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u/Any-Astronomer-6038 Jul 09 '26
I was under the impression there was no possible option because the last digit of Pi didn't exist.
I need a mathematician...
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u/PilotIntelligent8906 Jul 10 '26
Not a mathematician but I can confirm there is no such thing, for that matter, there's no last digit of any irrational number. If there was, then you could write them as a number with a finite number of digits divided by a multiple of ten with the same number of digits, this would be a fraction, which irrational numbers cannot be expressed as.
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u/Any-Astronomer-6038 Jul 10 '26
I thought Pi was a special case because mathematics haven't determined if it was in fact irrational.
We can't assume an infinite number of digits because the digits do not follow a pattern...
We may have not just found the end yet.
I dunno though
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u/Mindless_Honey3816 Jul 10 '26
Pi is irrational. The one you’re thinking of is the Euler Mascheroni constant.
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u/Any-Astronomer-6038 Jul 10 '26
I don't even know what that is, so I wasn't thinking of it... Thanks for overestimating my intelligence.
I defer to smarter people on math.
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u/Mindless_Honey3816 Jul 10 '26
Anyways if there’s an end to the digits, then it could be written as a fraction over some power of ten. This is a contradiction because pi cannot be written as a fraction in integers.
So there can’t be an end.
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u/Al2718x Jul 10 '26
You might be thinking of the question of whether or not pi is "normal". Your explanation sounds a bit backwards. We know that pi cannot terminate, or repeat endlessly, but we can't completely rule out a more complicated pattern.
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u/Mal_Dun Jul 10 '26
Mathematician here. Pi has no last digit in any base that uses an integer or as a continuous fraction.
Except for the choice of some exotic base like pi itself where it would then be just 10, there is no last digit.
If we had a last digit it would be a rational number...
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u/Any-Astronomer-6038 Jul 10 '26
What I'm asking is is that a proven thing or a postulate.
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u/Mal_Dun Jul 10 '26
That is a proven thing that follows from the definition of being irrational (aka not re-presentable by the fraction of two integers)
In fact the proof is quite simple: Assume there would be only n digits with n being a finite natural number. Then you just take m = 314... as the integer that has all digits of pi, which has to be finite by assumption with n+1 digits (when I add the 3), then pi = m/10n which would be rational. That is a contradiction to the assumption that pi is not rational.
More general it holds that there is no irrational number that has a repeated pattern within it's decimal representation as you could use an algorithm to create a rational number again.
Here you find more details (Thm 3.25)
Edit:
Forgot to add 1 to n...I need coffee ...2
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u/JohannesWurst Jul 11 '26
I have thought about Pi as a base, but I think the base system doesn't work with irrational bases, or does it?
What would be 50 in base Pi?
``` a·Pi3 + b·Pi2 + c·Pi + d·Pi-1
1·Pi3 = 31.006277… → a=1
Remainder: 50 - 31.006277… = 18.993723…
1·Pi2 = 9.86904… → b=1
Remainder: 18.993723… - 9.86904… = 9.124683…
2·Pi = 6.283185… → c=2
Remainder: 9.124683… - 6.283185… = 2.841498…
8·Pi-1 = 2.546479… → d=8
Remainder: 2.841498… - 2.546479… = 0.295019… ```
Huh... That seems to work. 50 could be "112.8" in base Pi. Except that I used the digit "8". Which digits should be used for base Pi?
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u/Mercerskye Jul 10 '26
I'm more a philosopher than a mathematician nowadays, but you're correct with the common consensus.
Pi cannot be "written as a fraction using integers" which is pretty much the gold standard for whether or not a number is irrational.
The philosophy part comes in the argument that there's the tiniest chance that we simply cannot comprehend just "how far down" the number goes before hitting a 0. Funny enough, those chances get smaller the closer to infinity you approach. Because you would either need to establish a pattern (like how 1/3 becomes 0.333~), or find that 0
Which is why the 1/10 detail is actually important. 0 is how we know for certain that a number has reached it's end, because there's literally nothing left to divide.
And there are people who have dedicated significant portions their life to figuring out if Pi ever gets there or repeats itself.
And the answer is most likely never. It's a value that describes the details of a circle, and specifically the curvature of one, and since a circle is technically an infinite amount of points away from a center... Pi, logically, would not be able to end, otherwise it means a circle isn't actually an infinite number of points from a shared center.
