What is a math concept or theorem that you wish there were a better explanation of?
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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]
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@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.
@Meowthias @futurebird I bring it up because of this question of pi's irrationality. I did physics as an undergraduate, which requires a lot of math, and I can kind of follow along with some of the proofs in this article. But they're definitely not gut level, and I don't come away with an intuitive sense of *why*.
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@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.
@dvandal @SeanPLynch @futurebird @Meowthias
I went looking on the 'tubes for a "simple proof that pi is irrational." This https://math.mit.edu/~poonen/papers/pi_irrational.pdf is the shortest one I found.
YMMV
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@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.
@SeanPLynch @futurebird @Meowthias Unless some component of that fractional base is itself related to pi (by a rational multipler), you're still going to end up with an infinite-length description of pi.
If you go for a multi-component base with non-transcendental components (say, the first digit is base 5, the second digit is base 3/2, the third is base sqrt(13), ...), then you'd still not be able to describe pi in a finite number of digits, even if your base has an infinite description.
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@Meowthias @futurebird I bring it up because of this question of pi's irrationality. I did physics as an undergraduate, which requires a lot of math, and I can kind of follow along with some of the proofs in this article. But they're definitely not gut level, and I don't come away with an intuitive sense of *why*.
@Meowthias @futurebird maybe part of the tradeoff of getting to know these facts is having specialists who dig very deeply into an area, such that they can tell us what they learned, but they can't exactly communicate why it's true. And we can't just chat about it over the campfire.
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@Meowthias @futurebird maybe part of the tradeoff of getting to know these facts is having specialists who dig very deeply into an area, such that they can tell us what they learned, but they can't exactly communicate why it's true. And we can't just chat about it over the campfire.
"And we can't just chat about it over the campfire."
I always take this as a challenge.
"watch me cook!"
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@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.
@dvandal @futurebird @Meowthias
Yes, that's why I first mentioned sponges.
We'd need something without distinct digits to develop a 'math' not based on distinct set of counting numbers. A non-real number system. Something more fluid.
It's not a matter of choosing a different base. Even choosing pi as your base won't help.
I like our math, and its unreasonable effectiveness...
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"And we can't just chat about it over the campfire."
I always take this as a challenge.
"watch me cook!"
@futurebird @Meowthias do it! I hope you can.
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People will try to blow this up into something much more complex but a proof is simply a convincing and correct *deductive* argument. It's a series of sentences (logical statements such as "If A then B") that you string together to justify a more concise and useful statement. "The sum of the interior angles of parallel lines is 180"
@futurebird that leads some people to say that logic (first order and higher order) are not part of maths, but is the language that allows maths to be done.
@jtnystrom -
@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.
@rallias @futurebird @Meowthias Would it be fair then to say, that the "infinite precision" of Pi could be read as a direct consequence of trying to calculate the ratios of an "infinite set" - that is, the set of all N-sided polygons?
That would make some sense as an explanation, to me.
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@dvandal @futurebird @Meowthias
Yes, that's why I first mentioned sponges.
We'd need something without distinct digits to develop a 'math' not based on distinct set of counting numbers. A non-real number system. Something more fluid.
It's not a matter of choosing a different base. Even choosing pi as your base won't help.
I like our math, and its unreasonable effectiveness...
@SeanPLynch @dvandal @Meowthias
Pi is defined as a ratio and the irrationality is a property of the ratio. I'm having trouble knowing how you could somehow have pi and it didn't have that property.
You could have some other notion of calculating where this didn't come up... but then you'd never define pi.
You see, to me, that pi is irrational is so intrinsic to what it is in a Euclidean space that I don't think it'd be "pi" anymore if it didn't have that property.
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@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.
@dvandal @futurebird @Meowthias
Correct on using a fractional base!
I had to think it through. Making a base pi will make pi rational, but you'll get irrational results for so many things.
Won't work with any base where the digits are distinct units apart.
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@rallias @futurebird @Meowthias Would it be fair then to say, that the "infinite precision" of Pi could be read as a direct consequence of trying to calculate the ratios of an "infinite set" - that is, the set of all N-sided polygons?
That would make some sense as an explanation, to me.
@seachaint @rallias @Meowthias
That explains why it *could* go on forever. It explains why it's possible to have an irrational number that isn't a nice ratio of integers... but it doesn't show that whatever process you use to estimate pi won't at some point down the line just start repeating.
You can define 1/3 as an infinite process too.
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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.
It's easier to reason about the area of a flat surface than the length of a curve, so instead of the circumference of a circle of diameter 1, let's look at the area of a circle (technically a disc) of radius 1.
If you truncate π at any digit, say to 3.141, then it's possible to construct a polygon (even a regular polygon inscribed in the circle) whose area is greater than 3.141 even though it fits entirely within the disc. If instead you truncate and round up, say to 3.142, then it's possible to construct a polygon (even a regular polygon circumscribed around the disc) whose area is less than 3.142 even though it entirely contains the disc. Therefore the area of the disc is between all of those rounded-down quantities and all of those rounded-up quantities, which ultimately is what it means to say that the area is given by this infinite sequence of digits.
