A Gentle Introduction to Learning Calculus BetterExplained

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A Gentle Introduction To Learning Calculus I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education. 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 lead to resistant 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. Calculus is similarly enlightening. Don‘t these formulas seem related in some way?

They are. But most of us learn these formulas independently. Calculus lets us start with ―circumference = 2 * pi * r‖ and figure out the others — the Greeks would have appreciated this. Unfortunately, calculus can epitomize what’s wrong with math education. Most lessons feature contrived examples, arcane proofs, and memorization that body slam our intuition & enthusiasm. It really shouldn‘t be this way.

Math, art, and ideas I‘ve learned something from school: Math isn’t the hard part of math; motivation is. Specifically, staying encouraged despite

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Teachers focused more on publishing/perishing than teaching

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Self-fulfilling prophecies that math is difficult, boring, unpopular or ―not your subject‖

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Textbooks and curriculums more concerned with profits and test results than insight

‗A Mathematician‘s Lament‘ [pdf] is an excellent essay on this issue that resonated with many people:

―…if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.‖ Imagine teaching art like this: Kids, no fingerpainting in kindergarten. Instead, let‘s study paint chemistry, the physics of light, and the anatomy of the eye. After 12 years of this, if the kids (now teenagers) don‘t hate art already, they may begin to start coloring on their own. After all, they have the ―rigorous, testable‖ fundamentals to start appreciating art. Right? Poetry is similar. Imagine studying this quote (formula):

―This above all else: to thine own self be true, and it must follow, as night follows day, thou canst not then be false to any man.‖ –William Shakespeare, Hamlet It‘s an elegant way of saying ―be yourself‖ (and if that means writing irreverently about math, so be it). But if this were math class, we‘d be counting the syllables, analyzing the iambic pentameter, and mapping out the subject, verb and object. Math and poetry are fingers pointing at the moon. Don’t confuse the finger for the moon. Formulas are a means to an end, a way to express a mathematical truth. We‘ve forgotten that math is about ideas, not robotically manipulating the formulas that express them.

Ok bub, what’s your great idea? Feisty, are we? Well, here‘s what I won‘t do: recreate the existing textbooks. If you need answers right away for that big test, there‘s plenty of websites, class videos and 20-minute sprints to help you out. Instead, let’s share the core insights of calculus. Equations aren‘t enough — I want the ―aha!‖ moments that make everything click. Formal mathematical language is one just one way to communicate. Diagrams, animations, and just plain talkin‘ can often provide more insight than a page full of proofs.

But calculus is hard! I think anyone can appreciate the core ideas of calculus. We don‘t need to be writers to enjoy Shakespeare. It‘s within your reach if you know algebra and have a general interest in math. Not long ago, reading and writing were the work of trained scribes. Yet today that can be handled by a 10-year old. Why? Because we expect it. Expectations play a huge part in what‘s possible. So expect that calculus is just another subject. Some people get into the nitty-gritty (the writers/mathematicians). But the rest of us can still admire what‘s happening, and expand our brain along the way. It‘s about how far you want to go. I‘d love for everyone to understand the core concepts of calculus and say ―whoa‖.

So what’s calculus about? Some define calculus as ―the branch of mathematics that deals with limits and the differentiation and integration of functions of one or more variables‖. It‘s correct, but not helpful for beginners. Here‘s my take: Calculus does to algebra what algebra did to arithmetic.

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Arithmetic is about manipulating numbers (addition, multiplication, etc.).

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Algebra finds patterns between numbers: a2 + b2 = c2 is a famous relationship, describing the sides of a right triangle. Algebra finds entire sets of numbers — if you know a and b, you can find c.

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Calculus finds patterns between equations: you can see how one equation (circumference = 2 * pi * r) relates to a similar one (area = pi * r2 ).

Using calculus, we can ask all sorts of questions:

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How does an equation grow and shrink? Accumulate over time?

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When does it reach its highest/lowest point?

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How do we use variables that are constantly changing? (Heat, motion, populations, …).

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And much, much more!

Algebra & calculus are a problem-solving duo: calculus finds new equations, and algebra solves them. Like evolution, calculus expands your understanding of how Nature works.

An Example, Please Let‘s walk the walk. Suppose we know the equation for circumference (2*pi*r) and want to find area. What to do? Realize that a filled-in disc is like a set of Russian dolls.

Here are two ways to draw a disc:

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Make a circle and fill it in

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Draw a bunch of rings with a thick marker

The amount of ―space‖ (area) should be the same in each case, right? And how much space does a ring use? Well, the very largest ring has radius ―r‖ and a circumference 2 * pi * r. As the rings get smaller their circumference shrinks, but it keeps the pattern of 2 * pi * current radius. The final ring is more like a pinpoint, with no circumference at all.

Now here‘s where things get funky. Let’s unroll those rings and line them up. What happens?

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We get a bunch of lines, making a jagged triangle. But if we take thinner rings, that triangle becomes less jagged (more on this in future articles).

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One side has the smallest ring (0) and the other side has the largest ring (2 * pi * r)

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We have rings going from radius 0 to up to ―r‖. For each possible radius (0 to r), we just place the unrolled ring at that location.

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The total area of the ―ring triangle‖ = 1/2 base * height = 1/2 * r * (2 * pi * r) = pi * r2, which is the formula for area!

Yowza! The combined area of the rings = the area of the triangle = area of circle! This was a quick example, but did you catch the key idea? We took a disc, split it up, and put the segments together in a different way. Calculus showed us that a disc and ring are intimately related: a disc is really just a bunch of rings. This is a recurring theme in calculus: Big things are made from little things. And sometimes the little things are easier to work with.

A note on examples Many calculus examples are based on physics. That‘s great, but it can be hard to relate: honestly, how often do you know the equation for velocity for an object? Less than once a week, if that. I prefer starting with physical, visual examples because it‘s how our minds work. That ring/circle thing we made? You could build it out of several pipe cleaners, separate them, and straighten them into a crude triangle to see if the math really works. That‘s just not happening with your velocity equation.

