Christopher Guest votes for base 11.
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What if when the get to 125.6 trillion digits they find out that it is a never ending repeat of the first 62.8 trillion digits after the decimal?
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Then we realize we really are in a simulation
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IFYP
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too true, too true
For irrational numbers to exist, the universe needs to have infinite memory, which canât exist in a simulation.
At some point in pi there is going to be a string of 62.8 trillion digits followed by an exact copy of those 62.8 trillion digitsâŚ
Why would the universe need to know the exact decimal expansion of an irrational number?
The universe doesnât. A simulation might need to.
Itâs a pretty old âproofâ, or paradox, if you will.
What does that mean for Reality? Well, since Pi is an irrational number, it automatically means that either the diameter (or radius), or the circumference of that circle has to be an irrational number as well. (You can check the details of this in the PS.)
It also means that a computer, which is a Finite State Machine, no matter how large and powerful, can not accurately represent the number.
Why? Because irrational numbers have an infinite amount of decimal points, and can not be represented any shorter.
So if the universe would be a simulation, the computer it is running on would have to regularly represent infinitely long numbers â for every particle, every field, every wave. Which, if the computer itself is finite , is actually impossible, because representing infinitely long numbers requires infinite amounts of memory â for each (!) irrational number in the universe.
Meh. It only really needs to get it down to the planck length⌠just need 62 digits of pi to pull that off for the circumfrence of the universe.
In the physical world, perhaps, you can argue that there is no perfect circle. But perfect circles exist in theoretical mathematics, which would be part of the simulation as well (assuming everything is simulated that is).
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I donât see why it canât generate the required digits just a few more decimal places than we have at any given time.
Itâs a fine enough argument - it just doesnât really convince me.
Yeah Iâm no expert on this. I think the argument is that, circles exist (physically or theoretically), and thus pi exists.
If the digits of pi is preset, but continues to updates itself to be ahead of our (simulated humans) calculation, that means every circle (real or theoretical) needs to be updated as well, because the circumference (or anything that depends on pi) will be ever so slightly be off otherwise.
And then there are the infinitely many other irrational numbers as well.
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Thatâs where I donât really see a problem happening. I may be missing something though.
And - like Morpheus says - we donât really know what the universe this simulation is taking place in would be like. Infinite memory might well be possible there. 
Well, think about how you graph (code) a circle in a computer (or a graphing calculator).
This is true.
In order for me to imagine a perfect circle - my brain doesnât need to contain all of the digits of pi. I only care about those digits if I want to measure it.
I can theoretically get it to whatever level of precision I want. Certainly enough to keep these pesky human programs happy.
But thanks to Real Analysis . . . we only need bounds . . . not exactness.
I think thatâs the thing though, if the circles are there to be measured out to infinity then something must store that infinite string since it presumably returns the exact same string for pi each time. Now perhaps pi in the simulation is constantly made more specific as we measure it out to more decimal places⌠sort of like a video game that could generate infinite squares but never does because you never actually require it to, but if you keep going in one direction then eventually your PC will run out of memory and the game will crash.
So really the pi researchers are trying to kill us all.
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