Of course, this is only more true if you take a terminating decimal that isn't even an approximation of π; you can use the same polygon as you would for the rounded-down or rounded-up approximation (as appropriate) with the same number of digits. It's because every terminating decimal gives us an area that is either too small or too large that π cannot be equal to any of them.
But don't ask me to prove this; it's actually hard prove it! Archimedes used all of his power just to find the areas of 96-sided polygons and demonstrate that π is between 223⁄71 and 22⁄7, which in decimals only gets you as far as 3.14+. Lambert proved it in 1761 using a continued-fraction expansion of the tangent function, which doesn't have much to do with the area of anything.
…
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It's easier to reason about the area of a flat surface than the length of a curve, so instead of the circumference of a circle of diameter 1, let's look at the area of a circle (technically a disc) of radius 1.
If you truncate π at any digit, say to 3.141, then it's possible to construct a polygon (even a regular polygon inscribed in the circle) whose area is greater than 3.141 even though it fits entirely within the disc. If instead you truncate and round up, say to 3.142, then it's possible to construct a polygon (even a regular polygon circumscribed around the disc) whose area is less than 3.142 even though it entirely contains the disc. Therefore the area of the disc is between all of those rounded-down quantities and all of those rounded-up quantities, which ultimately is what it means to say that the area is given by this infinite sequence of digits.
Of course, this is only more true if you take a terminating decimal that isn't even an approximation of π; you can use the same polygon as you would for the rounded-down or rounded-up approximation (as appropriate) with the same number of digits. It's because every terminating decimal gives us an area that is either too small or too large that π cannot be equal to any of them.
But don't ask me to prove this; it's actually hard prove it! Archimedes used all of his power just to find the areas of 96-sided polygons and demonstrate that π is between 223⁄71 and 22⁄7, which in decimals only gets you as far as 3.14+. Lambert proved it in 1761 using a continued-fraction expansion of the tangent function, which doesn't have much to do with the area of anything.
…
In principle one could turn this proof into an algorithm that takes any terminating decimal and gives you the number of sides of either an inscribed polygon with a greater area or a circumscribed polygon with a smaller area (and tells you which you have), although it doesn't seem well adapted to that. Still, it's because those polygons exist that π is either greater than or less than any given terminating decimal number. (And this works in any base, not just base 10: π is irrational.)
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People will try to blow this up into something much more complex but a proof is simply a convincing and correct *deductive* argument. It's a series of sentences (logical statements such as "If A then B") that you string together to justify a more concise and useful statement. "The sum of the interior angles of parallel lines is 180"
@futurebird @jtnystrom It's fairly important to note that there are always some (usually fairly simple) assumptions down at the bottom of everything. Like a+b is the same as b+a.
You can't dispose of or prove those assumptions. (Well, you can, but always by making others that you derive the original ones from).
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@SeanPLynch @futurebird @Meowthias Unless some component of that fractional base is itself related to pi (by a rational multipler), you're still going to end up with an infinite-length description of pi.
If you go for a multi-component base with non-transcendental components (say, the first digit is base 5, the second digit is base 3/2, the third is base sqrt(13), ...), then you'd still not be able to describe pi in a finite number of digits, even if your base has an infinite description.
@darkling @futurebird @Meowthias
What if, when our ancestors started to count, they decided...
Hey that middle finger is 'one' thing. The ones to the right and left of it are each almost 'one' thing.
That little finger is 2/3 of a thing.
The thumb is the same length as the little one, but it's fatter, so we'll call that 3/4 of a thing...
What would math be like?
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@futurebird @jtnystrom It's fairly important to note that there are always some (usually fairly simple) assumptions down at the bottom of everything. Like a+b is the same as b+a.
You can't dispose of or prove those assumptions. (Well, you can, but always by making others that you derive the original ones from).
@futurebird @jtnystrom In some cases, there are assumptions that lead to a load of interesting things -- like, if you've got a bunch of sets, you can make a new set by taking one thing from each one. There's whole branches of maths that only work if you assume that (it's the Axiom of Choice).
In some cases, you can even *delete* an assumption, and you get interesting things -- that's what non-Euclidean geometry is: the Geometry Expanded Universe when you remove Euclid's fifth axiom.
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@futurebird @Meowthias I don't think I have much that can help, but I feel like it's important to note that a regular hexagon doesn't have a consistent "diameter" (distance between two opposing corners is not equal to the distance between two opposing sides)
@Cheeseness @futurebird @Meowthias this is extremely important and it's how Archimedes did the first approximations of pi!
If you take a hexagon whose corner-to-corner length is the same as the diameter of a circle, it will fit inside the circle. It's "inscribed". Its perimeter is smaller than the circumference of the circle.
If you take another hexagon whose side-to-side length is the same as the diameter of the circle, it's entirely outside the circle. Exscribed! Its perimeter is bigger.
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@darkling @futurebird @Meowthias
What if, when our ancestors started to count, they decided...
Hey that middle finger is 'one' thing. The ones to the right and left of it are each almost 'one' thing.
That little finger is 2/3 of a thing.
The thumb is the same length as the little one, but it's fatter, so we'll call that 3/4 of a thing...
What would math be like?
@SeanPLynch @darkling @Meowthias
You could have a math where this thing that vexes us "why is pi irrational" isn't an important question I think. But if you dug deep it'd still be there.