A note on rigor (for the math geeks) I can feel the math pedants firing up their keyboards. Just a few words on ―rigor‖. Did you know we don‘t learn calculus the way Newton and Leibniz discovered it? They used intuitive ideas of ―fluxions‖ and ―infinitesimals‖ which were replaced with limits because “Sure, it works in practice. But does it work in theory?”. We‘ve created complex mechanical constructs to ―rigorously‖ prove calculus, but have lost our intuition in the process. We‘re looking at the sweetness of sugar from the level of brain-chemistry, instead of recognizing it as Nature‘s way of saying ―This has lots of energy. Eat it.‖

I don‘t want to (and can‘t) teach an analysis course or train researchers. Would it be so bad if everyone understood calculus to the ―non-rigorous‖ level that Newton did? That it changed how they saw the world, as it did for him? A premature focus on rigor dissuades students and makes math hard to learn. Case in point: e is technically defined by a limit, but the intuition of growth is how it was discovered. The natural log can be seen as an integral, or the time needed to grow. Which explanations help beginners more? Let‘s fingerpaint a bit, and get into the chemistry along the way.

Where next? My goal is to begin presenting a beautiful, oft-maligned subject in a new light. Many ideas are more intuitive than you think:

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Averages

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Pythagorean Theorem

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Imaginary Numbers

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Div, Grad, Flux and Curl (if you already know vector calculus)

My knowledge of calculus is still very mechanical, but I know this can change. As I explore this topic I‘ll cover the insights that worked, hoping you‘ll chime in with what has helped you. Here‘s the first:

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Prehistoric Calculus: Discovering Pi

Happy math. (PS: A kind reader has created an animated powerpoint slideshow that helps present this idea more visually. Thanks!)

Other Posts In This Series 1. A Gentle Introduction To Learning Calculus (This post) 2. Prehistoric Calculus: Discovering Pi 3. Learning Calculus: Overcoming Our Artificial Need for Precision 4. Why Do We Need Limits and Infinitesimals?

5. A Friendly Chat About Whether 0.999... = 1 6. A Calculus Analogy: Integrals as Multiplication 7. Calculus: Building Intuition for the Derivative 8. Understanding Calculus With A Bank Account Metaphor 9. How To Understand Derivatives: The Product, Power & Chain Rules 10. How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms

7 Category: Calculus, Guides, Math

Share what worked: Aha moments & FAQ Let's create a living reference for how best to understand this topic.

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The circle being unraveled and reassembled into a triangle. (More...)

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by Josie · full discussion page I knew circles and triangles were related, because my Geometry teacher told us they were, but..... That makes sense! Kalid Thanks Josie, seeing it unrolled was my big aha! too. Ben It's maybe a little outside the scope of the discussion, but what struck me about the circle -> triangle transformation was how elegantly it shows the superiority of tau as the circle constant. Typically, pi*r^2 is trotted out as the example of why circumference over diameter is preferable to circumference over radius, but that triangle just screams to me "I am half of a rectangle!" Kalid

Great point Ben! Ralph Brilliant, simple and fun explanation! It made me get up and walk around the room, beaming from ear to ear! Aha! kalid Awesome, so glad to hear... those are my favorite eureka moments! :) Zubair Wow nice topic..enjoying entire topic..!! Your name

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seeing the graph and relating that to the are of he triangle

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by pax · full discussion page Your name bruce how simple life could be if everything was explained as simply Your name

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"A premature focus on rigor dissuades students and makes math hard to learn." A ton of truth/insight packed into that well-articulated sentence.

by Charlie Flowers · full discussion page

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kalid Awesome, glad it clicked. To crib a programming phrase, premature rigor is the root of teaching evil. Your name

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Two insights from elsewhere that helped me (More...)

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by Charlie Flowers · full discussion page Two insights from elsewhere that helped me were: 1. Sylvanus P Thompson wrote about a person climbing a graph and said something to the effect that "he feels a particular steepness ... can we find out what it is"? Made the goal very concrete and intuitive. 1. When I realized that the function we were trying to find the slope of actually fully contains all the "knowledge" of every point that will be on the graph, and therefore, it already "contains" the "knowledge" of the "steepness" at every point. In fact, it contains the knowledge of how that steepness changes. And it became intuitive that the slope should be a "simpler" version of the original function itself that is "derived" from it. I wonder if that has anything to do with the terminology of "taking a derivative", but I have no idea whether it does. Great site, thanks for publishing this material! kalid Thanks Charlie, love the specifics about what's working. 2. Having a visceral experience of "steepness" is really cool, I like that. I'm almost feeling my stomach change by thinking about climbing a steep staircase.

3. Yes, all the knowledge is really linked. My analogy: If you know someone's bank account, you can monitor it and figure out their monthly income. And if you're monitoring their income, you can then figure out if they've received a raise. And if you know the raise is tied to the performance of the company, you can then figure out the performance of the company... all from looking at their bank account! Each step is "simpler", i.e. the bank account has the raw data, and each level has a slightly simpler look. Thanks for the support! Your name

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the uncurling of the circle

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by fred stelwagen · full discussion page Your name

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Aha! The insight that helped w as:

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1. Paul on May 2, 2008 at 12:43 pm said: 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.

2. Mike on May 2, 2008 at 12:50 pm said: 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! 3. Jesse Farmer on May 2, 2008 at 12:51 pm said: 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. 4. Prateek Sharma on May 2, 2008 at 12:57 pm said: 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!) 5. Justin on May 2, 2008 at 1:30 pm said: 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!) 6. James on May 2, 2008 at 2:16 pm said:

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? 7. Rodrigo on May 2, 2008 at 2:16 pm said: 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

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8. x on May 2, 2008 at 2:51 pm said: ―[...] 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. 9. Kalid on May 2, 2008 at 3:01 pm said: 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

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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 ([email protected]). @Justin: Thank you for the kind words, that really means a lot. Yeah, I wish I posted more frequently too

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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 10. Kalid on May 2, 2008 at 3:02 pm said:

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@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. 11. Zac on May 2, 2008 at 7:19 pm said: 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. 12. Kalid on May 2, 2008 at 9:37 pm said: 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!

