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 = ~~22 / 7 by definition~~ (the circumference of a circle divided by its diameter)
~~So whatever number comes up next in the string when you work out “twenty-two divided by seven” is the right number~~
All it is a division problem that never ends.
The next number will just be the next in the sequence. To that level of pi that number is accurate. The further you go, the more accurate you get. So if you are at the 5th number there will always be an option to go to the 6th….and the 7th…..and so forth.
Our methods of calculating pi also produce upper and lower bounds for what pi is. If the upper bound is 3.1416 and lower bound is 3.1415 for example then we know that the digits 3.141 100% are correct values in pi.
It’s decimal expansion might be infinite and non repeating. But it’s still computable.
We have multiple ways of creating sequences of numbers that have been proven to approximate π better and better as the sequence progresses. (https://en.wikipedia.org/wiki/Pi has several examples).
We can check a simpler irrational number: √2.
We know that √2 multiplied by itself gives 2, right? That’s kind of its definition.
Now, we know 1² = 1 < 2 and 2² = 4 > 2, so 1 < √2 < 2
And we keep going, 1.4² = 1.96, 1.5² = 2.25.
And we keep adding numbers. At each step we can find two numbers that keep sandwiching √2, so we know each new digit is the correct one.
There are similar processes for π. The exact process is (very very much) more complicated, but the principle remains the same.
pi divided by 4 is 0.78539816339…
You can approximate this value as 1 – 1/3. It’s 0.666, a bad approximation, so let’s add 1/5. That gives us 0.8666, closer but too high. Let’s subtract 1/7. That gives us 0.7238, too low now, so we add 1/9…
This sequence, **1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13…** is called the Gregory-Leibniz Series, and it gets a little closer to pi/4 each time you add or subtract the next fraction with an odd denominator. Mathematicians have proven this conclusively, via a proof too complex to give here.
To calculate the next (several) decimals of pi, computers can simply add the next number in the series. The actual method used by computers does not use the Leibniz formula, but it’s the same basic idea.
Similar functions exist to calculate other constants as well.
This is NOT how a calculator does it (the formula is different) but ancient mathematicians (before computers) used geometry to calculate approximations of pi.
First, start with the definition of pi: the ratio of the area of a circle to the square of its radius. This gives the geometric definition.
Now draw a circle (declare it a circle with 1 unit radius), therefore the area of that circle is (by definition) pi.
Within that circle draw a square that is fully within the circle and has the vertices on the edge of the circle. By Pythagoras theorem, we know that the side length of the square has to be 2/sqrt(2). and therefore the area is 2.
Now draw another square except the square encloses the circle and the midpoint of the edges of the square just touch the edge of the circle. We know that the square’s side length is 2 (twice the radius of the circle). The area of this square is 4.
Since the area of the circle must be greater than the smaller square and smaller than the larger square, we establish that pi must be between 2 and 4.
The trick is that we don’t have to use squares. By using to regular polygons with more sides (hexagon, heptagon, octagon…) as long as there are methods to calculate the areas (generally very simple formulas), it is always possible to bracket the lower and upper bounds of the value of pi by geometry.
As long as number of sides of the polygon are increased, the estimate for pi becomes more accurate to more decimal places.
Modern calculators use a different method (there is an infinite series expansion to give pi) but conceptually similar. As long as more and more terms of the infinite series are calculated, more and more accurate values of pi can be calculated.
First of all, pi is not infinite.
Pi < 4 < infinity and pi > 3 > -infinity.
Pi has an infinite decimal expansion, but so do plenty of other numbers. You’ve been given examples of other irrational numbers like sqrt(2), but plenty of rational numbers have infinite decimal expansions too. 1/3, for example, is 0.333… that goes on infinitely too (the digit just happens to be conveniently repeating, which pi does not).
Another thing to consider is that the fact that we express numbers in a base-10 system is pretty much solely due to the fact that we happen to have 10 fingers. For example, in base pi, pi itself is just 10.
It depends on the method you use to calculate it
Some methods use upper and lower bounds, and if both of the bounds have the first 100 digits in common, the actual value of pi must also have these 100 digits
A simpler example is √2. It’s also irrational, so its decimal expansion is infinite. 1.414² < 2 < 1.415², so 1.414 < √2 < 1.415. This means that 1.414… is definitely correct
For other methods you might be able to calculate the error in some way and get an upper bound on it. If the error is less than 0.00000000001, you know you have at least 10 correct digits
The decimal expansion for pi is not found by “testing in reality”. Using properties of circles and polygons inscribing and circumscribing a circle, mathematicians before the late 1600s found estimates on pi from below and above. For instance, if they found pi is between 3.1414 and 3.1416 then they knew pi starts out as 3.141… I am using decimals for their familiarity to you, but ancient mathematicians did not have the decimal system. But the idea of lower and upper estimates with fractions still made sense to them (a finite decimal is a special type of fraction.)
Once calculus was created, completely new methods were developed to rapidly compute pi to hundreds of digits. See the Veritasium video “The Discovery that Transformed Pi”.
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.
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