Pi Unraveled: Why It's Forever Irrational, Thanks to Lambert
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- Опубликовано: 2 окт 2024
- We think pi is probably a normal number, but we cannot yet prove it merely by looking at the frequency of the digits and strings of digits that appear in its decimal expansion. You might think that similarly, although we think pi is irrational since it doesn't happen to repeat through trillions of digits, maybe at some point it will start?
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It turns out, however, that that's not the case, and we've known for several hundred years now. Johann Lambert first proved that pi is irrational in the mid-18th century. And although the details are a little tough to work out in a 60-second short, the gist of it is that Lambert showed a particular infinite continuing fraction had to produce an irrational output if its input was rational (and non-zero). Contrapositively, if the output of that fraction was rational, that had to mean the input was irrational.
Conveniently, that infinite continuing fraction is equivalent to the tangent of the input. And since we know that various fractions of pi have tangent values of 1, we know it would produce a rational output in the infinite continuing fraction form. Therefore, that fraction of pi (π/4) must be irrational. And since we know it's not the 4 making it irrational, it must mean that pi is irrational.
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I understood all those words separately 😅
That's really funny! 🤣🤣🤣
There's nothing to understand as there's no proof.
He just gave a tool (the nested fraction) which can be used to approximate a value (here it's the tangent function).
There it's the premise that a rational x (except 0 as 0 divided by whatever value is 0) gives an irrational approximation and an irrational x gives a rational value.
Therefore if π was rational, we would get an irrational result from the approximation.
Great explanation!
Honestly I would have rather you proved that your statement wasn't circular, which is the argument of the comment. She didn't say "nuh-uh", she didn't even explicitly disagree with your conclusion, she simply said that your argument was bad. You can be right in your conclusion with bad arguments.
a circular argument would have been "she was wrong because pi is irrational, and we know pi is irrational because she was wrong." that is not the argument i made. i argued "pi is an irrational number, irrational numbers can't end in infinitely many 8s, therefore pi does not end in infinitely many 8s." in this video, i explained a bit more about how we justify the first premise in that argument, separate from just noticing that pi goes on for a really long time without repeating.
@@polymathematicI'm by no means qualified to refute any of this, but I'm genuinely curious how a computer actually interprets pi in a given equation. I assume it must depend on the software (and to some extent the hardware) how many decimal places it counts in the moment, because it breaks my brain to consider that it can make a calculation based on an infinite string in a finite time period. That would be incredibly cool, but I wonder if any uncertainty is introduced due to the infinite variability of pi as an integer string (my coding knowledge in context only goes about as far as knowing what an integer string is 😅)
@@0v_x0 i think most of the time it's just stored as a constant to some pre-determined level of precision. but there are infinite series you can use to calculate to any desired level of precision beyond that.
Good job, im not a big math guy but i watched until the end. Keep posting!
Poor explanation. You never answered the question, you simply stated we know it’s not because this random equation says it’s not (without explaining the logic of the equation or how it’s connected to pi)
2/10
This comment leaves a lot to be desired. It doesn’t take into account the limitations of a one-minute short. It misses the key connection between the continuing fraction, the tangent function, and tan(π/4)=1.
1/tau.
Math is weird man
Well explained!
Yes. Exactly. Yes.
I've been saying this for years! Well, at least for days...
It is too much for my little brain
Here’s a limerick: Lambert, you’re a
I may be wrong, but I thought that pi was an infinite series…..you can always add another term to it, even if the precision it gives you is so insignificant it would be smaller than the plank length on a real circle
Yes.
Any irrational number can be written as an infinite series. That doesn’t really mean anything in itself
Every number can be written as an infinite series.
Never saw this proof before. Interesting! Is there a proof that shows no p/q =pi?
Or is this the easiest way
I don't think there is a proof like that because π is a transcendental number that means it can not be expressed algebraically; unlike √2 and the Golden Ratio where a/b proof works
This is the first way, and it was proven by lambert in 1761. But we've known about pi for thousands of years. The reason it took so long is that it's a tough thing to prove. But now that we have more mathematical tools (eg calculus) we have come up with more proofs over time.
The channel mindyourdecisions has a proof that pi is irrational that proves it directly, but it's harder to understand than this proof.
This short does a really great overview of lamberts proof. For a more in depth view of it, check out mathologers video
They exist! Mathologer talked about what is perhaps the "simplest" proof of this form (though it requires, among other things, integration by parts twice!).
what if pi does have end, how do we know it goes on for infinite, im genuinely curious
If it had an end, then it could be re-written as a fraction making it a rational number. This video showed that it is not a rational number hence it is irrational. In other words, it has no end.
He just talked about this.
Mr. Panty dropper!
ok
this taught me nothing
Blind men can’t see how well you paint their houses.
it gives 0.0137 irrational so you are still wrong!