13. Viru on May 2, 2008 at 11:11 pm said: Another good explanation. Thanks Kalid. 14. Kalid on May 2, 2008 at 11:28 pm said: You‘re welcome Viru, glad you enjoyed it. 15. Doug Hogg on May 3, 2008 at 1:51 am said: 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‖. 16. Mark on May 3, 2008 at 6:12 am said: ―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. 17. Kat on May 3, 2008 at 6:36 am said:

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. 18. Zack on May 3, 2008 at 11:18 am said: @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? 19. Kalid on May 3, 2008 at 5:37 pm said:

@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 20. Mark on May 3, 2008 at 6:38 pm said: @Kalid:

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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. 21. Paul on May 3, 2008 at 9:03 pm said: 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. 22. Kalid on May 4, 2008 at 11:45 am said: @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

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23. Grey on May 4, 2008 at 5:43 pm said: 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!! 24. Kalid on May 4, 2008 at 8:45 pm said: 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. 25. Derive Host on May 4, 2008 at 10:28 pm said:

Many Thanks for Sharing, such a valuable information. Best Regards Team Web Hosting Sri Lanka 26. Martin on May 5, 2008 at 12:35 am said: 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 27. Kalid on May 5, 2008 at 11:34 am said: @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. 28. Pham on May 6, 2008 at 10:06 am said: 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. 29. Paul on May 7, 2008 at 1:44 am said: 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 30. Kalid on May 7, 2008 at 1:08 pm said: @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. 31. Interested Reader on May 8, 2008 at 8:21 am said: Keep up the fantastic maths analysis. Your diagramatic, pictoral explanations should be taught around the world. 32. Kalid on May 8, 2008 at 11:15 pm said: Many thanks, glad you enjoyed it! 33. Ferenc on May 11, 2008 at 2:02 pm said: Another great article from a great writer. 34. Kalid on May 11, 2008 at 5:38 pm said: Hi Ferenc, thanks for the support! 35. Goldust on May 11, 2008 at 9:35 pm said: 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! 36. Kalid on May 11, 2008 at 9:46 pm said: 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

.

37. Richard Bash on May 12, 2008 at 8:16 pm said: 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.

38. Kalid on May 12, 2008 at 9:12 pm said: 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. 39. Vasanth on May 13, 2008 at 10:30 pm said: 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. 40. Kalid on May 14, 2008 at 12:34 am said: 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

.

41. Kai on May 14, 2008 at 5:21 pm said: 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! 42. Kalid on May 14, 2008 at 10:29 pm said: 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! 43. Sarnath on May 15, 2008 at 5:26 am said: ― Did you know we don‘t learn calculus the way Newton and Leibniz discovered it? ― I have often thought about this one.. Thanks! 44. Paul on May 15, 2008 at 6:05 am said:

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. 45. Kalid on May 15, 2008 at 7:46 pm said: @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

.

46. Jeff on May 16, 2008 at 10:35 pm said: 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. 47. Gabe Murchison on May 17, 2008 at 2:51 pm said: 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! 48. Kelly on May 18, 2008 at 1:47 pm said: 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. 49. darwin on May 19, 2008 at 10:02 am said: 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. 50. wlid on May 19, 2008 at 5:37 pm said: excellent job on this one 51. Kalid on May 19, 2008 at 10:48 pm said: @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. 52. manu on May 21, 2008 at 4:21 am said: 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 53. Kalid on May 21, 2008 at 7:44 am said: 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

.

54. tiny on May 21, 2008 at 9:37 pm said: 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! 55. Kalid on May 22, 2008 at 5:06 pm said: Awesome, glad you‘re finding it useful

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56. Ferose Khan J on June 2, 2008 at 2:44 am said: 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 57. Kalid on June 2, 2008 at 11:35 pm said: 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! 58. Corey on June 5, 2008 at 12:31 am said: 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. 59. Kalid on June 5, 2008 at 11:35 am said: 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. 60. Sbs Matematik on June 6, 2008 at 3:52 pm said: Great thx for pictures 61. Anonymous on June 9, 2008 at 7:47 pm said: lol i already love math, and this article just made me love it even more

..

62. Kalid on June 9, 2008 at 7:51 pm said: @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 63. Hank on June 10, 2008 at 11:48 pm said:

.

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? 64. Kalid on June 11, 2008 at 11:18 am said: 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

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65. Pirx on June 13, 2008 at 1:32 am said: 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. 66. Tushin on June 20, 2008 at 4:33 am said: in all mi internet travails i hav never found such a clean and easy to understand explanation of calculus! 67. Kalid on June 20, 2008 at 10:16 am said: @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! 68. MRW on June 22, 2008 at 8:38 am said: 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. 69. Kalid on June 22, 2008 at 10:03 am said: @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. 70. TJ on June 24, 2008 at 5:04 am said: 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.

71. Kalid on June 27, 2008 at 4:39 pm said: 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

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72. Brendan on August 4, 2008 at 4:39 am said: That was beautiful. You have no idea how much this has helped me. Thank you. 73. Kalid on August 4, 2008 at 1:55 pm said: Hi Brendan, thanks for the note — always happy to help! 74. art on August 5, 2008 at 7:32 am said: too long 75. Hank on August 15, 2008 at 9:50 pm said: Still waiting for that next article. 76. Kalid on August 15, 2008 at 10:25 pm said: @art: 1600 words isn‘t that bad, is it? @Hank: Thanks for the encouragement, the next one is in the works as we speak 77. transpalet on August 20, 2008 at 12:21 am said: Beatiful post, thanks.. 78. Kalid on August 21, 2008 at 3:00 pm said: @transpalet: Thanks, glad you enjoyed it.

!

79. Dave Anderson on August 23, 2008 at 4:56 pm said: Kalid You are a gifted teacher. Thanks for your clear, concise explanations. I plan to visit your site often. Dave Anderson 80. Tyler on September 16, 2008 at 7:44 am said: Kalid, you are the man. The first illustration is perfect for a beginner. Cheers mate. 81. Kalid on September 16, 2008 at 6:23 pm said: @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. 82. resim on September 17, 2008 at 5:12 am said: thanks… perfect article 83. ankastre on September 17, 2008 at 5:13 am said: great post, thank again plase 84. forex on September 19, 2008 at 1:35 am said: great post thank for information 85. haberler on October 16, 2008 at 6:13 am said: Thanks so much for that. 86. Ben Waldman on November 2, 2008 at 10:02 pm said:

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. 87. bryan on November 19, 2008 at 12:08 pm said: 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. 88. Kalid on November 23, 2008 at 5:45 pm said: @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. 89. NOXmoony on December 19, 2008 at 12:48 pm said: 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 90. Kalid on December 21, 2008 at 7:54 pm said:

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

.

91. Val on December 26, 2008 at 1:17 am said: 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 =) 92. Sreenath Chary on January 9, 2009 at 1:31 am said: 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! 93. Piers on January 17, 2009 at 5:00 am said: As soon as you said unroll the rings I got it, fucking brilliant! 94. Victor on January 26, 2009 at 6:48 am said: Amazing post. 95. Hang Time on February 2, 2009 at 2:20 pm said: 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. 96. Kalid on February 18, 2009 at 1:37 pm said: @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. 97. Steve on February 18, 2009 at 9:28 pm said: 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! 98. Kalid on February 18, 2009 at 11:03 pm said: @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

.

