What is a math concept or theorem that you wish there were a better explanation of?
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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.
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@SeanPLynch @futurebird @Meowthias In that case, though, the description of the base would go on for ever.
@darkling @futurebird @Meowthias
Yeah some kind of fractional base. Maybe a tree, or a fern, with its fractal body design, would develop some kind of weirdly based counting system that could work.
Transforming to base 10, would still give irrational pi.
Great band name, irrational pi.
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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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This 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.
@futurebird best i (and apparently anyone at this point) can do is https://crypto.stanford.edu/pbc/notes/contfrac/cheat.html and continuous fraction expansions, which might be derived from pure geometry, but probably not in a way that makes it intuitive
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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.
@SeanPLynch @futurebird @Meowthias I think there is a fundamental misunderstanding of what an irrational number is going in here. Because regardless of the base that is being used, or the counting system at play, you can’t tweak how you count to make the irrational numbers suddenly rational.
The “ratio” in rational is about how the number can be described as a ratio of two other integers. To be irrational means that it “cannot be expressed as a ratio between two integers”
Whatever base you use does not get around this. Using a base that is fractional doesn’t change the fundamental definition of “expressed as a ratio between two integers” either, it just means that it is incredibly difficult to do math because you have to express things in complicated addition and subtraction chains to represent a whole integer.
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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 so, the short answer is, the more sides to an even-sided regular polygon that you have, the closer and closer you reach to a limit of the ratio between the distance between two oppos and corners and sum of side lengths. A circle is functionally an infinitely sided regular polygon. And so, with an infinitely sided regular polygon, the ratio of the distance between two opposing corners and the sum of the length of the sides happens to be that limit. That limit happens to be pi.
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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!
@SeanPLynch@mastodon.social @futurebird@sauropods.win @Meowthias@mastodon.world skeletal muscular biology has definitely impacted how we perceive the world 🧽 and therefore our mathematics
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@Meowthias @futurebird I've never seen an intuitive or visual proof that pi is irrational.
@Meowthias @futurebird an aside: we watched the film "Train Dreams" last night. There's one scene where the couple are discussing whether a puppy or a baby of the same age is smarter. And they come up with some pretty convincing theories about it, based on evidence they'd seen with their own eyes -- how independent a puppy can be after weaning, how dependent a baby is even when it can walk and talk.
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@johnzajac @leadegroot @futurebird @Meowthias I was talking about mathematical spaces; physical ones are not relevant to the technical definition of pi.
@khleedril @leadegroot @futurebird @Meowthias
So my much smarter husband just told me
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@Meowthias @futurebird if we lived in a simulation, somewhere, somehow, pi would be found to repeat, terminate, or crash the simulation with an unhandled floating point exception.
@llewelly @futurebird Thank you. I only understood half of this but the half I did understand is vaguely reassuring.
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@Meowthias @futurebird an aside: we watched the film "Train Dreams" last night. There's one scene where the couple are discussing whether a puppy or a baby of the same age is smarter. And they come up with some pretty convincing theories about it, based on evidence they'd seen with their own eyes -- how independent a puppy can be after weaning, how dependent a baby is even when it can walk and talk.
@Meowthias @futurebird it made me think about how science has crossed from rational examination and experimentation with our normal everyday sense experiences to extremely specialized equipment and methodologies. The question of whether puppies or babies have greater intelligence would be answered very differently in 2026 than in 1920, the setting of the film.
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@cford @futurebird I can't explain it, but I blame Kurt Gödel and the incompleteness theorem.
@llewelly @futurebird Imagine how much better off we'd be if Kurt had the persistence to finish his theorem.
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@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?
@independentpen @futurebird @Meowthias
"How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?"
[Albert Einstein]
