Learning Calculus: Overcoming Our Artificial Need for Precision

[…] I think the reason for this is that those of us in the digital generation (especially those of us who took too many discrete math classes in college, i.e. computer science majors), have a lot of philosophical trouble with inexact results. We studied the natural numbers and the integers, number theory as it relates to Gödel’s Incompleteness Theorem and Alan Turing and the Entscheidungsproblem (I love saying “Entscheidungsproblem”).  All of these theorems and results are about discrete, countable sets (the natural numbers, mostly).  The reals are left as kind of a messy exercise for the reader or the IEEE. […]

[…] Link RSS Filed under: math, software Democratic party » […]

Interesting perspective anh. I do like you approach to explaining precision. But one thing baffles me: how precise in approximations should we get when we whip out that calculator to crunch numbers when dealing with… say chemistry math? I remember struggling to decide to what decimal point we should use for calculations, how “precise” we should be, back in college. Any suggestions?

@Bruce, T.Gopalan: Ah, thanks for the clarification! Yes, that’s an interesting problem that calculus can help solve – I’ll keep it in mind as an example for upcoming articles!

@VT: Thanks Em, great question – it’s been a while since I’ve done any chemistry, let’s see if I can remember :).

In my head, I think you can only be as accurate as the “weakest” link in a chain. For example, someone said “Dinosaurs died 65 million years ago” in 1950, it doesn’t mean they died 65 million + 58 years ago :). The 58 years that have passed don’t even register, since 65 million is a very rough number (only 2 digits of accuracy) but 65,000,058 is a pretty accurate number (8 digits).

Intuitively, I look at how many digits “haven’t been rounded” so if you see 3.1 kg, it has 2 digits of accuracy (and 3kg has 1 digit of accuracy). When doing calculations, you need to keep that same number of digits that “haven’t been rounded” so you don’t give the impression of super-precise measurements. So after you do all the intermediate math, you round it back to the level of accuracy you started with. Hopefully this helps!

@Prateek: Thanks!

@Quan: Appreciate the comment. Wow, 60 digits of pi… building a ring around the universe? :slight_smile:

@biophonc: Glad you enjoyed it.

Nicely written. Concise, correct and clear. Always good to see old concepts reiterated in a helpful way.

Good job, I enjoyed reading it.

@Ethereal: Thanks for the comment!

I can’t really say much, except that I really enjoyed reading that :slight_smile: Nice work!

Only nitpick is that 1e27 / 1e-11 = 1e38 not 1e37 :wink:

You say that “It doesn’t matter how many infinitesimals we add — we’ll never detect them!” Calculus is in business of dealing with things to small to detect. But the most powerful part of calculus is the fact that you CAN add up enough “infinitesimals” to find the area of an irregular shape, or to find the rate of change of a quantity in a single infinitely small instant. If this is meant to be an introductory post to teach calculus, it seems to be off on the wrong foot

@CN: Whoops, thanks for the catch! Glad you liked it.

@Integrating in Illinois (love the name): Good point – I clarified the statement above.

Intuitively, I see infinitesimals separate from the “regular” small numbers we deal with (like 1E-30). That is, we can’t just multiply an infinitesimal by a large number (like 1E30) to bring it back to our scale.

To make infinitesimals useful, we have to add an infinite amount of them (integrate), which is a new operation in our math vocabulary.

So yes, infinitesimals can certainly be used to get real results (like areas of shapes, as you mention) but we need a new operation (integrals) to do it. I hope to make this more clear in future posts, thanks for the comment!

you should start talking about infinite series next. like, how e = x^n/ n! or something. nice post.

@SUM guys: Thanks for the suggestion, more articles on e are on the list :).

First of all, I’m sorry for my (I suppose) bad english writing. I do better with spanish :wink:

It’s amazing how your explanations make me understand so well the math’s background. I believe that your method could be the best alternative to teach the math in every school grade, at least here, in México. It looks so interesting for almost anyone, 'cause it’s not just “telling how to solve hypothetical problems” thing… thats boring for almost every student. There is philosophy behind any theorem, theory or whatever requires a mathematical analysis.

Thanks for all your posts.

@Felix: Thanks for the wonderful note! Your English is still better than my Spanish, and I studied it for a while :).

