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

- There are numbers
*between*the numbers we count with? (*Yes — decimals*) - There’s a number for nothing at all? (
*Sure — zero*) - The number line is two dimensional? (
*You bet — imaginary numbers*)

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

- Numbers don’t have to be perfectly accurate?
- Numbers aren’t just scaled-up versions of each other (i.e. 1 times some number)?

Today’s post introduces a new way to think about accuracy and infinitely small numbers. This is not a rigorous course on analysis — it’s my way of grappling with the ideas behind Calculus.

Contents

## Counting Numbers vs. Measurement Numbers

Not every number is the same. We don’t often consider the difference between the “counting numbers” (1, 2, 3…) and the “measuring numbers” like 2.58, pi, sqrt(2).

Our first math problems involve counting: we have 5 apples and remove 3, or buy 3 books at $10 each. These numbers change in increments of 1, and everything is nice and simple.

We later learn about fractions and decimals, and things get weird:

- What’s the smallest fraction? (1/10? 1/100? 1/1000?)
- What’s the
*next*decimal after 1.0? 1.1? 1.001?

It gets worse. Numbers like sqrt(2) and π go on forever, without a pattern. Numbers “in the real world” have all sorts of complexity not found in our nice, chunky counting numbers.

We’re hit with a realization: **we have limited accuracy for quantities that are measured, not counted**.

What do I mean? Find the circumference of a circle of radius 3. Oh, that’s easy; plug r=3 into circumference = 2 * pi * r and get 6*pi. Tada!

That’s cute, but you didn’t answer my question — what *number* is it?

You may pout, open your calculator and say it’s “18.8495…”. But that doesn’t answer my question either: What, exactly, is the circumference?

We don’t know! Pi continues forever and though we know a trillion digits, there’s infinitely more. Even if we knew what pi was, where would we write it down? We really don’t know the *exact* circumference of anything!

But hush hush — we’ve hidden this uncertainty behind a symbol, π. When you see π in an equation it means “Hey buddy, you know that number, the one related to circles? When it’s time to make a calculation, just use the closest approximation that works for you.”

Again, that’s what the symbol means — we don’t know the real number, so use your best guess. By the way, e and √(2) have the same caveat.

## 40 digits of pi should be enough for anyone

We think uncertainty is chaos: how can you build a machine unless you know the exact sizes of its parts?

But as it turns out, the “closest approximation of pi that works for us” tends to be surprisingly small. Yes, we’ve computed pi to billions of digits but we only need about 40 for any practical application.

Why? Consider this:

- Atoms are about 1e-11 meters across
- The universe is about 90 billion light years (1e27 meters) wide

Dividing it out, it takes about 1e38 (1e27 / 1e-11) atoms to span the universe. So, around 40 digits of pi would be enough for an exact count of atoms needed to surround the universe. Were you planning on building something larger than the universe and precise to an atomic level? (If so, where would you put it?)

And that’s just 40 digits of precision; 80 digits covers us in case there’s a mini-universe inside each of our atoms, and 120 digits in case there’s *another* mini-universe inside of that one.

The point is our instruments have limited precision, and there’s a point where extra detail just doesn’t matter. Pi could become a sudoku puzzle after the 1000th digit and our machines would work just fine.

## But I need exact numbers!

Accepting uncertainty is hard: what is math if not *accurate* and *precise*? I thought the same, but started noticing how often we’re tricked in the real world:

Our brains are fooled into thinking 24 images per second is the same as fluid motion.

Every digital photo (and printed ones, too!) are made from tiny pixels. Pictures seem smooth image until you zoom in:

The big secret is that **every digital photo is pixelated**: we only *call* it pixelated when we happen to notice the pixels. Otherwise, when the squares are tiny enough we’re fooled into thinking we have a smooth picture. But it’s just smooth for human eyes.

This happens to mechanical devices also. At the atomic level, there limits on measurement certainty that restrict how well we can know a particle’s speed and location. Some modern theories suggest a *quantized universe* — we might be living on a grid!

