# Prehistoric Calculus: Discovering Pi

Pi is mysterious. Sure, you “know” it’s about 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?

This is a companion discussion topic for the original entry at http://betterexplained.com/articles/prehistoric-calculus-discovering-pi/

Great post. I really liked the formatting too, and the calculator at the end. Worthwhile talk about how you might estimate pi in other ways, such as estimating the number co-prime numbers, or the Buffon Needle problem.

Just some ideas.

Again, great post.

Matt

I like the article, and will definitely use the ideas in my classes, but first there is a minor problem to solve. You have the inside perimeter as the geometric mean of the previous estimates, and the outside perimeter as the harmonic mean. The problem is, the harmonic mean is always /smaller/ than the geometric mean. Maybe they just switch, but I don’t see how yet.

The page to which you link contains the same mistake. I have no idea where that guy got the trig identities he cites, but I’ll keep working on it.

Update: Okay, I now see I was reading the formulae incorrectly (one refers to newIn rather than Inside), but the formulae themselves are wrong as well (on your page; the page you link to gets them right). They should be

newOut = harmonicMean(Inside, Outside)
and
newIn = geometric(Inside, newOut)

which is what you use in the spreadsheet.

The comment form just ate my last comment; I have no idea why that is.

I continued on from there:

223/71

Okay, I figured it out. It misinterpreted what I typed as html.

In my email to you about writing a guest article, I had one that this article just destroys. I talked (a lot) about Archimedes’ discovery that 223/71 is less than pi which is less than 22/7, though I focused more on the concept than the mathematics behind it.

Although, your square root comment made me think: I have a better explanation for Newton’s method than you had in the Quake Square Root article, so maybe I should write about that…

@Matt: Thanks, glad you enjoyed it! Those are great suggestions, I think it’d be great for a follow-up. I didn’t want to distract from the calculus roots too much in this post, but the needle approach is a fun way to look at probability.

@Chad: Sorry about the confusion there! Yes, I made a major flub and miswrote the equations (just corrected it), the spreadsheet should have the correct ones.

@Zac: No worries – I should probably install a live preview plugin so people will know when their comment is getting eaten / mistaken for HTML. Sure, if you have ideas for the square root method feel free to write them down – once the contribution wiki is up I’m sure it’ll be a nice addition :).

hey nice one there for a quick look , although we know the value of pi after all those yrs of forced insertion of the value into our heads, but this gives a better insight to the derivation in a way,
appreciate the effort!

Thanks Brijesh! Yep, we know pi because we’ve seen it before, but it’s nice to see how we came to that result.

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Descubriendo el valor de Pi…

Prehistoric Calculus: Discovering Pi es una anotación del siempre recomendable sitio sobre matemáticas Better Explained donde se narra cómo a lo largo de los tiempos diversos matemáticos fueron aproximando el valor de π de forma cada vez más y más …

>Whatever the circumference is, it’s somewhere between the perimeters of the squares: more than the inside, less than the outside.

Why is it obvious that the outside square and futhermore 512-sided-thingy has bigger perimeter?

[…] Prehistoric Calculus: Discovering Pi | BetterExplained Warning math ahead, but if you stick with it, you’ll find out how Archimedes found pi to 99.9% accuracy 2000 years ago. (tags: history mathematics pi) […]

Reading a little more into pi and the ways of calculuating it seem to always lead me to Taylor Series. It would be nice to really understand what’s going on there.

Pi is a fun number. For some reason, I decided to memorize it to 50 decimal places. The fact that it’s impossible to calculate exactly just makes it even more fun to try and find more.

Well,I’m from Lima,Perú.And I never going to understand the way americans do math.For us "PI"is=3.1416.And if I’m not wrong 22/7 is not a correct anwser.not even that 223/71.Also I found and america you guys solve math problems outside down.My favorite subject is MATH,and II want tobe a math teacher.

this guy says pi is 3.154700 www.dinbali.com

Dude the chinese made a much better version than Archimedes sooner. They made ones with roughly 30-100 sides according to my research.

@tekumse: That’s an interesting question, sometimes it’s good to break down these assumptions. The formal name for the inside shape is “inscribed” and the formal name for the outside shape is “circumscribed”.

The area of the inscribed shape is less than or equal to the area of the circle, since all points are inside the boundary.

The area of the circumscribed shape is greater than or equal to the area of the circle, since all points are outside the boundary. Therefore, the area of the inscribed polygon is less than or equal to the area of the circumscribed shape.

For similar shapes, the greater area corresponds to a greater side length (see the Pythagorean theorem for more details). Since we are using similar shapes (squares, octagons, 16-gons, etc.) the circumscribed shape will have a larger side length (and perimeter) than the inscribed one. Hope this helps.

@Zac: Yep, the Taylor series will be fun. I want to think about it more to see if I can find some insights that link it to everyday analogies :). And 50 digits of pi is pretty precise, enough to estimate the size of the universe to 1 atom’s precision, I think.

@Miguel: Thanks for the comment, glad you like math. I think most students know 22/7 (or 3.1416) is just an estimate for pi, not an exact value.

@Anonymous: Don’t believe everything you read!

@Cheeseburger: That’s interesting, Archimedes made this technique famous but others may have used it as well.

Descubriendo el valor de Pi [eng]…

"Prehistoric Calculus: Discovering Pi" es una anotación del sitio sobre matemáticas Better Explained donde se narra cómo a lo largo de los tiempos diversos matemáticos fueron aproximando el valor de π de forma cada vez más y más exacta. Relac…