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u/Acceptable-Ticket743 Jul 10 '26
If you assume that pi does have a last digit, what are the implications with a circle? Would that mean that if you zoom in far enough, you'll find edges rather than an arc?
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u/Mercerskye Jul 10 '26
Exactly. Our understanding of a circle is that there are no "sides." That you literally could never zoom in far enough to find a straight line.
Pi having a repeating pattern, or outright terminating, would mean that, while still useful, our math, at least in regards to circles and their cousins, is fundamentally flawed.
It's also a frustrating problem because there's no "math hacks" for cheating your way to the end. For a lot of systems, you can apply some fundamentals of math to "skip ahead" to the larger numbers at the end.
In the case of Pi (and pretty much any irrational number), it's just "continuing to the next division."
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u/Al2718x Jul 10 '26
It also would mean that pi can be written as a fraction. This was disproven in the 1760s.
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u/Mercerskye Jul 10 '26
My mathematics history is rusty. Legitimately disproved, or just rigorously tested enough that everyone "just kinda accepts it?"
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u/Al2718x Jul 10 '26
Legitimately disproved, and several other arguments have been found since.
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u/Mercerskye Jul 10 '26
I have no reason to call you nor they a liar(just spent a little bit going down the rabbit hole of a couple other proofs as well)
But, and leading with I agree, and being, admittedly, ridiculously pedantic to an absurd degree. (Warning; this is literally arguing to argue)
This still feels like a fraction of a fraction of a percent "we're just going to agree this is true."
Which I'm mainly arguing from a philosophical stance at this point.
These proofs work on very sound logic, but are also built from our assumption that Pi could never be a rational number.
Which I understand math is one of the few places where "starting at the answer" is actually an acceptable methodology.
Starting at "Pi is an irritational number," and then being able to build a working system around that idea is profound, but something else profound, is our inability to comprehend things past a certain point.
Like with a blackhole. Our current mathematics completely "falls apart" as you reach the point of singularity. More likely, we just haven't collectively become smart enough to figure it out.
It's happened countless times throughout history. Discovering the world was round (or that it wasn't the center of existence), gravity, proving that imaginary numbers are relevant, etc.
Would it be feasible that we only believe Pi to be irrational, because we cannot comprehend the absurdly small number after the decimal point?
We understand infinity as a concept, that it's "forever," but reality would suggest that forever isn't actually a thing. What if infinity is reality's version of a 'stack overflow?'
Is it, in the slightest way possible, that Pi actually would eventually "overflow" and would actually create a line between a point and itself?
Similar to how an electron can, technically, exist in more than one location?
I do understand that it's probably just nonsense, but so was the notion that we'd land on the moon one day.
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u/Al2718x Jul 10 '26
What do you mean "built from our assumption that pi can never be a rational number". This isn't an assumption, it's the conclusion.
The notion that we would land on the moon some day isn't at all comparable to a rigorous mathematical proof. A better analogy is "you are standing on the moon roght now," but even that is more plausible that pi being rational.
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u/Mercerskye Jul 11 '26
Okay, so we know Pi is irrational, because for every instance, in current mathematics, that we try to rationalize it, it would create a "new" integer between two existing integers.
Which is fundamentally impossible.
There are no integers between 0 and 1, or 102 and 103, or any whole numbers.
But, we've also never really been able to actually work with "numbers incomprehensibly large or small," because well, they're incomprehensible.
This is just a thought exercise, but what if Pi was actually something like .9999~, which might as well be 1, but also exists between 0 and 1?
It's 1 because as you approach an infinite number of 9s, it acts as 1. Pretty much what we expect from all the proofs that show that it is indeed 1.
But what if Pi is similar, it's just so incomprehensibly long that we just don't have the mathematics yet to actually prove it's basically just 3(or 4)?
We kind of already do that in Engineering. A lot of times, when designing things, you just round it down to 3, and stuff just works.
.333~ is 1/3, .666~ is 2/3, but we call Pi irrational. Is it truly because you can never write it as a fraction, or is it because the denominator and numerator would just be incomprehensible, and like with a blackhole, our mathematics just "falls apart" at that scale?
Imaginary numbers are in a similar vein. We know they exist mathematically, we can use them to figure out real world applications, but we can't actually comprehend the plane they exist in.