99. aleemb on February 25, 2009 at 1:46 am said: ―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. 100.

Kalid on February 27, 2009 at 1:44 am said:

@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. 101.

Deryk on March 7, 2009 at 5:31 pm said:

http://simple.wikipedia.org/wiki/Differential_calculus is another excellent beginners article that is good companion to this article. 102.

Shashank on March 18, 2009 at 11:16 pm said:

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 . 103.

Larry Johnson on April 3, 2009 at 1:25 pm said:

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 104.

balakrishnan on April 20, 2009 at 12:40 pm said:

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

105.

wolfizzi on May 4, 2009 at 12:58 am said:

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. 106.

Shuhel on May 8, 2009 at 12:16 pm said:

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 107.

Sumit Rai on June 8, 2009 at 7:52 am said:

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. 108.

boss on June 22, 2009 at 4:34 am said:

excellent article. i love it.

109.

Chris on June 27, 2009 at 1:10 am said:

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 ―semiformal‖ 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! 110.

Anonymous on July 3, 2009 at 2:10 pm said:

111.

Kalid on July 21, 2009 at 1:23 pm said:

@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! 112.

Mike on July 24, 2009 at 2:24 pm said:

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. 113.

Kalid on July 24, 2009 at 4:09 pm said:

@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. 114.

AZEEZ on August 5, 2009 at 5:50 am said:

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 115.

Kalid on August 6, 2009 at 1:03 pm said:

@Azeez: Thanks! Glad you enjoyed it 116.

.

x to the nth on August 13, 2009 at 4:12 pm said:

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. 117.

steven on September 11, 2009 at 1:44 pm said:

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. 118.

Kevin on September 18, 2009 at 5:26 pm said:

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. 119.

Kalid on September 20, 2009 at 12:20 am said:

@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! 120.

Ron on October 10, 2009 at 12:56 pm said:

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 121.

Kalid on October 10, 2009 at 5:30 pm said:

@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 122.

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Wilshire on November 30, 2009 at 12:49 pm said:

Wow! I must admit that I am very bad at maths. But after coming here and looking the enthusiasm and the way it is shown here I don‘t think maths is a boring subject at all! Now I need a miracle to pass this year. But I‘m sure I‘ll give it my best! Thanks a million yaar! I always knew maths as a boring subject. But not anymore. Again, thanks a lot! 123.

Kalid on December 1, 2009 at 1:12 am said:

@Wilshire: You‘re welcome! Really glad you‘re starting to enjoy math again 124.

.

JaySicks on December 19, 2009 at 1:19 pm said:

Hi Kalid! I was really surprised, how similar your article to my thoughts. Because of education, I always realize the ―gotcha‖ things years after I learn in school/university. A few weeks ago I was thinking on the relation between a function and it‘s derivative function. And when I realized how they logically relate to each other, how the derivative function describes the original, I had the same dual feeling as you; calculus is beautiful, and it is really sad, that students have to realize it by themselves. Teachers prove things by 2 whiteboard long equations, instead of explaining things from the scratch – the path how to ―find out‖ things for ourselves. It seems teachers want us to learn, not to understand. But I think people forget things fast if they don‘t understand it (at least I do

).

I‘m really happy that there are more people out there who see things like this. (I wonder if this way of thinking is somewhat related to the fact that we‘re both interested in programming

)

Keep up the good work, and save the world! 125.

hitendra on January 18, 2010 at 1:25 am said:

hey kalid, sorry to bother you again. but i have thought of another way of finding area of circle assuming 2*pi*r is the circumference. If we take r to be 100. then we can have hundred circles of radius going from 1 to 100. now if we consider their circumference to be part of area of bigger circle except the last circle of radius r we get following equation 2*pi*1 + 2*pi*2 +….. 2*pi*(r) =2*pi*(1+2+…r) =2*pi*r*(r+1)/2 =pi*r2 + pi*r (we can discount pi*r for being circumference of the biggest circle). so we get pi*r2 as area taking it as simplistic deduction. It‘s just another attempt towards thinking abstract. hope i‘m close. Thanks 126.

Seamus on January 19, 2010 at 11:25 am said:

The slideshow really helped

and can you tell me how negative numbers cancel each other out?

Because I read your post on imaginary but don‘t understand how (-3)x(-3)=9 Until someone told me that they cancel each other out, but how? (:-O my ―aha‖ moments are more like ohhhhhhhhhhhhhhhh moments) 127.

Kalid on January 30, 2010 at 4:20 pm said:

@Seamus: Glad the slideshow helped, I was very happy for the contribution. For negative numbers, I consider -3 to be a shortcut for 3 * -1, which means ―Take 3, and then flip it‖". So -3 * -3 means to me 3 * -1 * 3 * -1 which is ―take 3, flip it, make it 3x larger, flip it again‖ … the two flips cancel each other out in that case. I like to visualize numbers like that, but other people may have analogies that help too 128.

.

Jasper on February 24, 2010 at 4:03 pm said:

False sense of knowledge here, nice you non-rigorously showed the the area of the circle is πr^2 now good luck with the sphere. Doesn‘t work very easily with geometry, does it? 129.

Ketil on February 26, 2010 at 9:55 am said:

Thank you for this and many other great articles. I am currently learning Pre-Engineer Math and Physics in a 5 month course. It is incredibly difficult because I start from scratch. It feels a lot like I am being taught to be a robot feeding numbers into mysterious equations. Your writings really help me see the meaning and beauty behind it all. I like learning but not without understanding. You help me understand Math the same way reading Feynman helps me understand Physics. Thanks! 130.

Kalid on February 27, 2010 at 1:11 am said:

@Ketil: Thanks for the note! I completely know what you mean about the robot being fed numbers — it‘s just so frustrating not understanding _why_ something is happening. I love Feynman‘s level of insight, he‘s one of my explanation heroes that I greatly admire 131.

.

Shankar on February 27, 2010 at 3:59 am said:

Hi Kalid……..beautiful article.. I study in Class 10 and I don‘t have to learn calculus but I guess the beauty in the name ―Calculus‖ and my curiosity took me deeper. And u have helped me a lot in understanding it. Your method of deriving the area of a circle from its circumference is cooool. It inspired me to derive the volume of a sphere from its surface area. Doing it the same way as you did, we get a cone whose volume can be found by using the formula : (1/3)*(Base area)*(height)…….. Hope u continue the good work ……….!! 132.

shory on May 27, 2010 at 12:29 am said:

This article is absolute genius as opposed to the contrived genius that pen math text books. True genius is the ability to explain PhD level stuff to a 10 year old. Contrived genius is the ability of a textbook author to make a PhD candidate feel like a 10 year old.