I’m glad you’re able to enjoy the style of the site. Yes, I feel education has veered towards things that are easily testable/measurable (memorizing facts & plugging items into equations) instead of focusing on the deep insight which is more important, but can’t be measured. Thanks again for the comment!

Brilliant. One of the best blogs I came across explaining fundamentals of math. Your thinking seems related to/influenced by (?) the Pythagoras “Science of numbers” and using numbers to understand the universe around us.

@Mitra: Thanks for the kind words! I hadn’t heard of the “Science of numbers” but it sounds interesting. I have a curiosity about how things “really” work at an intuitive, not formulaic level. Thanks for dropping by.

Great article with a lot of valid statements.

However, the way you state your main thesis ultimately boils down to a confusion. You conflate numbers and numerals, ie. the symbols we use to represent numbers. This is an extremely fine distinction to make, and often a very difficult one, because often there is no consequence to confusing them, and humans are not good at abstraction.

By way of example, consider “1”, which is the symbol we use to represent the unit number. This number is absolutely exact. It is exactly one number, and it is not identical with any other number. However, there is a second numeral for it! You can write the very same number as “0.999999…” with an infinitely continuing series of 9s. This is easy to show: both 0.999999…/3 and 1/3 are equal to 0.3333333…. So both “0.999999…” and “1” are the same number, even though the two representations look drastically different, and one of them goes on forever, so that it cannot be written down exactly.

Likewise, “π” is the symbol we use to represent the ratio of the circumference of the unit square and the unit circle. This number is absolutely exact. It is exactly one number, and there is no other number that is identical with it. F.ex., you can say that a circle with radius 3 has exactly 3 times the circumference of a circle with radius 1. Both “3.141592…” and “π” represent the same, exact number… even though the two representations look drastically different and one of them goes on forever… so that it cannot be written down exactly. :)

There is merely no way to represent π with a finite decimal numeral, but this does not mean that the number π is somehow any less exact than 1.

Numbers are mathematical entities, so are absolutely exact. Numerals, in contrast, are human conventions invented in trying to use the finite, discrete materials of this universe (ink, luminescent dots on a screen, whatever) to represent numbers. (Calculus adds another distinction on top of that, but that is yet another issue.)

@Aristotle: Thanks for the detailed comment! Yes, I agree with your clarification – there is a difference between the concept (the number or idea of 1) and our representation of it (the numerals). Speaking of which, I was planning on using the .9999… = 1 as an example of limits.

The higher-level thought, that I may need to back and revise to make clear, is this: our numerals are finite, and thus have limited ability in describing certain numbers. That is, we cannot finitely describe pi using our numerals except to abstract it into a symbol, or give it as the result of a limiting process.

Calculus helps us recognize that that our numerals are “limited”. Thanks again for the comment!

[…] popular public links >> decimals Basic Monte Carlo First saved by tigerlily963 | 7 days ago Lack of privacy, Breach Candy etc. First saved by 1nkubo | 8 days ago Fractions, Decimals and Percentages Are Just Coded Maths Games First saved by pitbree | 12 days ago Learning Calculus: Overcoming Our Artifical Need for Precision First saved by jmiguelrodriguez | 14 days ago Decimals & Money Gr.3 4 Book by Teacher Created Resources - Low Price! First saved by garyduke | 18 days ago Fractions, Decimals and Percentages Are Just Coded Maths Games First saved by cdwarren | 27 days ago Birthdays and the Days in Between First saved by xXD4RwDXx | 51 days ago How do you format a number to 2 decimals in java? First saved by Muil | 56 days ago Rational and irrational numbers First saved by lonelygirl15 | 66 days ago Decimals First saved by probo | 69 days ago FRACTIONS TO DECIMALS TO MILLIMETERS CONVERSION CHART First saved by WindDance | 85 days ago Rock ‘N LearnFractions & Decimals Auctions On Ebay First saved by johncollins | 92 days ago Motorola clings to number one spot in US sales, RIM still rocking First saved by olinerd | 102 days ago binge thinker First saved by kmitaksov | 103 days ago 20080815 all rationals are periodic decimals First saved by sgirlcharm | 104 days ago Bargain Deal on Scholastic Success With Fractions & Decimals Workbook First saved by greendaydude34 | 114 days ago […]

Wow!! Thanx a Lot Sir!!!