What is a math concept or theorem that you wish there were a better explanation of?
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Is every regular polygon perimeter-to-radius ratio rational?
If so, then could you show Pi is irrational by solving a polygon, adding another side, then solving it, and adding another side, so the student understands that with infinite sides, the fine adjustments go on forever?
"Is every regular polygon perimeter-to-radius ratio rational?"
Oh no no no. A triangle and a square will produce irrational ratios.
But there are two kinds of irrational numbers. Some can be represented as roots. It makes sense that the root of a square would be the ratio of the diameter of a square to the perimeter... these are numbers that go on forever like pi.
But pi is even more irrational than roots... it can't even be written using roots. It's "transcendental."
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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 well polygons are made of straight line segments
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@Meowthias @futurebird I wasn't worried about that... until now
@willyyam @futurebird You should be worried because a few of these have almost made sense.
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@futurebird @Meowthias my theory for a while now, has been that the value of pi is a result of the curvature of space - somewhere else pi might be a whole number
@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.
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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?
It's because we have ten fingers.
That's why we use base 10 numbers. It's also why numbers are called digits.
If we were intelligent sponges, or smart coral, we'd probably see quantities in some less distinct way and wouldn't run into the irrational division results.
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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 A good intuition here is that every polygon with less than infinite sides/vertices can only *approximate* the circle. There will always been a bit more circumference than you can account for with an integer number of sides… and because there is always a tiny bit that can't fit, the decimal representation of pi continues forever.
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"Is every regular polygon perimeter-to-radius ratio rational?"
Oh no no no. A triangle and a square will produce irrational ratios.
But there are two kinds of irrational numbers. Some can be represented as roots. It makes sense that the root of a square would be the ratio of the diameter of a square to the perimeter... these are numbers that go on forever like pi.
But pi is even more irrational than roots... it can't even be written using roots. It's "transcendental."
@futurebird @Phosphenes @Gustodon So, somehow adding more sides transitions in the limit from roots to transcentants?
Doesn't sound like a subject that can be "answered" simply. -
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.
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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 idk what's so complicated about adding fractions? Or substracting them even.
E.g. 49/14-25/10 = (49-25)/(14+10), easy
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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
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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.
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!
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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!
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.
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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).
