How are new PI digits discovered?
In: 5
They aren’t exactly discovered. They are computed.
We have exact formulas that allow us to compute Pi, and the number of “discovered” digits only depends on how precise your computing is.
They’re not exactly discovered, they’re calculated.
There are lots of formulas that produce progressively more digits of pi the more you do them. You just keep doing them longer than anyone else did before and you get new digits.
π is irrational real number. it means that it can’t be written as a/b (please dont say π/1 is a ratio for π lol). so it goes infinitely, like √2.
new digits are not discovered, but calculated.
one way to calculate π is to use infinite sum formula
π/4=1-1/3+1/5-1/7+1/9…….
rearange it to get
π=4-4/3+4/5-4/7+4/9…….
if you keep adding and subtracting more 4/odd number you get more digits of π. you approach it. although it worths noting that approximating pi to 4 decimal places is pretty much enough for most of the people, meaning that π≈3.1415 will give reasonably accurate answer for almost all problems that you’ll see in textbooks.
I cant remember all the details, but [this](https://youtu.be/gMlf1ELvRzc) video was really cool and explained how pi is calculated, along with some other interesting stuff
While we can’t cleanly express the value of Pi, we can express an infinite series that converges to Pi.
An infinite series is just a continuous sum of smaller and smaller terms. Eventually the terms being added are so small that we can be certain of the values of the higher digits.
For example, we know that Pi/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11… and so on. If you decide to cut off that formula at some point, you can just run the calculations and figure out an approximate value. The larger more terms you have, the more precise you are. There’s a secondary calculation you can do that shows how many digits you can trust from your original answer (i.e., how big the portion you chopped off was).
They’re computed. We have exact formula for pi, since as early as Archimedes. Now we have much better formula that are faster to compute with and we have supercomputers.
But you can have a formula for pi without anything fancy. “Knowing” the digits of pi just means being able to approximate pi to certain precision: for example, knowing 6 digits of pi (after the decimal dot) just means knowing that 3.141592 is less than pi, but 3.141593 is more than pi. So a simple way of calculating this is to just find the area of a circle of radius 1, using a grid. If you take a square grid where each cell has very small side length and you count the squares that lie strictly inside the circle you get a lower bound for pi, and if you count the squares that overlap with the circle you get an upper bound of pi. Pick a fine-grained grid (the side length is small) and you will get a good approximation for pi, which gives you the digits.
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