What is a math concept or theorem that you wish there were a better explanation of?
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@leadegroot @futurebird @Meowthias While you can find curved spaces in which the ratio of diameter to circumference is different (like exactly 3, or even 4), the definition of pi is that it is the ratio specifically of a circle in a flat space.
@khleedril @leadegroot @futurebird @Meowthias
So it's a category error, since any time you're experiencing gravity of any strength at all you're within curved space?
Essentially, Pi is not infinite somewhere not influenced by the Great Attractor. *If* space itself isnt curved by nature, which is an open question
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Pi goes on forever because if you take the diameter of a circle and try to wrap it around the circle there is no simple ratio between these lengths.
Now why isn't there a simple ratio? With a hexagon the diameter fits three times. So, why can't exactly three diameters make up the circumference of a circle?
I'm thinking about how to answer this without just going "it's Euclidian space" which isn't a real explanation.
Maybe someone else can help here.
@futurebird
@Meowthias a first, usually non satisfying answer: if you pick a number uniformly between 3 and 4 (which is easy to show that's where pi lives), the probability of landing on a rational number (or even an algebric irrational like sqrt(11) is 0), so for pi to be irrational was very likely. And now I'm trying to think of a more satisfying answer before looking up what others said
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Pi is still irrational in other bases, though. Because if you have a circle and flatten it out, and you have the diameter of that circle and you make exact copies of these two lengths and lay them side by side one line of diameters and one line of repeated circumferences they will never ever ever ever perfectly match up no matter how many you lay down.
It's like the lengths come from two incompatible lego sets. There's no ratio to make them perfectly even.
But if you don't care about "perfect" 22 diameters will match up almost perfectly with 7 circumferences.
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Think about the sponges you were posting about a few days ago ...
If they were intelligent they wouldn't use base 10 because they don't have 10 digits (fingers).
Sponges might develop some way of counting quantities that wasn't based on distinct numbers, but was more fluid and could handle irrational division.
We are trapped in our 'digital' world by our own biology!
Using base 6 (ants?), or base 2 (binary), or base 16 (hexadecimal) doesn't help the pi issue because you still get an irrational ratio.
The distinct digits of any rational number set will always produce an irrational pi.
So maybe something that is more fluid in its own biology would develop a math where pi would not go on forever.
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@futurebird
@Meowthias a first, usually non satisfying answer: if you pick a number uniformly between 3 and 4 (which is easy to show that's where pi lives), the probability of landing on a rational number (or even an algebric irrational like sqrt(11) is 0), so for pi to be irrational was very likely. And now I'm trying to think of a more satisfying answer before looking up what others saidThis makes sense but we know circles are important and not just "random" so I think that's why this fails to feel like it really explains it.
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Pi is still irrational in other bases, though. Because if you have a circle and flatten it out, and you have the diameter of that circle and you make exact copies of these two lengths and lay them side by side one line of diameters and one line of repeated circumferences they will never ever ever ever perfectly match up no matter how many you lay down.
Yes, that's why I mentioned sponges.
You'd want something that isn't going to count in distinct digits.
Like 10 for us, 8 for an octopus, maybe 6 for an insect?
You'd want something with no digits.
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@futurebird idk what's so complicated about adding fractions? Or substracting them even.
E.g. 49/14-25/10 = (49-25)/(14+10), easy
@IngaLovinde @futurebird That's a good one
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Pi goes on forever because if you take the diameter of a circle and try to wrap it around the circle there is no simple ratio between these lengths.
Now why isn't there a simple ratio? With a hexagon the diameter fits three times. So, why can't exactly three diameters make up the circumference of a circle?
I'm thinking about how to answer this without just going "it's Euclidian space" which isn't a real explanation.
Maybe someone else can help here.
For Pi, there is the "proof for 5-year-olds": Putting little boxes in a circle until it is completely full because the free spaces are so small that they can't be seen anymore.
And if you spin that idea further, zooming into those empty spaces, you will see zones that are maybe nice straight lines on one or two sides, but a little curvy thing on the remaining side. Which doesn't go away no matter how far you zoom in (don't forget to bring really small boxes).
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@khleedril @leadegroot @futurebird @Meowthias
So it's a category error, since any time you're experiencing gravity of any strength at all you're within curved space?
Essentially, Pi is not infinite somewhere not influenced by the Great Attractor. *If* space itself isnt curved by nature, which is an open question
@johnzajac @leadegroot @futurebird @Meowthias I was talking about mathematical spaces; physical ones are not relevant to the technical definition of pi.
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@IngaLovinde @futurebird That's a good one
'good'
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@futurebird I would like an explanation for why pi goes on forever. Is it evidence we are living in a simulation? Is it because if you trace the circumference of a circle with your finger you never reach a beginning or an end? Is it a message from the gods?
