Awesome, glad you’re finding it useful :).
[…] Pi is mysterious. Sure, you “know” it’s 3.14159 because you read it in some book. But what if you had no textbooks, no computers, and no calculus (egads!) — just your brain and a piece of paper. Could you find pi? […]
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
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!
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.
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.
Great thx for pictures
lol i already love math, and this article just made me love it even more :P…
@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 :).
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?
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 :).
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.
in all mi internet travails i hav never found such a clean and easy to understand explanation of calculus!
@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!
It seems that maths and science are taught in completely contrasting styles, as the article (which I’d like to add was fantastic, I wish I was taught like this) and several posts point out. Science does teach complicated things in a historical fashion: In England at least, Bohr’s outdated, yet simple, model of the atom is taught up to age 16, then 17-18 a summary version of the quantum atom is taught. I wish this was true for maths.
@MRW: 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.
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.
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 :).
[…] Time for a beefier example. Calculus is about many things, and one is what happens when numbers get really big or really small. […]
That was beautiful. You have no idea how much this has helped me.
Thank you.