What is a math concept or theorem that you wish there were a better explanation of?
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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.
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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.
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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
