What is a math concept or theorem that you wish there were a better explanation of?
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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 How a proof is both irrefutable and can have mistakes.
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@futurebird Why e is special. I understand why, but I've never seen a good short explanation, nor do I have one.
@jmax @futurebird I think that is the nature of the understanding.
We have to work through the layers to get to the understanding.
Sometimes it stays built, and sometimes we have to rebuild it N + 1 times
There is something amazing though, when one of mine *gets* a thing.
Tuesday I had a "Memorized it all" student talking to a "reason it all" and the second one built the comprehension for dividing fractions and then MARVELED at when he just multiplied by the reciprocal without visualizing it etc... it still worked!!! -
@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?
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.
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@futurebird How a proof is both irrefutable and can have mistakes.
What proof are you thinking of that's like this. I tend to think a proof with "mistakes" is simply not a proof.
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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 what precisely constitutes proof? (I know to some degree now but remember that when we first encountered the idea in school, proofs weren’t defined, just illustrated by example.)
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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 I have a story about someone who believes the repeating .333 needs to be freed from that repetition. I wrote it because as a kid I couldn't believe that it would never end, even though it manifestly never ended. Similar sort of preoccupation.
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@futurebird How a proof is both irrefutable and can have mistakes.
@cford @futurebird I can't explain it, but I blame Kurt Gödel and the incompleteness theorem.
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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 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.
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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 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
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@futurebird what precisely constitutes proof? (I know to some degree now but remember that when we first encountered the idea in school, proofs weren’t defined, just illustrated by example.)
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"
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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 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.
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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 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.
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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.
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?
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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 I wasn't worried about that... until now
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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 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)
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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 To answer the part about running your finger around the circle: despite the fact that pi goes forever, it is clearly bounded above by 3.2 (for example; 3.15 is another bound), so if you move your finger 3.2 diameters around the circumference, you will have gotten back (and past) where you started, no infinities involved.
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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.