133.

Kalid on May 31, 2010 at 10:43 pm said:

@Shory: Thanks, glad you liked it! Yes, a huge part of teaching is just trying to explain without getting your ego in the way. It‘s not about you, it‘s about whether you can develop an idea in someone else‘s mind. 134.

Indrani Sengupta on June 13, 2010 at 1:46 am said:

Amazing explanation. Very lucid . If maths is taught like this then 90% of students who are scared about maths and its formulas will grow aa interest towards it. I wuld like to go through some more articles like this. 135.

Chetan Anand on June 26, 2010 at 7:06 am said:

Explicate bellemente!!! Nunc, io non pensa que le mathematica es un subjecto difficile! 136.

Jang on July 2, 2010 at 9:16 pm said:

Thank Kalid for absolutely superb posts! Yesterday, I accidentally saw one of your articles and have kept reading your posts. And I feel so lucky to find your site and am excited to teach my son sometime later. Keep your good work up! 137.

Kalid on July 2, 2010 at 10:27 pm said:

@Jang: Awesome, thanks for the encouragement! Glad you were able to browse around 138.

.

Utpal Dasgupta on July 15, 2010 at 3:42 am said:

The calculus the differential & Intregal both are tough, this page does not answer the question how to make the calculus easy for begineer. 139.

Meg on July 30, 2010 at 5:16 am said:

I have to take Calculus this fall and I was practically in tears over it until I saw this website! Now I want to go out and buy pipe cleaners…and probably some finger paints too 140.

Kalid on July 30, 2010 at 8:56 am said:

@Meg: Nice, happy it was able to help! 141.

mathews museneni on August 10, 2010 at 4:46 am said:

I have been thinking of doing a course that involves alot of mathe matics of late but i have never been good in it. but now after looking at your definations, i know am going to change my thinking and just go for it. 142.

Kalid on August 16, 2010 at 3:48 pm said:

@mathews museneni: Awesome, good luck! 143.

Duylam on August 27, 2010 at 7:59 pm said:

can you be my math teacher instead? 144.

Kalid on August 28, 2010 at 6:19 pm said:

@Duylam: Hah, I can try one article at a time 145.

.

Bron on September 27, 2010 at 3:00 pm said:

Arrg!!! I want to learn calculus, but I‘m still in eighth grade learning algebra!!!!!!! Grrrrr! 146.

Kalid on October 8, 2010 at 10:16 am said:

@Bron: Don‘t worry, once you learn algebra, calculus will be there for you 147.

Taneja on October 26, 2010 at 8:40 pm said:

God bless you! Or Nature bless you! Or fate bless you! Or whatever….. Thanks!!

.

148.

Kalid on October 27, 2010 at 3:23 pm said:

@Taneja: Thanks! 149.

Wolf on November 2, 2010 at 6:10 pm said:

This article is very helpful, but I‘m still having a hard time with deciphering intermediate forms. Could you post an article about L‘Hospitals rule? 150.

Kalid on November 3, 2010 at 6:28 pm said:

@Wolf: Thanks for the comment — that‘d be a great topic for an upcoming article. 151.

Anonymous on November 30, 2010 at 1:16 pm said:

awesome!!!!!!!!!! 152.

JF on December 22, 2010 at 5:20 pm said:

I have a question. In the triangle, I can see why 2*pi*r is the height but can you better explain to me why the base is r? Thanks. 153.

Kalid on December 22, 2010 at 5:45 pm said:

@JF: Great question. There‘s a visual and algebraic way to look at it, let‘s look at the visual way first. If you have the triangle (made of some gummy substance) and want to roll it into a circle, how thick should the bottom be? Well, if the very bottom doesn‘t bend at all, then it should be as thick as the radius. I imagine the straight edge of the triangle, with radius 2*pi*r, being able to bend perfectly into a ring of radius r (after all, a ring of radius r can be unbent into a straight line of height 2*pi*r, right?). So, my visual interpretation is ―the triangle needs to be large enough to account for the very outside of the circle‖. Here‘s a more algebraic way to see it: every point on the circle needs to be ―covered‖ by a matching point on the triangle (they are the same shape after all, just bent, right?).

Well, every point on the circle can be described using two coordinates: 1) how far from the center are you? and 2) what angle are you? For example, the very edge of the circle on the right side is ―distance = r (the full distance), angle = 0″. The very top of the circle is ―distance = r (full distance), angle = 90 degrees‖. The very middle of the circle is ―distance = 0, angle = 0″. If we draw a ring on the circle, we keep a certain distance (like distance = half the radius) and take every angle we can, 0 to 360. This ends up being a straight line on the triangle — go out some distance, and draw a line up as far as we can. The length of the line varies, though — the further away we get, the more we have to travel to get the full ―0 to 360″ coverage. But, the key is that we need to go out the full ―r‖ in order to have lines on the triangle that match up to points on the circle. Phew! Hope this helps. 154.

Kalid on December 22, 2010 at 5:49 pm said:

@JF: I forgot that I had color coded the lines. To make it more clear: In the circle, we have the largest ring being dark blue. On the triangle, where should this shop up? Well, the radius of the largest ring is 2 * pi * r, and it is at distance ―r‖ from the center of the circle. Thus, there needs to be some part of the triangle which as height 2 * pi * r (and there is — the blue strip). There aren‘t any more rings after the blue one, so we can stop building the triangle there. A giant caveat is that we‘re ignoring the thickness of the ring — a bit like how we ignore the thickness of a line when doing geometry. 155.

Tomer on January 13, 2011 at 6:02 am said:

it seems to me that you only got half the area. Shouldn‘t the base of the triangle be 2r? It looks like you only ―unrolled‖ half the sphere from the middle up – what about the other half?

Where have I got it wrong? Thnx 156.