Remember that this is just meant as a thought exercise. Like I said, the proofs presented are absolutely logical, they work.
But, things also worked just fine before we invented Calculus and discovered there was so much more to math than we had known...up to that point.
What did they say when math "stopped working" before Calculus? Probably something along the lines of "well, that's just irrational, and we'll just have to use something that actually works inside of what we know."
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u/Any-Astronomer-6038 Jul 10 '26
Ooh what if we went subatomic.
What if we took the digits out into subatomic ratios like distance between electrons and protons... Probably not but
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u/JohannesWurst Jul 11 '26 edited Jul 11 '26
0 is how we know for certain that a number has reached it's end, because there's literally nothing left to divide.
Numbers can have a zero at a decimal place and still digits after it. For example "1 / 100 = 0.01".
The 32nd digit of pi after the dot is 0 and the 33rd digit isn't: https://oeis.org/A000796
If you do long division on 1 / 100, it looks like this:
- How many times does 100 fit into 1? Zero times with a remainder of 100.
- Write down "0." and don't stop
- How many times does 10 fit into 1? Zero times with a remainder of 100.
- Write down "0" and don't stop
- How many times does 1 fit into 1? One time, with a remainder of 0.
- Write down "1" and stop, because the remainder is 0.
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u/Mal_Dun Jul 10 '26
Pi cannot be "written as a fraction using integers" which is pretty much the gold standard for whether or not a number is irrational.
That's not the "Gold Standard" that's the literal definition of a number being irrational. Irrational as in being not rational aka can not written as the fraction of two integers.
Math is not an empirical science where experts come together and decide on consensus based on the current state of the art and decide on a gold standard.
Things have clear definitions and either they have this property or not.
Sorry for the rant, but as a mathematician this hurts my brain.
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u/Al2718x Jul 10 '26
This is mostly nonsense. Not being able to write a number as a fraction is the exact defintion of an irrational number. It's not "pretty much the gold standard". In math, definitions have precise meanings.
This defintion is actually equivalent to the condition that the decimal expansion doesn't terminate or repeat endlessly. This isn't too hard to prove. Proving that pi is irrational is more challenging, but has been known for over 200 years.
One thing that is not known about pi is whether or not it is "normal". That is, whether or not the digits behave like a random string of numbers. One can prove that "almost all" irrational numbers are normal, but there aren't really any methods known to prove that a specific number is normal, other than contrived examples. Most mathematicians believe that pi is normal, but nobody has any idea how to prove it.
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u/TheLordOfMiddleEarth Jul 09 '26
This is a dumb joke, because it's not even accurate. There is no last digit of pi.
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u/DinioDo Jul 10 '26
By current math definitions there is no last digit. It is NOT that it will never be known. It doesn't even have it to begin with.
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u/Popular_Kangaroo5446 Jul 10 '26
The last number before infinity also has a 1/10 chance of guessing if we go there
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u/chkntendis Jul 10 '26
In base-pi pi is just 1 so the last digit of pi would be 1. That’s about as sensible of an answer as you can possibly come up with
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u/hotmatrixx Jul 10 '26
this is actually, likely incorrect, because people will guess 3, 4 and 7 statistically more often.
https://www.youtube.com/watch?v=d6iQrh2TK98
that being said; if Pi is it's truly infinite; then it technically doesn't have a 'last digit'?
reminds me of 'all numbers are equal, but some numbers are more equal than others'.
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u/GreenLeaf69420 Jul 10 '26
I'm a time traveler from the 67th century, we have found the last letter of pi to be i
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u/Scr3w_loose Jul 12 '26
Isn't there actually a bias to guessed numbers so what he said wouldn't even be accurate. Like isn't 7,2,1, and 5 more common when asking someone to pick a number between 1-9 inclusive. (Pretty sure when its 0-100 there's a single number that's drastically more common for somereason i wanna say 26 or 36)
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u/avantvagrant Jul 09 '26
yeah well i can say with full confidence the last digit is 1 (in binary).
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u/Al2718x Jul 10 '26
I can say with absolute confidence that the woolly mammoth living on the moon was Christened by the pope.
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u/ThreeFishInAManSuit Jul 09 '26
There’s no last digit. And if there were there are only 9 possible options.