π = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631...
What do we know about π?
- It has to do with circles - something something 2πr
- It tastes delicious
- It's infinite
- It goes on forever
- It has infinitely many decimal digits
- It's random
- There's no pattern to its digits
- It's irrational
- It's transcendental
Measure a pendulum's swing
\[ \pi = \frac{T}{2} \sqrt{\frac{L}{g}} \]
Embrace the noise
\[ x^2 + y^2 \leq 1 \]
π ≈ 4 × 0 / 0 ≈ ?
Draw a polygon which looks like a circle
\[ P_{2n} = \frac{2p_nP_n}{p_n+P_n}, \\[1.5em] p_{2n} = \sqrt{p_nP_{2n}} \]
Give up and legislate it
"the ratio of the diameter and circumference is as five-fourths to four."
Machin's formula
\[ \frac{\pi}{4} = 4 \arctan \left(\frac{1}{5}\right) - \arctan \left(\frac{1}{239}\right) \]
Rewrite as an infinite series
\[ \pi = 4 \sum_{n=0}^{\infty} \frac{(-1)^n)}{2n+1} \left( 4 \left(\frac{1}{5}\right)^{2n+1} - \left(\frac{1}{239}\right)^{2n+1} \right) \]
William Shanks, Local Hero
William Shanks spent over 20 years calculating π to 707 decimal places.
But he made an error at the 528th digit.
William Shanks, Local Hero
William Shanks spent over 20 years calculating π to 707 decimal places.
But he made an error at the 528th digit.
William Shanks, Local Hero
William Shanks spent over 20 years calculating π to 707 decimal places.
But he made an error at the 528th digit.
Use a machine
D.F. Ferguson spotted Shanks's error and broke his record with a mechanical desk calculator.
Since then, computers have gone on to compute a stupid number of decimal places of π.
(Image by Wikipedia user Nageh)
Spigot algorithms
(A bit like this bear, but for digits of π)
Spigot algorithms
Iterative algorithms which produce digits one at a time, and never reuse a digit in a later step.
And they only use a constant amount of working memory.
The Bailey-Borwein-Plouffe formula
\[ \pi = \sum_{k=0}^{\infty} \frac{1}{16^k}\left( \frac{4}{8k+1} - \frac{2}{8k+4} - \frac{1}{8k+5} - \frac{1}{8k+6} \right) \]
You can extract any digit of π without working out any of the previous ones.