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
It’s like a game of hot/cold that you played as a kid.
Suppose you start off somewhere in the country playing a game of hot and cold. When you get closer, someone tells u are warmer, and when u get further, someone says you are colder. Your target could be anywhere in the country when you start.
You start in new york. After a while, you realize that you are only getting hotter when you head west. Eventually, after many rounds, you end up in seattle. After more rounds you narrow it down to a specific neighboorhood, to a specific house, and finally to a specific room. Now, you don’t know where it is yet, but you know it’s in this room, cause whenever you are outside the room they say you are getting colder.
Now, after all this work, someone comes to you and asks -how do you know it’s not in florida? The answer is obvious to you. Well, it’s the same way that we know the first digit of pi is a 1. We have a formula that gets us closer and closer to the answer the more we use it – and after you have gotten 10 digits in deep, you know the answer isn’t in florida any more.
This is not how mathematics works. Mathematics is all about rigorous proofs. You cannot proof that a decimal representation of π is correct up to say 5 places by testing it a bunch of times. That doesn’t proof anything, maybe you just never found a counterexample even though one exists.
Instead you would show that a way of computing π computes a number that fulfills all defining characteristics of π.
Mathematics is not really about doing computations yourself. Instead it is about building “chains” of logical arguments. True statements lead to true statements which then lead to true statements and so on.
This is what computers do while looking for new decimals of π. Mathematicians have developed computational steps which can be formally (not by experiments!) proven to result in the correct decimal representation of π.
Reading these replies makes me realize I need a sub called “explain it to me like I’m five weeks”.
You verify your answer.
Simply put 1+1=2. take a 1 pen, bring in an additional pen. how many pens? 2. Repeat with another item. same answer? Have someone else attempt it. same answer? then 1+1=2.
With Pi, and other irrational numbers. they start the calculation, then attempt the calculation a second, then a third and fourth time,. Mathematicians repeat and if the answer comes back every time with the same answer then we know that the original calculation is correct.
Note that calculating larger values of pi doesn’t have a practical value. It’s now a way to show technical chops in how the program is written, the hardware used to perform the calculation, or if a new algorithm is faster than others.
NASA doesn’t use more than 15 digits of pi (including the 3 in front of the decimal point) in their most precise calculations: [https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/](https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/)
>”Let’s say through thorough testing in reality, we can prove with certainty Pi is correct up until 5 decimal places”
This is the source of your confusion. Pi does not *exist* in reality. It is just a concept; the algorithms to compute it are derived by inexorable logic following from the definition of the concept.
You might think that Pi exists in reality because of its relation to circle geometry, but remember, perfect circles don’t exist in reality either.
The number Pi is not infinite. It just has an infinite number of decimals.
Pi is >3.14 and is <3.15
I am curious if there are any real life scenarios where we actually have a use for the hundreds or thousands of non-repeating digits after the decimal place for pi in base 10.
I mean, for any reasonable calculation where pi is needed, there is literally zero purpose for most of those digits. But I wonder if there are times when those digits are helpful to us.
Maybe something involving cryptography?
It seems no one has addressed the obvious, which is that the question from the get-go is wrong: **Pi is not infinite.** Pi is a finite number, somewhere between 3 and 4, but has an infinite number of digits in its decimal representation.
Pi is indeed an infinite number with an endless string of decimal places. It’s true that we cannot calculate or write down all of its digits because they go on forever. However, mathematicians have developed clever ways to make sure that each new decimal they calculate for Pi is valid.
They use something called algorithms, which are like special step-by-step instructions, to calculate Pi. These algorithms are based on mathematical formulas and rules that have been proven to work correctly. Mathematicians have checked and double-checked these algorithms many times to make sure they are accurate.
When a computer calculates Pi, it follows these algorithms and performs lots of calculations. The calculations become more and more precise with each step. While it cannot calculate all the digits of Pi at once, it can keep going and generate more and more accurate digits as it performs the calculations.
To make sure the new digits are correct, mathematicians use something called mathematical proofs. These proofs are like logical explanations that show why something is true. For Pi, mathematicians have proven that the algorithms used to calculate it will always produce the correct digits.
So, even though we can’t calculate all the digits of Pi at once, we trust that each new digit calculated using these algorithms is valid because they have been tested, verified, and proven to work correctly. This is how mathematicians and computers keep generating more and more accurate decimal places of Pi.
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