Mike on January 17, 2011 at 9:22 pm said:

Thank you for an excellent article. I just started Calculus I and have been nothing but frustrated by exactly what you describe here. I am very good at problem solving, especially physical, mechanical type problems and for many years made my living as a ship‘s engineer where I did nothing but repair, maintain and implement all sorts of machinery and electro-mechanical systems. I am also good at math… but the approach to calculus (that I have experienced so far) is to basically leave out all of the information necessary to solve a problem and to focus on one small step of the process that has been emptied of all of it‘s meaning. The student has no way of knowing if s/he is proceeding in the right direction or even what it is that s/he is trying to figure out. It is maddening!!! I explained this approach to a friend of mine as trying to learn a foreign language, but not being taught the meaning or translation of any of the new, foreign words. The student is expected to simply just memorize the sounds that have been detached from any kind of meaning and become proficient at making those sounds… and then after a few years of ―learning‖ this way the meaning of the words is slowly revealed (if the student has not already forgotten 95% of the meaningless sounds that they ―learned‖). Your fingerpainting analogy is even more succinct. It is encouraging to see that I was not mistaken when I perceived Calculus as a simple form of problem solving that has been made extremely difficult (almost unapproachable)by the standard, backwards approach that is presented by most textbooks and professors. Are there any other good sources for learning Calculus from the intuitive point of view that I can reference as I try to make sense of the garbage they are force feeding me in this class? Thanks for your time. 157.

Wolfy on January 27, 2011 at 8:14 pm said:

I stumbled upon this in a desperate search to understand what was going on in my Calc I class. It‘s my dream to become a doctor and the only thing seemingly standing in the way is my longstanding difficulty with higher level math. Yet in 20 minutes with just this page, I not only understood something that looked completely foreign to me, but actually enjoyed it. Who‘d have thought it? You, and your explanations both completely rock.

158.

Kalid on January 28, 2011 at 10:33 pm said:

@Mike: Wow, thanks for the note! What you say about the process really rings too — it‘s so frustrating to go through math as a series of mindless steps without knowing *why* we‘re doing what we‘re doing. I fear that many people don‘t realize how much more there is, but get discouraged and just go through the rote memorization to get through the class (and never touch the topic again). I haven‘t looked deep enough at other Calculus resources but I like this book: http://www.math.wisc.edu/~keisler/calc.html It teaches calculus using the techniques that its inventors used, not the mathematicians who made it ―rigorous‖ in the 1800s. 159.

Anonymous on February 8, 2011 at 5:35 pm said:

If my AB calculus class was half as good as this, I swear to god I‘d know calculus in an afternoon. People talk about how oil‘s a limited resource. People talk about how money is a limited resource. Thank you for proving as definitively as ever that the only limited resource is human intelligence, and the creativity to do a little critical thinking and generate something as lucid and sensible as this site. 160.

Kalid on February 8, 2011 at 5:57 pm said:

@Tomer: The triangle is only ―r‖ wide because an individual ring goes all the way around the circle and is counted on both sides. For example, looking at the outermost ring (in blue), you can see that it starts at position ―r‖ away from the middle but can loop all the way around. Its height is 2*pi*r. I should make a quick video with pipe cleaners to show what I mean @Wolfy: Thanks! Good luck with your class

.

. It‘s funny, I think most mathematical subjects can be

learned in hours or days, not weeks/months, if given the right approach. @Anonymous: Thank you! I deeply believe that it‘s not technology or money which is holding back education — it‘s simple, heartfelt explanations and an encouraging attitude which actually help us learn. Appreciate the support! 161.

Anonymous on February 25, 2011 at 7:00 am said:

Thanks I leanrnt the basics of Calculas here now……. 162.

Ken on February 26, 2011 at 4:18 am said:

The only thing that threw me is, how do you know the unrolled rings will create a straight lined hypotenuse rather than some sort of curve? 163.

Joe on March 6, 2011 at 7:40 am said:

Ummm, I‘m in 8th grade (taking 9th grade math) so can someone introduce calculus for my grade level? If not, that‘s fine. 164.

Kalid on March 7, 2011 at 2:22 am said:

@Ken: Great question. Intuitively, I see the rings as being very, very thin lines. Then the question becomes ―Should the lines get larger at the same rate?‖. i.e., should there be a smooth, straight line following the ―tops‖ of the lines making the triangle? Well, the lines come from 2*pi*r — that is, the length of each line is the circumference at that radius (for example, if r = 2, the length of the line would be 2 * pi * 2 = 4 * pi). We can see as the radius increases smoothly (2, 2.1, 2.2, 2.3) then the circumference should increase in that same progression (2*pi*2.1, 2*pi*2.2, 2*pi*2.3, and so on). Basically, because the circumference is directly proportional to the radius, as the radius increases in a straight line (from 0 to the full radius), the circumference should increase also. Hope this helps! @Joe: I have another post in the works which takes another approach to introducing calculus. Should be out this week. 165.

darien on March 9, 2011 at 11:54 pm said:

this made me make it 1 step closer to being as good as my dad he confuses every one with calculus hes a genius and hes mabye the smartest guy at calculus in the world and im only 10!!!!!!!!!!!!!!!!!!!!!!!!!! 166.

James on March 16, 2011 at 5:05 pm said:

Not only does the cutting up the rings stuff work in 2D to find area. But you can extend to 3D to find volume. Only this time it‘s like peeling a layer off an onion and stacking up the peelings. Then to get the volume, find the volume of this stack of peelings using the formula for a pyramid. This means you do 1/3 * r * surfaceArea for the 3d case. Infact as long as you know the radius of the largest sphere you can fit in a shape and the perimiter/surface area of that shape, you can use this method to find the volume or area of pretty much anything. Peace. 167.

Ann on March 23, 2011 at 5:39 pm said:

I totally agree with the way math is taught makes students say that math doesn‘t relate to the real world. Our educational system just focuses on memorization and not real thinking or problem solving, because that is what they test on. As a result we are now proving just how much our students have been taught that they really DON‖T know how to think for themselves or solve problems. We‘ve all learned exactly what the system has taught us – Just to follow directions and do without thinking. It is all very sad. Your visual explanation of the circle to triangle was beautiful. It was a huge moment of clarity for me. I would love more of this for calculus. I took it and passed many, many years ago and do not remember much. Your article just makes me know I could understand the meaning of what I had memorized long ago and since forgot. If I could read more of this, I know it would give meaning and understanding to all that long lost information. I noticed how long ago this was posted. Do you have any other articles on calculus with other visuals to explain concepts? I‘ve noted that it was commented that this was actually a visual for integrals. Some have commented that derivatives were simple to understand. Well, for one who still needs more visuals, can you provide either more comments or point me to another article that will help me to see how simple they are too? I need to understand what they really mean. I know it relates to looking at the slope at a given point on the curve. But I feel like I need more substance. How does that help? Or relate? It makes me want to say, so what? What do I do with that now? Can you help? Thank you. 168.

Humberto on April 30, 2011 at 3:01 am said:

Hi! Your site is great! Congratulations!

