What is a math concept or theorem that you wish there were a better explanation of?
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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.
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@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.
@Cheeseness @futurebird @Meowthias so the circumference is between the perimeter of the inside and outside hexagons.
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@Cheeseness @futurebird @Meowthias so the circumference is between the perimeter of the inside and outside hexagons.
@Cheeseness @futurebird @Meowthias the neat thing about hexagons is that they can be made up of six equilateral triangles. So, the inside hexagon (six sides) has a perimeter that's six times the radius, or 3 times the diameter. So, the circumference is more than three times the diameter.
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@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.
@SeanPLynch @futurebird @Meowthias a number system in a base of Pi is still expressible in other number systems.
The thing about number bases is that they are still expressing the same numbers, so you inevitably run into the same issue of “ratios between integers”. Math is describing something outside of the context of the counting system.
The counting system is just the tool used to express the concepts, but they aren’t intrinsic to math itself. They are one of the many symbolic vehicles we use to express and share concepts and discoveries about in math, but they aren’t math itself.
If the number system shifts as it goes along, it doesn’t change that we can express in using conventional maths, integers, and symbols in base 10. So we always return back to the “cannot be cleanly defined as a ratio between two integers”
Integer has a specific meaning and definition, that isn’t just “a digit expressed as a whole unit in a particular counting base”.You kinda get what I’m getting at? No matter the base, even one that changes the “step” as it goes along, the same fundamental constants of relationships between the geometric expressions of the radius to the circumference will never be a clean ratio between two whole Integers.
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@futurebird I'm a little nervous that if you explain it in a way that makes sense to my English major brain the universe might get unplugged.
@Meowthias @futurebird if you do fiber arts "non euclidean" is actually not that scary.
Like, you can crochet a non-euclidian shape really easily.
https://m.youtube.com/watch?v=w1TBZhd-sN0Anyways, as an artist, my answer is straight lines can approximate curves, but if you want a real curve it'll be a little bit more material than if it was just straight lines. So, pi is 3-and-a-bit because it's got curves.
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What proof are you thinking of that's like this. I tend to think a proof with "mistakes" is simply not a proof.
@futurebird I've scraped together enough maths education to realise that, but it still feels intuitive to me that we might say something like "X thought they had proved the theorum, but it turns out they made a mistake so that the thing they published that everyone thought was a proof actually wasn't", especially when the common way of explaining what a proof is is "a watertight argument".
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@futurebird I've scraped together enough maths education to realise that, but it still feels intuitive to me that we might say something like "X thought they had proved the theorum, but it turns out they made a mistake so that the thing they published that everyone thought was a proof actually wasn't", especially when the common way of explaining what a proof is is "a watertight argument".
@futurebird For example, Wiles' first proof of Fermat's last theorem. You could argue that it's not actually a proof, but I don't think you had trouble understanding what I meant when using the word.
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@futurebird I've scraped together enough maths education to realise that, but it still feels intuitive to me that we might say something like "X thought they had proved the theorum, but it turns out they made a mistake so that the thing they published that everyone thought was a proof actually wasn't", especially when the common way of explaining what a proof is is "a watertight argument".
When that has happened in math we call it a massive error in peer review. And it's generally been VERY rare compared to other areas of study.
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@futurebird For example, Wiles' first proof of Fermat's last theorem. You could argue that it's not actually a proof, but I don't think you had trouble understanding what I meant when using the word.
I would say it was not a proof since it didn't prove it? IDK maybe I've bought the math orthodoxy too much.
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I would say it was not a proof since it didn't prove it? IDK maybe I've bought the math orthodoxy too much.
@futurebird I agree with your definition, but it's my impression that "How can a proof have mistakes?" is a source of confusion to non-mathematicians. And telling them they've made a category error doesn't seem to help.
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@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.
@futurebird @rallias @Meowthias Well, the number of sides never repeats in the set of all N-sided polygons?
(Edit: why'd I say vertices?)
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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 it took me many years of teaching to get an answer I like to the first question: adding fractions is very simple, what's complicated is writing a given number as a fraction
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@futurebird it took me many years of teaching to get an answer I like to the first question: adding fractions is very simple, what's complicated is writing a given number as a fraction
@futurebird like if i give my student a quarter of an apple and a third of an apple, it's very intuitive that they now have 1/3+1/4 of an apple and that this is a completely fine way of writing that number. And once that's intuitive we can move on to the hard task of finding how to cut an apple in pieces of equal sized so that a whole number of such pieces is the same amount of apple as that first 1/3+1/4
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@Cheeseness @futurebird @Meowthias the neat thing about hexagons is that they can be made up of six equilateral triangles. So, the inside hexagon (six sides) has a perimeter that's six times the radius, or 3 times the diameter. So, the circumference is more than three times the diameter.
@Cheeseness @futurebird @Meowthias for the outside hexagon, figuring out the perimeter is a little harder. But you can use the Pythagorean Theorem on this equilateral triangle. The distance from the centre to the edge is 1/2 the diameter of the circle (by definition). That makes a triangle that splits the equilateral triangle in two. If we say the edge length is x, we have a right triangle with one side x, one side d/2, and one side x/2. We know (x/2)^2 + (d/2)^2 = x^2 so x^2 = d^2/3
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@Cheeseness @futurebird @Meowthias for the outside hexagon, figuring out the perimeter is a little harder. But you can use the Pythagorean Theorem on this equilateral triangle. The distance from the centre to the edge is 1/2 the diameter of the circle (by definition). That makes a triangle that splits the equilateral triangle in two. If we say the edge length is x, we have a right triangle with one side x, one side d/2, and one side x/2. We know (x/2)^2 + (d/2)^2 = x^2 so x^2 = d^2/3
@Cheeseness @futurebird @Meowthias that means x = d/sqrt 3.
Since the perimeter is 6x, it's 6d/sqrt 3, or 2 times the sqrt of 3. That's around 3.4.
So, we know pi is between 3 and 3.4.
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@Cheeseness @futurebird @Meowthias that means x = d/sqrt 3.
Since the perimeter is 6x, it's 6d/sqrt 3, or 2 times the sqrt of 3. That's around 3.4.
So, we know pi is between 3 and 3.4.
@Cheeseness @futurebird @Meowthias Archimedes did this with polygons up to 96 sides (!), and came up with a pretty good approximation of pi.
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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 FWIW, √2 is also irrational, so the length of the diagonal of a square has the same problem. Irrational numbers pop up everywhere in geometry, not just with circles.
(Square roots are actually a different kind of irrational than π, in the sense that π cannot be reached from the integers using algebra. Numbers like that are called "transcendental". And unlike square roots but like some other transcendental numbers, π appears to be "normal": every possible digit appears equally often in whatever base you write it out with. IIRC we haven't actually proven π is normal, because proving a number is normal is really hard.)
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@llewelly @futurebird Imagine how much better off we'd be if Kurt had the persistence to finish his theorem.
@cford @futurebird Kurt Gödel never finished his theorem, which led to Alan Turning's halting problem, which in turn led to an industry full of software whose correctness cannot be determined until after it blows up in our faces. Kurt Gödel, look what you've done!
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@Cheeseness @futurebird @Meowthias Archimedes did this with polygons up to 96 sides (!), and came up with a pretty good approximation of pi.
@Cheeseness @futurebird @Meowthias I think the math is a lot easier if you use an inscribed hexagon and an exscribed square, but I wanted to use your insight first.