Here’s the point: approximations are a part of Nature, yet everything works out. Why? **We only need to be accurate within our scale**. Uncertainty at the atomic level doesn’t matter when you’re dealing with human-sized objects.

## Every number has a scale

The twist is realizing that even *numbers* have a scale. Just like humans can’t directly observe atoms, some numbers can’t directly interact with “infinitesimals” or infinitely small numbers (in the line of 1/2, 1/3… 1/infinity).

But infinitesimals and atoms aren’t zero. Put a single atom and on your bathroom scale, and the scale still reads nothing. Infinitesimals behave the same way: in our world of large numbers, 1 + infinitesimal looks just like 1 to us.

Now here’s the tricky part: **A billion, trillion, quadrillion, kajillion infinitesimals is still undetectable!** Yes, I know, in the real world if we keep piling atoms onto our scale, eventually it will register as some weight. But not so with infinitesimals. They’re on a different plane entirely — any *finite* amount of them will simply not be detectable. And last time I checked, we humans can only do things in finite amounts.

Let’s think about infinity for a minute, intuitively:

- Infinity “exists” but is not reachable by our standard math. No amount of addition or multiplication will take you there — we need an infinite amount of addition to make infinity (circular, right?). Similarly, no finite amount of division will create an infinitesimal.
- Infinity and infinitesimals require new rules of arithmetic, just like fractions and complex numbers changed the way we do math. We’ll get into this more later.

It’s strange to think about numbers that appear to be zero at our scale, but aren’t. There’s a difference between “true” zero and a measured zero. I don’t fully grasp infinitesimals, but I’m willing to explore them since they make Calculus easier to understand.

Just remember that negative numbers were considered “absurd” even in the 1700s, but imagine doing algebra without them.

## Life Lessons

Math can often apply to the real world. In this case, it’s the realization that accuracy exists on different levels, and perfect accuracy isn’t needed. We only need 40 digits of pi for our engineering calculations!

When doing market research, would knowing 80% vs 83.45% really change your business decision? The former is 100x less precise and probably 10x easier to get, yet contains almost the same decision-making information.

In science, there’s an idea of significant figures, which help portray uncertainty in our measurements. We’re so used to contrived math problems (“Suzy is driving at 50mph for 3 hours”) that we forget the real world isn’t that clean. Information can be useful even if it’s not perfectly precise.

## Math Lessons

Calculus was first developed using infinitesimals, which were abandoned for techniques with more “rigor”. Only in the 1960′s (not that long ago!) were the original methods shown to be justifiable, but it was too late — many calculus explanations are separate from the original insights.

Again, my goal is to understand the ideas behind Calculus, not simply rework the mechanics of its proofs. The first brain-bending ideas are that **perfect accuracy isn’t necessary** and that **numbers can exist on different scales**.

There’s a new type of number out there: the infinitesimal. In future posts we’ll see how to use them. Happy math.

## Other Posts In This Series

- A Gentle Introduction To Learning Calculus
- Understanding Calculus With A Bank Account Metaphor
- Prehistoric Calculus: Discovering Pi
- A Calculus Analogy: Integrals as Multiplication
- Calculus: Building Intuition for the Derivative
- How To Understand Derivatives: The Product, Power & Chain Rules
- How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms
- An Intuitive Introduction To Limits
- Why Do We Need Limits and Infinitesimals?
- Learning Calculus: Overcoming Our Artificial Need for Precision
- A Friendly Chat About Whether 0.999... = 1
- Analogy: The Calculus Camera
- Abstraction Practice: Calculus Graphs

## Leave a Reply

68 Comments on "Learning Calculus: Overcoming Our Artificial 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.

Thanks!

-Mahesh

Celebrating Life…

Hi,

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).

Hi!

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?

Regards,

T.Gopalan

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,

T.Gopalan

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

Kalid,

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!

[…] 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? :)

@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.