Just a small remark: the process of unrolling a ring does not preserve area, that is, the original ring and the final rectangle don‘t have the same area. This put an extra difficulty in your visual proof. An alternative proof (with no area deformation) is given by Kepler: http://www.matematicasvisuales.com/english/html/history/kepler/keplercircle.html See also (in portuguese): http://www.uff.br/cdme/dsp/dsp-html/dsp-ac-br.html All the best! 169.

Anirudh on May 1, 2011 at 10:37 am said:

Absolutely Brilliant…After a long time I reread this article and its just as its whats needed to be done in any teaching.Your way of explaining is like breathing fresh air. Keep up the good work. ―Awesome‖ 170.

Kalid on May 1, 2011 at 4:51 pm said:

@Humberto: Thanks for the note! I‘m not clear that the transformation deforms area, but I suppose that makes sense. I do like the alternative proof though! 171.

Kalid on May 1, 2011 at 5:03 pm said:

@Anirudh: Thank you! 172.

Rupe on May 17, 2011 at 9:08 am said:

Cool article I love the way you explain the relationship between the circle and triangle area ,wish more books would start off like that . After twenty years i still groan at doing calculus for a new course and I am doing one now with a lot of vector based calculus for Electromagnetism ,surfaces Gausses Theorem, Gaussian surfaces ,your explanation of divergence clicked right away. may I recommend a book to your readers which although old is freely available and makes calculus

simple for simple minded folk like me . The title is ―Project Gutenberg‘s Calculus Made Easy, by Silvanus Phillips Thompson ― Details: Title: Calculus Made Easy Being a very-simplest introduction to those beautiful methods which are generally called by the terrifying names of the Differential Calculus and the Integral Calculus Author: Silvanus Phillips Thompson Release Date: July 28, 2010 [EBook #33283] Language: English Here is a quote from the prologue: ―Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks. Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics—and they are mostly clever fools—seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way. Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.‖ Powerful stuff . I am reading your articles with great interest , thanks for sharing. 173.

Kalid on May 19, 2011 at 11:22 pm said:

@Rupe: Thanks for the wonderful comment! That‘s a great quote, sometimes math is made more complex than it needs to be. That book is been on my list

.

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Chase on June 3, 2011 at 12:52 pm said:

Nice Bruce Lea quotation!! 175.

Tom on June 22, 2011 at 4:33 pm said:

Re: your nested rings exercise proving area of circle formula… either I don‘t get it or it is truly illogical. The concentric rings in your example have a conveniet thickness. Instead, let those thickness be close to zero. Or a mile thick and in both cases the answers pertaIn not at all. What am i missing here please? 176.

Kalid on June 22, 2011 at 4:55 pm said:

@Tom: Great question! The key idea is that instead of measuring a wiggling shape directly, we break it into easy-to-measure pieces and measure those. Check out the diagram on this page for an example: http://betterexplained.com/articles/why-do-we-need-limits-and-infinitesimals/ We can model a shifting wave with a bunch of smaller rectangles. The key is the finer-grained our measurements (mini-shapes), the closer we match the real shape. So, taking measurements a mile-wide would give a pretty poor approximation

.

You‘ve hit the heart of calculus with ―close to zero‖ though. The idea is to make measurements so fine, so close to zero, that we can‘t tell it apart from the original. (This happens all the time, by the way… we watch movies and think we‘re seeing fluid motion, but it‘s 24 frames per second. We don‘t need perfectly fluid motion, we just need something ―good enough‖ that we can‘t tell the difference). Calculus is about finding that threshold for ―good enough‖ where there‘s no detectable mathematical difference from the real thing. In this case, Calculus tells us there‘s no detectable difference between the unrolled circle and the triangle. (There are much more formal ways to state this, but it‘s how I think about it). 177.

mitrajyoti on July 17, 2011 at 9:09 pm said:

Thanks Kalid, reading your blog is much like a dream come true for me like most of the others. Today I am going to Start teaching Calculas to my my first student. He is just 2 yeas younger than me and I am thinking how to show him that differentiation is just the opposite of integration. But how to show that slope calculation is the opposite of calculating the area under the same curve? I will post as soon as I find some analogy. By then,Everyone‘s view is welcome for me. 178.

Kalid on July 18, 2011 at 9:02 am said:

@mitrajyoti: Thanks for the kind words! I‘m working on an analogy for differentiation too — I think I‘d like to avoid explicit mention of slope (at first) just because it‘s another concept to learn. I don‘t think people are super-comfortable (intuitively) with graphs, so using this as a building block might be tricky. But you never know, if you find an analogy that works, use it! (and share!) 179.

Peter Farrell on August 18, 2011 at 9:11 am said:

Kalid, your approach is refreshing and enlightening! The way Calculus is taught is wrong, wrong, wrong. My lawyer says I need your permission before tattooing the entire ―A Note on Rigor‖ paragraph on my back. 180.

Kalid on August 18, 2011 at 11:19 am said:

@Peter: Thanks for the note! Hah, I don‘t have any tattoos but a tirade in favor of intuitive math education is definitely a contender 181.

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CD on August 19, 2011 at 9:36 pm said:

This all seems so easy, over night i decided to brush up on algebra and it took less then an hour to remember everything, afterwards i decided hell i will take a crack at calculus, after reviewing only 3 pages and about 2 hours of writing things down i find myself stuck… what else is there, i get the concept of familiarity to algebra and using the concept to solve equations from calculus, but what i don‘t get is what now… i feel i must be missing something so i am now looking at physics which seems to be using calculus to make up things about physical aspects. if any ones has any advise where i can look further about calculus and math in general to further expand my curiosity please let me know. one thought i wanted to work on is my idea for the ever expanding universe, but not in an explosion but in multiple explosions that further expand the universe more and more, and pushing the universe

further and further and all of its matter threw a very delicate process of gravity+force… best way to describe would be taking a pebble and dropping it in a pond and watching it ripple…. again just doing this all for fun but would love to try and test this theory with math 182.

CD on August 19, 2011 at 9:43 pm said:

i guess what i am trying to say with this is that force; eg. explosions have gravity and explosions would also die out over time, so depending on how much force is in the explosion there could very well be gravity or some other force constructed by this explosion, thus would be causing the universe to act like a ripple and would send galaxy‘s and other things in space in an up and down motion while expanding… again this is just a thought i had and would like to work on it 183.

CD on August 23, 2011 at 6:10 am said:

so no advise on any other learning points? 184.

CD on August 23, 2011 at 6:12 am said:

well if anyone can give me any tips on how to move forward my email is ranmalrac@yahoo 185.

