Learning Calculus: Overcoming Our Artificial Need for Precision

Accepting that numbers can do strange, new things is one of the toughest parts of math:

Calculus is a beautiful subject, but challenges some long-held assumptions:

This is a companion discussion topic for the original entry at http://betterexplained.com/articles/learning-calculus-overcoming-our-artifical-need-for-precision/

Hi! Kalid

I have been following this Blog for some time. Really appreciate the time and effort you are putting into this to help us better understand the concepts and create “Aha moments”

I’m learning and getting a lot of interesting insights from this blog.

Celebrating Life…


Nice article, clears a couple of questions of my own.

@Mahesh: Thanks for the note! Glad you’re enjoying the site, it’s been a lot of fun to make so far. Always happy to have you drop in.

@Calin: Glad you found it useful! I’ve always been befuddled by the inner workings of Calculus, but I think infinitesimals can make it a lot more clear.

[…] Learning Calculus I recently stumbled over an article in the BetterExplained blog (Learning Calculus: Overcoming our artificial need for precision). The blog itself contains a lot of interesting articles. However, the blog has a tendency to bash “school mathematics” in favor of “insight”. The subtitle of the blog “learn right, not rote” says it all. […]

Nice article with lots of things to discuss and think about! I left an answer to this on my math blog. See the URL above.

Sorry, I noted that the URL is below, not above. Click on my name!

Hi Rene, thanks for the detailed response! I’ve left a reply on your site :).

> my goal is to understand the ideas behind Calculus

Get the book “Calculus: The Elements”. See this review: http://www.worldscibooks.com/mathematics/4920_rev01.html if you need convincing.

@asdf: Cool, thanks for the pointer.

I used to suffer from the need for precision… Thanks for yet another amazing article!

Hi Kalid, nice post, as always. I like the title “Overcoming the Artificial Need for Precision.” I’d like to add one more point to that, not about Math really.

I believe that I should try for precision in order to get to the closest possible approximation. It’s like striving for the perfect software with no bugs knowing that there is no software that has “zer0” bugs. Deadlines and other external forces can resist (or end) our quest for perfection. OK, What am I trying to say here? Oh yes, this is what. Even if we strive for perfection, time will stop us and it’s not that bad to strive for perfection in that time frame.

And BTW, I liked that pixelation example very much. Thanks for post.

Hi Srikanth, thanks for the comment. I agree – we should always strive for the maximum precision possible, while realizing that our precision is limited (not infinite).


I welcome your slow cooking approach to the introduction of calculus which is ultimately going to be delicious.

Occasionally, I suggest, you throw a A4 piece of paper to non-calculus mortals and ask them to come up with largest volume of rectangular box in conventional way and show us , it can be done with one stroke through calculus in your unique way.

Is it a long way?



Hi T.Gopalan, thanks for the comment & kind words. I’m not sure I understand the question – were you thinking of ways to make the biggest box possible given some constraints? Or a box with infinity on each side? :).

Dude! As usual, you ROCK!

I really like the approach of your article - thumbs up + cheers.

Hi! Kalid,
What I mean is the A4 size paper has constraint
dimension of 210 x 297 mm. I hope I am making myself clear.
Thanks for your response,


Cool post!! I personally like to use 60 digits of pi in everyday applications.


My son was given the problem that T.Gopalan mentions - take a sheet of paper, cut equal-sized squares out of each corner, and fold the sides up to create a box (with no lid). His assignment was to find the size of the squares that would make a box with the largest volume, through trial and error, to the nearest tenth of a cm. I told him that his teacher would probably relate this to calculus and show how to find the exact answer., but his teacher never did - an opportunity missed!