Prehistoric Calculus: Discovering Pi

Blog Articles
61

Great article. I think one of my face-palming moments was when I realized that pi was the result of infinitely improving the number. (This also helped me to understand transcendental numbers, since you need an infinite series of algebraic formulas to reach it.)

What I think is particularly interesting is how something infinitely complex can make formulas so simple. Instead of picking an approximation (since, a lot of the time, we don’t know ahead of time what this should be), we use the pure number “pi” to allow somebody else to approximate later. Not only that, but it makes the formula easier to read as well by encapsulating the complexity in a single constant. Truly beautiful.

(Side note: working with image processing and other forms of computer graphics, I sometimes wish “pi” was initially measured with the radius instead of the diameter. That way, we could use the constant itself instead of writing “2*pi” everywhere. The constant really only represents half of the shape of a circle.)

62

I really enjoyed this article, and it makes complete sense why Archimedes used this method, although i would have never thought of it on my own. I liked the style of the writing too, very easy to understand. The one thing is didn’t understand was the formula for perimeter of the inside and outside shapes. I don’t understand why we use sin. Other than that great article

63

Why was this never explained like this in high school?

64

@Joe: Thanks for the comment! Yeah, one of the weird things about pi is that it’s never “done” – i.e., when does a shape with “infinite” sides become a circle? It raises all sorts of interesting philosophical questions too – i.e., we use pi for calculations but will never encounter a perfect circle in the real world. But the beauty, as you say, is that we encapsulate this whole concept into a symbol which is “use the best approximation of the perfect circle that you can…”.

I agree on the pi vs. 2*pi thing – have you seen http://tauday.com/?

@Matthew: Thanks! Great question on the formula – there’s an explanation on why sine is used here:

http://personal.bgsu.edu/~carother/pi/Pi3b.html

but I’d like to cover it in more depth myself. Thanks again for the note!

65

[…] Pi is the most famous example of similarity — all circles share the same ratio (circumference / diameter = pi). Again, no matter how much we zoom to make one circle appear like another, every circle has this fundamental trait. […]

66

[…] Pi iss thee moost famous sample of similarity — alll circles share thee same ratio (circumference diameter = pi). Again, Noo matter How many wwe zoom tto earn 1 circle apear love another, everi circle hass this fundamental trait. […]

67

[…] a formula to make smooth changes…). Eventually, we'll understand the foundations intuitively (e, pi, radians, imaginaries, sine…) and they can be mixed into a scrumptious math salad. Enjoy! 0 […]

68

this is a wonderful article. thank you.

there is only one place where i disagree. I would say that pi can ‘hide but not run’ instead of the other way around.

cheers

69

@eczeno: Thanks! Yep, to each their favorite phrasing :).

70

[…] http://betterexplained.com/articles/prehistoric-calculus-discovering-pi/ http://mathforum.org/~sanders/exploringandwritinggeometry/polygons.htm […]

71

Wow!great information on pi this has really widen my view abt maths.maths is becoming interesting to me.thanks 4 making it interesting.i’ll love 2b a mathematician.

72

funny, i just only understood the point of taylor series while reading your article on intro to calculus. and its right on this page! thanks for giving me a wonderful “Aha!” moment. love your site.

if you ever watched the movie 3 idiots, you remind me of one character, Rancho. I hope that turned out as a compliment. :slight_smile: more success on this and other ideas of yours!

73

@mel: Thanks for the kind words! Really happy the site is helping with those ahas. I haven’t seen the movie but have heard much about it!

74

Hi Ali, no problem! Glad you enjoyed the article. I might like to do a follow-up to really understand how those formulas came about.

75

Is there another article after this? I’m probably not looking properly, but I want to read on!!

76

the equation above is wrong dipshit, the inner squares side doesnt equal .7, it equals the square root of .5 . pythagoras states in the above equation that a²+b^2=c^2 〖.5〗^2 〖+ .5〗^2=c^2 .5=c^2 c= √(.5) not .7 as stated above. just thought id let you know

77

Dear Kalid,

I am now in my 60s, but majored in Mathematics and the History and Philosophy of Science at Melbourne University (Australia) some 30 years ago. The latter major - a fascinating study - focused on the development of several different sciences, including mathematics, physics, chemistry, biology, etc. from times BC to the 20th century.

Motivations underlying early mathematical development in different areas BC (Eastern Europe/Middle East, India, China) included fascination in number theory and algebra, astronomy (“understanding the movements of the planets or heavens”), religion or belief in divinities (predicting or setting auspicious dates for appeasing the gods or for festivities), agriculture (predicting the seasons), and taxation (calculating approximate land areas under plot, based on shapes and dimensions).

It therefore should come as no surprise that I appreciated your paper on estimating a “best value” for PI (Discovering PI, 2008). However, to avoid confusion and unhelpful feedback, it may be worthwhile clarifying the dimensions of the inner square, especially for busy teachers and students.

The inner square has a diameter or hypotenuse of 1, as is obvious from the diagram of the inner square within the circle and within the outer square. Based on elementary geometry for right-angle triangles in a square, we can calculate a^2 + a^2 =1, where “a” is the length of a side of the inner square.

Simplifying and transposing, we get:
x) 2a^2=1,
y)a^2 =1/2 =0.5, whence
z)a=1/SQRT(2) or a=SQRT(0.5)

It may be stating the obvious to record these arithmetic identities, but even chess Grand Masters can make blunders, let alone hasty readers!

PS: My 3 sons have all been strong in Mathematics, and my youngest (now aged 16) is excelling in the subject and is currently doing early-entry University study. He combines Mathematical study with Musical study (playing piano and percussion, and conducting his school orchestra).

78

Sorry for jumping the gun… My question got answered following the link you have already referred here: http://personal.bgsu.edu/~carother/pi/Pi3b.html#geometry

Thanks again!

79

Sides = 4037146 is the first here to show up as 100% Accuracy

Just so you don’t keep trying
LIKE I DID.

80

eu amo estudar sobre o numero pi!!!1