Today I'd like to teach you all about one of my all time favorite constants: π!
Most likely, all of you know that π is the ratio of a circle's circumference to its diameter, and probably only know only 3 to 5 decimal digits of it. However, π is an irrational number, meaning it has no end! Let us go through all of the different formulae of π I have programmed, and explain what they mean! Let's dive in. (These are numbered in the order that they print when the attached code is run)
- The Monte Carlo Method
One way you can do this is by drawing a square, with a circle whose edges touch the 4 edges of that square within. Now imagine you were to draw many many dots randomly within the square-circle hybrid. If you take the number of points that fell within the circle, divide it by the total number of points, and multiply it by 4, you get (you guessed it) π!! More info can be found here:
- The Chudnovsky Algorithm
This is one of the fastest methods out there, being used in the world record, to calculate 50 trillion digits of π! I already explained it in a previous post, but the formula (and more cool information!) is here:
- The Basel Problem
This problem, posed by a man named Pietro Mengoli, asked for the exact sum of an infinite series, with proof. Mathematician Leonhard Euler answered this, with proof, finding it to equal exactly π^2/6. Further info and interesting facts here:
- The Wallis Product
Unlike the other formulae here, this one uses the Product Operator in its equation. If repeated over an infinite number of times, it will equal π/2. More information here:
- The Leibniz Formula
Last but certainly not least, I have here probably one of the slowest π-convergent methods out there. In fact, to get π accurately to 10 decimal places takes about 5 billion iterations, according to the Wikipedia page! This formula alternates between adding and subtracting fractions with odd denominators (meaning this is an example of an alternating series), and converges on π/4. For extra information, go here:
I plan to make more π approximation programs in the future, so stay tuned if these kinds of things interest you as much as they interest me!
Thank you. ^ ^
@JosiahKnisely Oh, that's alright. I can explain. ^ ^
No, there are not multiple values for π. There is only one, that being '3.14159...' and so on. These are merely methods of calculating π, albeit with varying amounts of accuracy at certain numbers of iterations.
π is quite interesting and cool, in that it can show up within the solutions to various mathematical problems that appear to have absolutely nothing to do with circles (at least at first glance)!
The things that the program displays are not π, in that, based on the accuracy I've asked for, the digits it shows aren't EXACTLY π.
For example, a common value that we use when approximating π is 22/7. This is only accurate to 3 digits (3.14), and even closer is 355/113, which is accurate to 7 digits (3.141592).
While they aren't equal to π, we often use them, since they are close enough to be somewhat useful for our purposes!
We don't know what π equals exactly, since it is infinitely long, but these equations can hep us learn more and more.
Here's a helpful link that non-math people can understand ^ ^