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
Pi is not infinite. It’s a very specific number, just like 0, 1, or 2 are specific numbers. It’s just if you try to write pi with decimals, then it’s “infinite”. You could also say 1 is infinite if you wrote it as 1.00000…. and never stopped writing zeros. We don’t have to do that with 1, but it doesn’t mean pi isn’t just “pi” like 1 is “one”.
Pi is a very specific number about the ratios of the parts of a circle and it is true and the same for all circles of all sizes, etc. Like other similar irrational numbers (they have “infinite” digits in decimal form), we can try to find the digits by doing two similar calculations above and below what we think the target number is. As we get closer and closer, we have to get more and more specific, and therefore we know that all our previous work that matches between our two test numbers is definitely right, and we’re just figuring out the end piece.
Since Pi is a bit tricky, let’s look at √2 for how this works:
We know √2 * √2 = 2 but how could we turn √2 into a decimal? Well, first we know that √2 must be below 2, and so then we can look at squaring other numbers. First, the most obvious ones would be 1*1 = 1 and 2*2 = 4. Those two results (1 and 4) are on either side of 2, so it must be that √2 is between 1 and 2, and likely a bit closer to 1 (since 1 is closer to 2 compared to 4).
Then we do the same steps a few times, trying to get more specific (and making some lucky guesses to make this faster). First, we look at 1.4*1.4=1.96 and 1.5*1.5=2.25. Both near 2, but definitely closer to 1.4 than 1.5. How about 1.41*1.41=1.9881 and 1.45*1.45=2.1025. We can go on and on with this, but just from these few steps we KNOW that at least the 1.4 part is right. It’s a bit too low, but whatever we end up with is going to start with 1.4 because we know the “full” answer is somewhere between 1.41 and 1.45.
Eventually we will know something like √2 = 1.414213… To get more beyond that, we would check 1.4142131^(2) = 1.99999869… and 1.4142139^(2) = 2.000000954… And this would go on and on. We’re not sure about that 1 or 9 on the ends, but we know the 1.414213 is right because adding those 1 and 9 put us above and below. The “answer” is somewhere between those two so everything before it won’t change.
With pi, the math isn’t as simple as just picking two numbers, squaring them, and seeing if they are higher or lower than 2, but there are ways to do similar checks. Using that method, at each step we would have two numbers where we KNOW that one is bigger and one is smaller than pi and are just making those more and more and more specific which tells us pi is somewhere in the middle (and all the earlier decimals before the parts we are adding/adjusting at the end match with pi).
It would be something like… we know it’s between 2 and 4, then we know it’s between 3 and 3.5, then we know it’s between 3.1 and 3.3, then we know it’s between 3.12 and 3.15, then we know it’s between 3.141 and 3.149 (and on and on). Any parts that match up between the two below and above numbers we know will not change as we go further and so we know pi at least that far (already locked in for 3.14 here). Do this long enough and with crazy enough computers and you can have the value out to thousands or millions of decimals and then you’re still just arguing over the next/last decimal you’re adding on.
Latest Answers