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 Why e is special. I understand why, but I've never seen a good short explanation, nor do I have one.
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@futurebird
Last summer I read "Is math real?", by #eugeniacheng with great pleasure. A key takeaway for me is to ask "when", instead of "why". The question becomes more open and interesting. E.g. "when is 1+1=2?" is open for discussing why one apple plus one apple is two apples, but one pile of sand plus one pile of sand is still only one pile of sand.@futurebird
This relates to your question. For example, instead of a teacher correcting a student for saying that sqrt(x^2)=x, it is better to ask when it is true. If you already know that x is positive, you can ignore the negative solution, as we routinely do. -
@futurebird Why e is special. I understand why, but I've never seen a good short explanation, nor do I have one.
@futurebird A young lady I was trying to impress at the time asked me that question during her struggles with calculus. I would greatly have preferred an alternative to waving my hands. The desire for a simple answer has bugged me ever since.
(She was not, in fact, impressed.)
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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.
Exponential growth is growth that explodes. The rate an exponential is growing is increasing.
How fast the exponential function is growing is larger when the function is larger. The bigger x, the steeper it gets.
This is true for all kinds of exponentials with different (positive >1) bases but if you want the function where the rate of growth is *exactly* the value of the function that is e^x
That's what I think of first but I don't think it's simple enough.
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Exponential growth is growth that explodes. The rate an exponential is growing is increasing.
How fast the exponential function is growing is larger when the function is larger. The bigger x, the steeper it gets.
This is true for all kinds of exponentials with different (positive >1) bases but if you want the function where the rate of growth is *exactly* the value of the function that is e^x
That's what I think of first but I don't think it's simple enough.
@futurebird Yeah. It's not hard, but it's very resistant to extensive simplification.
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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 logarithms! -
Exponential growth is growth that explodes. The rate an exponential is growing is increasing.
How fast the exponential function is growing is larger when the function is larger. The bigger x, the steeper it gets.
This is true for all kinds of exponentials with different (positive >1) bases but if you want the function where the rate of growth is *exactly* the value of the function that is e^x
That's what I think of first but I don't think it's simple enough.
I have a math degree and did not understand e until now.
(tbf to me, I mostly studied computer-related stuff that doesn't use e.)
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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 yeah, I echo a lot of above. I could answer test questions correctly about Euler's identity but I didn't *get* it in my bones. Always felt that if I could understand it that I could UNDERSTAND.
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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 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?
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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?
This is so cool .. you guys are talking about pi so early in the morning. Makes me hungry
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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"