Michelle on August 30, 2011 at 12:00 pm said:

Thanks for a wonderful and well-written article! As both a college student and a math tutor, I have found many of your posts helpful. I don‘t normally read the comments, but this time I was curious to see what others thought of the evolution-calculus parallel. And after reading the comments, I could not resist putting in my own two cents. 1) The end result of evolution is to render God superfluous. Religion is defined (according to the Oxford dictionary) as ―the belief in and worship of a superhuman controlling power, especially a personal God or gods.‖ Evolutionists maintain that their conclusions are purely scientific, but their conclusions are premised on the belief that God does not exist, or at the very least, was not vital to the creation of the universe. However, I do not have a problem with mixing science and religion. To my mind they are inseparable, for religion is the lens through which we interpret the world.

2) I find it very interesting that the ―fathers of calculus‖ (Newton and Leibniz) both believed God had major role in the creation of the universe. ―In whatever manner God created the world, it would always have been regular and in a certain general order. …‖ – Leibniz ―Gravity explains the motions of the planets, but it cannot explain who set the planets in motion. God governs all things and knows all that is or can be done.‖ – Newton How peculiar that two men who were so ―wrong‖ about a subject as crucial as the origin of the universe would be able to formulate a subject as intricate as calculus. And, just for the record, here is a quote from Einstein: ―In view of such harmony in the cosmos which I, with my limited human mind, am able to recognize, there are yet people who say there is no God. But what really makes me angry is that they quote me for the support of such views.‖ 186.

Erica on September 10, 2011 at 5:44 am said:

you have no idea the ―aha‖ moment I had just reading this 1 article. THANK YOU THANK YOU THANK YOU. T 187.

Kalid on September 11, 2011 at 11:59 pm said:

@Erica: Yay! Happy it worked 188.

Scott on September 20, 2011 at 3:35 pm said:

Great article! I struggled with calculus until a class in Statistics, all of a sudden the area under the curve made sense. Learning computer programming simplifies this entirely. Take a for-loop, all this is is integration from one value to another. Calculating the area under the curve is exceedingly easy when plugging the iterator value into the function.

189.

Kalid on September 20, 2011 at 3:39 pm said:

@Scott: Thanks! I hadn‘t thought of the programming for loop, but that‘s a great analogy! 190.

Don Nyberg on October 17, 2011 at 2:07 pm said:

WOW! er a-ha never took trig or calculus in high school because the theoms and postulates (spelling?) really turned me off. I loved and still do love physics, but was so discouraged by memorization of formulas without any practical examples, that i could not continue. that was 30 years ago, and i can honestly say that if I had had an instructor/professor that explained it like you did my life would be completely different. the first time i was ever told what pi is, and how it was/is derived makes it so easy. Thank you 191.

Kalid on October 17, 2011 at 2:47 pm said:

@Don: Thanks for the note, I‘m really happy it helped! One of my biggest insights was that once you get over the frustration, learning can become truly, genuinely enjoyable. So much education is focused on memorization because that‘s the easiest thing to measure (and usually the first thing we forget!). Thanks again for the comment. 192.

Aj meunier on October 29, 2011 at 9:20 am said:

So im only in grade 8 but i really wanna know calculus because i hate when people know it and it makes me feel dumb. This page has sorta helped me understand it but i still want to know how to do an equation that someone gives me. My math teacher teaches all levels of math from 10-2 to calculus and he teaches me 10-2 now so im ready because i asked him for help but its still hard to understand calculus and what the equation equals to. 193.

Gordon Knight on November 14, 2011 at 5:55 pm said:

By Page 2 I was laughing my head off, then crying with joy. I have begun the understanding of calculus. God knows how long it will take me to get through the rest of the pages but I am expecting the finest of steak dinners. Thank you for being. 194.

kalid on November 14, 2011 at 7:45 pm said:

@Gordon: Wow, thank you for the heartfelt comment, it made my day

. I‘ve realized there‘s no

race in math, we learn what we can as we can (just like there‘s no race to read every book). Really glad it clicked for you. 195.

rash on November 30, 2011 at 3:36 pm said:

THANK YOU! i‘ve always been a bit disappointed that maths (at pre-uni and uni level) never seems to be intuitive to me the way it was when i was a kid. i still occasionally get an ‗aha!‘ moment, and it‘s the greatest feeling when i do, but it‘s often after months (or years) of using them to calculate things without really understanding. (the point in implicit differentiation when you could separate dx and dy was a complete shock to me, i thought it was like splitting sin into si * n or something). I had my aha! moment for what dy/dx really means probably over a year after i started using it, when i noticed the slope of a straight line was just the differential which was just difference in y/ difference in x and the one for integration was only a couple of months ago (over two years since i learnt it). BOOKMARKED, looking forward to more aha! moments from this site 196.

kalid on November 30, 2011 at 6:51 pm said:

@rash: Awesome! Glad it was helpful

. I totally know what you mean, I love the excitement of

having a tough idea finally click. That‘s a good point about separating dy/dx into ―dy‖ and ―dx‖ — in physics you are allowed to, but in ―rigorous‖ math you aren‘t (as you say, it‘d be like separating sin into s*in)! But the intuition comes when you can separate the ideas and play with them a bit. I‘d like to write more about this topic. Thanks for the comment. 197.

Marvin K. Mooney on December 4, 2011 at 6:24 pm said:

WoW! This is an exceptionally ―cool‖ way of looking at this subject that is typically considered lackluster and dry. If more kids could be exposed to this article alone I‘m sure they would be given hope in terms of their interest in Math. I know I was!! Thanks dude. 198.

kalid on December 4, 2011 at 6:30 pm said:

@Marvin: Thanks (and please don‘t go home). I find anything can be fascinating if presented properly!

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Bill on December 21, 2011 at 9:10 am said:

Hi, I like the idea. I don‘t know if you are familiar with tau (tauday.com) but it is a good way to introduce new people to the concepts being discussed without an errant factor of 2 in there (or a 1/2 missing). Let tau=2pi. 200.

kalid on December 22, 2011 at 10:45 am said:

@Bill: Thanks, I like tau as well. It helps people break away from a memorized formula and think about what the concept of pi really is. 201.

Husam on January 10, 2012 at 12:46 am said:

good work, i was thinking about calculus, it is very interesting subject to learn, but my book defines calculus concepts so confusingly e.g. here is definition of continuity as per my book, ―A f(x) is said to be continuous at a point x=c in its domain if for a given Є > 0, there exists δ > 0, such that |f(x)-f(c)|