simply figure out why they work. The core ideas behind all of these discoveries just shed new light on how you view everything. I talk to a couple of kids in my math class, and the majority of them hate math, but if I ever try to get in deeper than the memorized formulas and ideas they’re taught to know and not really understand they never seem to have any idea what’s going on beneath it all.

    I probably seem like I’m rambling now, but that’s because it’s 2:40 AM where I am, and I’m pretty tired.

    Anyway, these two article’s have been great, and I’ve completely clicked on nearly everything you’ve said. I especially liked what you wrote about the epiphany like moments when you finally have an intuitive grasp over the concepts, because I end up having those a lot whenever I’ve been thinking into an idea for a while. It was also pretty interesting looking through that proof of the area of a circle where the one I had learned was completely different. It had to do with the an equation of the area of a regular polygon with n sides (1/2 * perimeter * apothem) and if you imagined adding sides to a polygon until it was a circle, the apothem would the the radius, and the perimeter would be the circumference, and you plug that in and then you get pi*r^2.

    This was a pretty great find for such a late internet excursion, and I have to say I’m looking forward to the next one, and have to ask when’s it coming?

    Hank — June 10, 2008 @ 11:48 pm

64. Hi Hank, thanks for the wonderful message! I’m really impressed that you’re searching for intuitive insights this early in school, as you mention most people just want to memorize the formula and move on. But that attitude will really help you in learning, so congratulations!

    That’s an interesting proof for the area as well — one thing I like about math is that there’s so many ways to understand the same result. So part of what I try to do is collect the various insights that worked for me, since it’s not always explained in that way. I haven’t set a date on the next calculus article but would like to have it out in a week or two .

    Kalid — June 11, 2008 @ 11:18 am

65. To me it always seems as though creationists refute their own arguments. I enjoy finding these posts where they seem willing to talk more about their beliefs, letting us see more about the thought processes behind their opinions.

    Corey, when you say, “I wish more people would realize how much claimed “knowledge” really is just opinion,” who do you believe is claiming unsubstantiated facts? Is it the biologists, who consider any refutable theory that supports the observations? Or the creationists, who claim to know the designer of the universe, no matter what the observations might reveal? Which one of these (the biologist or the creationist) will readily tell you, as you suggest, the details of exactly how much they don’t know about the origins of life?

    When you wrote, “many of the most educated and intellectual people (including many, many scientists) do not believe in that theory,” I found myself at first doubting your sincerity. This claim has been made in the past by some disreputable people, but has been thoroughly debunked. You probably are sincere, but just don’t realize that you have been mislead.

    If you doubt this fact, I urge you to check out Project Steve, from the National Center for Science Education. Although the Discovery Institute has A PAGE of signatures from scientists of various qualifications, the statement they signed does not suggest that they “do not believe in that theory.” It is a vague, thick statement that does not include the word ‘evolution’, and states that study of “Darwinian theory should be encouraged.” Project Steve sets forth a very clear, unequivocal, concrete statement supporting evolution and specifically against ‘Intelligent Design’. The only people eligible to sign this statement are scientists named ‘Steve’. (This is estimated to be approximately one percent of all scientists who are eligible to sign this statement, should they choose.) To date there are EIGHT HUNDRED EIGHTY NINE signatures!

    When you say, “Einstein professed belief in intelligent design,” I suspect you are repeating fabricated talking points used in church groups to convince the ‘believers’ that they are smart, and right. Albert Einstein wrote, “The word god is for me nothing more than the expression and product of human weaknesses, the Bible a collection of honourable, but still primitive legends which are nevertheless pretty childish. No interpretation no matter how subtle can (for me) change this.” He also wrote many other things expressing disdain for religion. And the phrase “intelligent design” is something the Discovery Institute just dreamed up recently, long after Einstein’s death.

    Your point that very little is understood about “first cause” is of course true, and how exciting! There is so much left to learn! But clearly, the origin of life and the origin of the universe are completely different subjects (except to theists).

    Corey, I hope that you and everyone reading this can see that there’s nothing wrong with being wrong. We are ALL wrong about many, many things. I myself have discovered I was completely wrong about some things which I was utterly sure of. Being mistaken, or wrong, or holding false beliefs is part of being human. The ability to RECOGNIZE our mistakes, and LEARN from them is perhaps our greatest strength as a species. It is also the basis of the scientific method.

