Not a mathematician here at all so perhaps my question is phrased incorrectly.
Let’s say through thorough testing in reality, we can prove with certainty Pi is correct up until 5 decimal places,
3.14159
The computers that are calculating Pi to an endless degree, how do they validate new values that are calculated as correct and cannot be otherwise?
In: 434
Algorithms for pi often give an *upper bound* and a *lower bound* respectively. As these converge, we get a smaller and smaller range – i.e. more decimal places – within which we are certain pi lies.
Pi isn’t infinite: it’s greater than 3.1 and less than 3.2.
What it isn’t is rational: it’s not equal to a ratio of integers. “3.14159” is, in fact, a ratio of integers: it’s 314159/100000. When we calculate pi to one more decimal place we’re still getting a number that is not equal to pi; it’s just that now we’ve limited how far off we are by another power of ten.
It’s been 30 years since I took calculus, but this is what I recall:
When people wanted to find the area of a circle, they could draw and measure a square with its four points touching the circumference of the circle and another square on the outside that contains the circle. The area of the circle is somewhere in between the area of those squares. This allowed a rough approximation.
The next step in the logic is that if I use increasingly smaller squares on the inside my measurement will be more accurate.
One of the basic elements of Calculus is the “limit” – it allows a calculation of how a number behaves as it gets closer and closer to a given number (usually 1 or zero). This principle allows you to calculate the area of all the squares above as the length of their sides gets closer and closer to 0.
Archimedes calculated pi (250BC) with a similiar approach by using polygons with ever increasing numbers of sides.
A thing I would like to point out is that technically speaking to a computer at least pi is a fixed number.
Every equation we do with pi, is only valid to a certain level of accuracy. In short, certain tasks will require a certain level of accuracy for the calculated answer to be valid. For higher thresholds you must use pi to a larger number of decimal places.
An example of this could be using pi to work out the radius of a pizza so you can serve equal amounts. In this situation using something as low as pi = 3 will still yield an accurate answer. In regards to astrophysics however, this level of inaccuracy would result in drastic changes to the result.
Ok. So in order to calculate pi you need an algorithm that converges to pi. So some series that’s limit is pi.
One algorithm would be its definition, the circumference of a circle devided by its diameter. Using a circle with radius ½ its circumference is pi.
So you approximate a circle with polygons with more and more sides and add the side lenghts. This approches pi.
Now there are other definitions, and way more efficient algorithms but the key is convergence:
In certain spaces, like what we are using now we can say that every series converges if its a Cauchy series. A Cauchy series is a series where the distance between the n -th and the n+1 -th element is smaller than between n and n-1. So lets say we have some distance function d so d(n,n+1)<d(n,n-1) for every n. If any series with this property converges we say our space is complete in this sense. If our space is a healthy vector space, we call this kind of space a Banach space.
We are going to find pi in our healthy Banach space so all we need to show is that a series we use has the Cauchy property. There are many series that converge to pi or something proportional to it. And we can call any such series the definition.
So how do we know that a Cauchy series approches pi and not something that really close to it? Well because the limit of many such series can be found analytically. So you don’t need to represent pi in its decimal form, you can find the limit analytically and pi would appear as a symbol. But if so this means that for any finite step this will be a better and better approximation as long as you actually care about the decimal representation.
The formulas for calculating pi come with error bounds that shrink as you calculate more. If you’re partially calculating it and you have 3.1416… with an error bound of 0.0001, then you know that pi is between 3.1415 and 3.1417. You’ve now guaranteed that the first 4 digits are 3.141. If you calculate more to shrink the error bound further, then you can be sure of more decimal places in the same way.
The only difficulty is with the digits 9 and 0, because 0.99999 is extremely close to 1.00000, even though they have all different digits. But there are other independent proofs that pi is irrational, so we know that 9s and 0s will not repeat forever. So if we ever get a long string of 9s, we just need to keep calculating for longer and eventually they’ll stop.
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