@Meowthias @futurebird this isn't easy or intuitive! The key property is that pi can't be represented as a fraction or ratio, a/b. If it could, its decimal representation would eventually stop (a = all the digits, b = 10^number of digits). But it can't, so they don't.
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@khleedril @leadegroot @futurebird @Meowthias
So it's a category error, since any time you're experiencing gravity of any strength at all you're within curved space?
Essentially, Pi is not infinite somewhere not influenced by the Great Attractor. *If* space itself isnt curved by nature, which is an open question
@johnzajac @khleedril @leadegroot @futurebird @Meowthias If you actually *measured* a circle in that kind of space, then yes, you'd get different answers. (Note that you probably can't measure beyond a few digits of precision, though, so it's a pretty pointless approach).
However, the "standard" (Euclidean) geometry that we work with in maths isn't like that, and it's in *that specific geometry* that we have the result about the ratio of circumference to diameter being transcendental.
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Pi is still irrational in other bases, though. Because if you have a circle and flatten it out, and you have the diameter of that circle and you make exact copies of these two lengths and lay them side by side one line of diameters and one line of repeated circumferences they will never ever ever ever perfectly match up no matter how many you lay down.
@futurebird @SeanPLynch @Meowthias how does a mathematician know such a thing? ... that they will never match up? Is it because a repeating pattern is found? But I thought pi does not repeat?
But wait how can we be sure that pi never will repeat?
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@futurebird I'm sorry if this question is boring but I'm a simpleton.
Can you "fool" pi with a circle that is distinctly a shape with 360 sides? I remember making clocks with LOGO and some of the circle discussions were interesting.
@Gustodon @futurebird You can get arbitrarily close to pi with shapes with a large number of sides, yes. In fact, this is how Archimedes is famed to have gone about his calculations. (Though I'm not sure that is actually true.) Not all regular N-gons will have nice formulas for their perimeter or area, though.
It's not the best way to approximate pi but it could be done. There are far better ways based on infinite series. (The Taylor series expansion of the Gamma function being one of them.)
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Using base 6 (ants?), or base 2 (binary), or base 16 (hexadecimal) doesn't help the pi issue because you still get an irrational ratio.
The distinct digits of any rational number set will always produce an irrational pi.
So maybe something that is more fluid in its own biology would develop a math where pi would not go on forever.
@SeanPLynch @futurebird @Meowthias In that case, though, the description of the base would go on for ever.
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@Meowthias @futurebird this isn't easy or intuitive! The key property is that pi can't be represented as a fraction or ratio, a/b. If it could, its decimal representation would eventually stop (a = all the digits, b = 10^number of digits). But it can't, so they don't.
@Meowthias @futurebird why is pi irrational, that is, can't be represented as a fraction? That was not clear for a long time. People kept doing rational approximations, and they weren't exactly pi. So they started guessing that it might be irrational.
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Pi goes on forever because if you take the diameter of a circle and try to wrap it around the circle there is no simple ratio between these lengths.
Now why isn't there a simple ratio? With a hexagon the diameter fits three times. So, why can't exactly three diameters make up the circumference of a circle?
I'm thinking about how to answer this without just going "it's Euclidian space" which isn't a real explanation.
Maybe someone else can help here.
The explanation isn’t in the math, it’s in us… I would reverse the question:
Why *should* there be a neat and tidy ratio?
Because it would be satisfying to our very peculiar little minds. Humans find simple relationships and tidy explanations very rewarding, we look for and will try to force story and relationship onto things that don’t have them… that’s about human psychology, not how the universe works.
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@Meowthias @futurebird why is pi irrational, that is, can't be represented as a fraction? That was not clear for a long time. People kept doing rational approximations, and they weren't exactly pi. So they started guessing that it might be irrational.
@Meowthias @futurebird the first proof came in 1764 from Johann Lambert. He showed that if a number were non-zero and rational, its tangent was irrational. Because we know that the tangent of pi/4 is 1, then pi/4 can't be rational, so pi can't be rational. The first part is kind of hard, though!
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What is a math concept or theorem that you wish there were a better explanation of?
It could be from arithmetic: Why is adding fractions so complicated?
From grade-school algebra: Why does the teacher get so sad and angry if I just √(x²+y²)=x+y
From the calculus: Why do I need to write dx with the integral?
or beyond.
@futurebird I love how you start with "fractions are hard" and then next thing you know you're in deep discussions about pi. :
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@Meowthias @futurebird the first proof came in 1764 from Johann Lambert. He showed that if a number were non-zero and rational, its tangent was irrational. Because we know that the tangent of pi/4 is 1, then pi/4 can't be rational, so pi can't be rational. The first part is kind of hard, though!
@Meowthias @futurebird I've never seen an intuitive or visual proof that pi is irrational.