    While I very much appreciate having my opinions respected, and living in a time and place that such a thing is possible, opinion really doesn’t enter into evolution, or science in general, in a very significant way. That’s the beauty of it (in my opinion).

    Now I’m sure that nothing I could possibly write here could change your beliefs, and that is as it should be. But in much the same way that what you wrote sparked an interest and a new understanding in myself, I hope that you can also derive some satisfaction from this submission.

    Pirx — June 13, 2008 @ 1:32 am

66. in all mi internet travails i hav never found such a clean and easy to understand explanation of calculus!

    Tushin — June 20, 2008 @ 4:33 am

67. @Pirx: Thanks for the eloquent comment. I agree — my focus is understanding ideas via unambiguous, falsifiable theories that make testable predictions.

    People can believe what they like, but understanding the (currently known) mechanisms behind phenomena, such as evolution or gravity, helps understand more about the world. The orbits of the planets are not a mystery but follow a predictable pattern. And yep, a huge realization is knowing that our understanding may not be fully correct will constantly improve — actually, that’s why I called this better explained not “best” explained .

    @Tushin: Thanks!

    Kalid — June 20, 2008 @ 10:16 am

68. It seems that maths and science are taught in completely contrasting styles, as the article (which I’d like to add was fantastic, I wish I was taught like this) and several posts point out. Science does teach complicated things in a historical fashion: In England at least, Bohr’s outdated, yet simple, model of the atom is taught up to age 16, then 17-18 a summary version of the quantum atom is taught. I wish this was true for maths.

    MRW — June 22, 2008 @ 8:38 am

69. @MRW: Ah, that’s a great insight. Yes, understanding the historical context can help refine understanding (similar for Newton’s laws to relativity).

    The nice thing about math is that it never gets outdated or incorrect, better techniques just come along. So we can learn that Archimedes was developing ideas that led to the theory of Calculus, like Bohr made a model that led to quantum theory (it would be hard to jump from nothing to quantum theory, or from nothing to Calculus, but that’s how it’s often taught!). Thanks for writing.

    Kalid — June 22, 2008 @ 10:03 am

70. Okay, I jumped to the comment section to leave a comment before reading the rest of your wonderful article. Sue me

    I’m not particularly great at math, but far along enough to realize how stunningly beautiful the insights gained through math may be.

    I doubt much has changed since I was a student, and here’s a little something every student should know. The educational system is generally not structured to teach you much of anything. It exists to discover and promote students with the promise to thrive in corporate ranks. Nothing more and nothing less. Really, “When was the last time you heard the word entrepreneur mentioned in an educational setting? ” Think about it.

    TJ — June 24, 2008 @ 5:04 am

71. Hi TJ, thanks for the comment! Yes, unfortunately the educational system doesn’t seem focused on real insights (more test memorization, which is quickly forgotten) and the reward system for professors in universities is not geared to reward the best teaches (publish or perish). This site is just my little candle in the darkness .

    Kalid — June 27, 2008 @ 4:39 pm

72. That was beautiful. You have no idea how much this has helped me.

    Thank you.

    Brendan — August 4, 2008 @ 4:39 am

73. Hi Brendan, thanks for the note — always happy to help!

    Kalid — August 4, 2008 @ 1:55 pm

74. too long

    art — August 5, 2008 @ 7:32 am

75. Still waiting for that next article.

    Hank — August 15, 2008 @ 9:50 pm

76. @art: 1600 words isn’t that bad, is it?

    @Hank: Thanks for the encouragement, the next one is in the works as we speak !

    Kalid — August 15, 2008 @ 10:25 pm

77. Beatiful post, thanks..

    transpalet — August 20, 2008 @ 12:21 am

78. @transpalet: Thanks, glad you enjoyed it.

    Kalid — August 21, 2008 @ 3:00 pm

79. Kalid -

    You are a gifted teacher. Thanks for your clear, concise explanations. I plan to visit your site often.

    Dave Anderson

    Dave Anderson — August 23, 2008 @ 4:56 pm

80. Kalid, you are the man. The first illustration is perfect for a beginner. Cheers mate.

    Tyler — September 16, 2008 @ 7:44 am

81. @Dave: Thank you for the kind words! Running the site is a lot of fun.

    @Tyler: Thanks for the feedback — I was very surprised that this relationship between formulas we learned in Geometry wasn’t shown until much later. Appreciate the comment.

