Learning Calculus: Overcoming Our Artificial Need for Precision

Blog Articles
41

I suppose the epsilon-delta example of limit is perhaps another good example besides the pixel example.

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[…] Quick answer: Small enough where the value looks the same for the entire duration. We don’t need perfect precision. […]

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Quick Question:

Is this article saying that being precise is good but unneccesary?And isn’t pi 3.14596 ect?

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@Seamus: Great question. Precision is great, but infinite (perfect) precision is not necessary. Basically, the idea is to find what level of precision is “good enough” for your needs (and you can’t say you need perfect precision! :)). For pi, 30 digits is far more than anyone needs for engineering purposes. Even 10 digits is extremely accurate – it’s far more likely that errors are being introduced by other measurements/tolerances, and not the imprecision in your expansion of pi.

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Brilliant. This site is a work of genius, it really is. Not just because you describe things well. Especially because you had the balls to put your intuitions about math online in a clear way that challenges most of our dearly held conventions. This article is not simply mathematics, it is philosophy. You correct many misconceptions students have about numbers and how they relate to life. Awesome work.

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i understand what you’re going for here, but don’t agree that proofs and concepts are separate

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[…] “Learning Calculus: Overcoming Our Artificial Need for Precision” by Kalid Azad, Better […]

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[…] "Learning Calculus: Overcoming Our Artificial Need for Precision" by Kalid Azad, Better […]

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i love this site!!! i can’t sleep right now cause i can’t stop reading your articles! i’m currently struggling with calculus. but i used to love math… before there was calculus. i wasn’t able to fully understand it before, since my professor did nothing to help me appreciate the subject. oh, and she failed more than half the class, including me. i freaked out, and i hated school. so, uhmm, thanks a lot!! now i’m beginning to appreciate the beauty of calculus. and once again, i love math! :))

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@shelly: Awesome!!! I’m really glad it helped (and that you got addicted to the articles, heh), one of the best feelings is starting to overcome our inhibitions about a subject. Not everyone needs to be in love with math, but we can at least appreciate it as much as we do a good song or nice poem :).

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[…] http://betterexplained.com/articles/learning-calculus-overcoming-our-artifical-need-for-precision/ […]

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Thanks!

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the whole idea of “an infinitesimal” is so badly conceived as to be impossible to rehabilitate. so you have a positive number that, when added to some other number, leaves that number alone? OK, so what happens when we add “N” of them together, where “N” is an integer > 1/x (your “infinitesimal”)? yeah, suddenly the real numbers aren’t a field anymore, hey what’s the worst that can happen?

there’s a reason the formal definitions of various limit statements were developed, and it has nothing to do with any kind of neurotic need for complication. moreover, my experience is that a ninth-grader in algebra two can master the implementation, despite the intimidating burden of notation.

and the “infinity exists” is even worse- much worse. if you want to treat your own headaches by decapitation, that’s one thing, but let’s not advocate the procedure to impressionable young people, ok?

54

Bump, just stumbled across this site and it looks great. Much respect for people who share their insights (whilst yearning for more) and help to educate others. Great work!

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@TJ: Thank you!

56

This is a very precise article.

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Hey Kalid,

Great blog. I’ve been reading it for a few days now. One question though.
I read your blog post about “e” and how it’s connected to growth. It made perfect sense. But now you say that the size of the atom is 1e-11, and the universe 1e27. The size of the atom is not changing over time at some rate, so I can’t visualize this as growth. How should I exlplain to myself that “e” fits so perfectly for descriping the size of the atom and universe?

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Math is a subject that never seizes to captivate if done the right way.

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@Ankur: Totally agree – anything can be enjoyable if explained intuitively.

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Hi Jules, great example. Every decimal point is 10x the precision, so might not be justified for all scenarios.