    Kalid — September 16, 2008 @ 6:23 pm

82. thanks… perfect article

    resim — September 17, 2008 @ 5:12 am

83. great post, thank
    again plase

    ankastre — September 17, 2008 @ 5:13 am

84. great post
    thank for information

    forex — September 19, 2008 @ 1:35 am

85. Thanks so much for that.

    haberler — October 16, 2008 @ 6:13 am

86. the dissecting the circle proof of the area of a circle was published in the Talmud 1500 years ago in much the same way you describe here.

    Ben Waldman — November 2, 2008 @ 10:02 pm

87. i have had these math courses in high school algebra 1,2 and geometry and in college i had stats and physics. i would like to start over so i can build on a strong foundation and eventually get to calc and other higher maths with the desire to career change from social services to perhaps actuarial science fields or strategic management. any suggestions as to where to begin again and what books can u suggest, specific authors. thanks for any help.

    bryan — November 19, 2008 @ 12:08 pm

88. @Ben: Interesting note!

    @Bryan: That’s great about revisiting. Unfortunately I don’t have any specific book recommendations, but here are some people found helpful (comment #33):

    http://betterexplained.com/articles/how-to-develop-a-mindset-for-math/

    In general I would suggest always looking for the “big picture” behind the concepts as they are presented. And always look for another explanation if the one in the book doesn’t make sense.

    Kalid — November 23, 2008 @ 5:45 pm

89. mahn..that was something!!

    awesome job Kalid and this article is just like so
    like the ones ive been searchin for all my life..

    ahh great job mister, i wonder what you’re doin though..are you a scientist or sth?

    

    anyhoo..awesome article agon

    NOXmoony — December 19, 2008 @ 12:48 pm

90. @NOXmoony: Thanks, glad you liked it! Nope, not a scientist, currently working at a startup with friends .

    Kalid — December 21, 2008 @ 7:54 pm

91. Hi anh. Just finished reading this article. Really awesome!! I really like the “big pictures” that you put side-by-side together. Math sounds much more interesting the way you see it. I really wish I was taught by you in my previous math classes (or at least become my cute tutor). hee hee. And you’re so right about the the velocity equation (I had to wiki it). I am similar to #2, where I get very intrigued by the root/origin of something rather than the nitty-gritty details (essential, yet… difficult to grasp). Keep up the fantastic work!

    PS remind me to give you a relaxing massage whenever you write another article =)

    Val — December 26, 2008 @ 1:17 am

92. This is one of the best readings I have done on Calculus in a long long time. I used to enjoy doing calculus when I was in college…now I have a son who is in 10th grade and hates math. I just wanted to find out if there was a better way of getting him to understand the beauty of calculus….and your page is brilliant. Thank you very much for making this page!

    The ‘aha’ value I got from seeing the Area of a circle derived…I wish you were my teacher when I was studying! I probably would be in academics instead of being a salesman!

    Sreenath Chary — January 9, 2009 @ 1:31 am

93. As soon as you said unroll the rings I got it, fucking brilliant!

    Piers — January 17, 2009 @ 5:00 am

94. Amazing post.

    Victor — January 26, 2009 @ 6:48 am

95. Another great article – keep on changing the world one article at a time.

    And great points about motivation. Reminds me of my high school physics class…one of the times I was most motivated was when I was trying to calculate Michael Jordan’s hang time when dunking from the free throw line. The interest in the subject comes first, the learning second.

    Hang Time — February 2, 2009 @ 2:20 pm

96. @Val: Thanks for the encouragement Em!

    @Sreenath: You’re welcome, I’m really happy you were able to find the page useful and share it with your son. It’s never too late to tinker around with numbers .

    @Piers: Glad you enjoyed it.

    @Victor: Thanks.

    @Hang Time: Heh, I’ll do what I can in my little corner of the ‘net. I completely agree — you can only push a rock uphill so long, when there’s interest the learning comes easily.

    Kalid — February 18, 2009 @ 1:37 pm

97. Brilliant!

    I was just perusing Google for a quick refresher on elementary Calculus and this article came up. Never before have I heard such a clear and concise explanation of the fundamentals… I seriously could have saved hours of hair-pulling in university had I had access to this article years ago.

    Keep up the fantastic explanations!

    Steve — February 18, 2009 @ 9:28 pm

98. @Steve: Thanks for the kind words and encouragement! I was in the same boat — it was years after I “learned” Calculus until I started seeing what it was really about. And I’m still finding out .

    Kalid — February 18, 2009 @ 11:03 pm

99. “Unroll the rings”. This single picture if shown to students of Calculus would set off a lot of light bulbs. I saw that and I am still in awe of how simple it is.

    Am also terribly peeved at the academia for sapping away the joy of mathematics and not providing more motivation.

    aleemb — February 25, 2009 @ 1:46 am

100. @aleemb: Thanks, really glad it clicked for you! Yes, it’s funny how a complex idea can just be unraveled when you look at it differently.

     Kalid — February 27, 2009 @ 1:44 am

101. http://simple.wikipedia.org/wiki/Differential_calculus is another excellent beginners article that is good companion to this article.

     Deryk — March 7, 2009 @ 5:31 pm

102. I recently read your article on calculus and it was amazing . Hats Off to you . While I was reading a book on Sir Issac Newton I found out about calculus . I wanted to learn about it so the very next day searched for it on the net and I got it .Also could you send me the url of the sites where one can learn calculus .

     Shashank — March 18, 2009 @ 11:16 pm

103. Amazing…Thank you
     I enjoy this type of material
     Makes the most complicated, unknown so easy to
     comprehend and store…
     I have read a lot on higher mathematics but this is
     a very refreshing approach..so sensible.
     Your insights are mind boggling…
     Thanks again

     Larry Johnson — April 3, 2009 @ 1:25 pm

104. try this book.It has clear explanations of basics – “idiots guide to calculus”.
     It is available as a torrent download.
     check this link http://www.mininova.org/tor/2414500

     balakrishnan — April 20, 2009 @ 12:40 pm

105. A little more than thirty years ago I won an award at my high school for being the top math student. A couple of years later, I abandoned the study of mathematics. You see, I could make good grades in my Calc classes, but I had absolutely no idea why I was memorizing how to do it. It was no fun anymore.

     My 77 year old father has cancer, but he has always been my inspiration in science and math. He is one of the minority scientists who disputes a “big bang” origin of the universe. (Basically, there is no expansion of the universe, only local contraction as a result of the constant and continual creation of energy, resulting in the gravity phenomenon.) To fully understand what he is trying to tell me, I need to understand calculus. By that, I mean that I don’t need to know how to do calculus, I need to see what it is about. I won’t have my dad for much longer, so an article like this is invaluable for someone like me.

     One final comment: There is no science of “intelligent design” unless its proponents are willing to admit and believe that it might be wrong.

     wolfizzi — May 4, 2009 @ 12:58 am

106. I am here in this blog for about 2 hours, moving post to post. I am just loving it. This is something i was looking for.

     Respect to Khalid

     Shuhel — May 8, 2009 @ 12:16 pm

107. Hey it’s a really cool article. Im currently doing my masters in regenerative medicine but my interest in Nanotechnology leaves me no choice to know this subject of calculus. Could you please let me know how should I go about it in detail and also about articles that are as visually appealing as yours so that i can easily understand rather enjoy the subject. Superb work by you.

     Sumit Rai — June 8, 2009 @ 7:52 am

108. excellent article. i love it.

     boss — June 22, 2009 @ 4:34 am

109. I was googling “learning calculus”, seeing as how I’ve also been quite the frustrated math student. I took BC calc my senior year of high school and absolutely hated the way it was taught. The book used was simply terrible, as other users on amazon would attest to that as well. It skipped out on all the insightful moments leaving that solely to the reader and focused instead on equations and a “semi-formal” approach to proofs. I’m starting college this fall and need to seriously brush up on my calculus with the intent of pursuing physics.

     Your article was quite insightful and what I needed, thank you!

     Chris — June 27, 2009 @ 1:10 am

110. 

     Anonymous — July 3, 2009 @ 2:10 pm

111. @Shashank: Thanks! Dr. Math has some very good discussions on math that may help.

     @Larry: Thank you, really glad it was useful!

     @wolfizzi: Wow, I’m happy the article was able to help you in this way. And I agree — if a theory can’t be refuted, it isn’t science.

     @Shuhel, @Boss: Thanks.

     @Sumit: I don’t have any detailed advice; if you need to learn calculus for a course a professor & book are probably your best bet. I’ll have a series of articles which should help provide some intuitive insights about what’s being taught.

     @Chris: Thanks!

     Kalid — July 21, 2009 @ 1:23 pm

112. Appreciate your intent Kalid but fail to see what these commenters are rhapsodizing about,you’ve taken 1600 words to convey a simple geometry lesson,even the crux of it was too lengthy/convoluted,just say to transcribe/reassemble a circle to a right triangle having the same area,make the radius the base,circumference the height and connect the hypotenuse,and seriously,calculus is needed to show that a disc and a ring are related?
     Please enlighten me if i’ve missed the point.

     Mike — July 24, 2009 @ 2:24 pm

113. @Mike: If you haven’t been taught calculus in a rote, dull manner, this post may not resonate as much for you. Unfortunately, many calculus introductions jump into definitions and symbol manipulation, without shedding light into the bigger picture of what calculus is for. As a result, students get discouraged, and only see the underlying themes if they happen to stick with the subject to Physics or other “applied” uses.

     For the circle/triangle example, it’s just tangible example of calculus in action. Sure, you can solve it using pure geometry, but calculus gives you a step-by-step method that uses formulas to get to the same result. Finding the surface area of a sphere using geometry alone would be pretty challenging, but calculus makes it simpler.

     You might have several “one-off” geometric proofs, each with their own quirks, but calculus can directly show how the various formulas are related and variations of the same theme. I haven’t seen many calculus introductions discussing this use of calculus, which is one reason I made the intro.

     Kalid — July 24, 2009 @ 4:09 pm

114. WOW! THIS IS BRILLIANT, I LOVE THIS ARTICLE OF YOURS… I MEAN THE EXPRESSIONAS, ILLUSTRATONS AND ALL I CAN SAY IS BRAVO! I’VE ALWAYS TRIED TO BUILD-UP MY SELF IN THIS ASPECT BUT WHERE I HAVE PROBLEMS IS THE APPLICATION… HOPE YOU UNDERSTAND WHAT I MEAN, I’LL REALLY LOVE TO CONTINUE. PLEASE HOW CAN YOU PROCEED THIS YOUR LECTURE TO ADVANCE THE LEARNING… BRAVO!!! again

     AZEEZ — August 5, 2009 @ 5:50 am

115. @Azeez: Thanks! Glad you enjoyed it .

     Kalid — August 6, 2009 @ 1:03 pm

116. I see your point but when you strip-out the largest circumference you’ve got 3 points = triangle = the total area that the circle had. If you actually stripped-out a 4-sided segment you would have to incorporate equations involving subtracting the radius of 1 concentric circle from another or something.

     x to the nth — August 13, 2009 @ 4:12 pm

117. I didn’t read all the way to the bottom, so sorry if i’m offering something that has been said.

     both creationism and evolution receive equal arguments in your article. one would have to be very sensitive and polarized to extract bias from your writing (regarding evolution/creationism).

     I don’t really believe in God, but I don’t not believe in God. Maybe I should write “god”.

     Idk, i thought i would let you know how I feel about the accusations of subtle religious bias.

     steven — September 11, 2009 @ 1:44 pm

118. I’m 13, and I think this article was amazing. I read “A mathematician’s Lament” when it was on slashdot a few months ago and until now I think that was the greatest mathematical paper I’ve ever read. I’ve only found this site a few minutes ago, but the explanations are so clear and elegant. I love the evolution analogy. I think you might be like the second feynman or something.

     Kevin — September 18, 2009 @ 5:26 pm

119. @Kevin: Thanks for the kind words! I really like that paper as well . I’m a huge fan of Feynman, I love reading/listening to the way he explains things, he’s an inspiration for me. Thanks again for the comment!

     Kalid — September 20, 2009 @ 12:20 am

120. Kalid, you’re the teacher’s teacher.

     You have very rare gifts.

     Last but by no means least, I’m sure I speak for most people here when I say that you come across -more than anything- as a caring friend.

     Best wishes from Downunder

     Ron — October 10, 2009 @ 12:56 pm

121. @Ron: Thank you for the wonderful comment! It really means a lot, my goal is to write things as if I were just having a fun chat about them, just person to person. I’m happy that is coming through .

     Kalid — October 10, 2009 @ 5:30 pm

RSS feed for comments on this post. TrackBack URI

Leave a comment

Have a question? Know an explanation that caused your own a-ha moment? Write about it here.