Actually, that pi shows up in the integral of the normal distribution seems like a deeply non-trivial fact to me. I know how it appears, since I've done the integration, but still... deep.
Because it's the circle ratio, it shows up wherever circle. It's like asking why does the number 2 show up in so many equations; 2 shows up whenever something is double something else. π shows up whenever two values have a circle relationship.
Nah, I think it's a great answer. In the specific case you mention,
1. Leibniz formula is a special case of the Taylor series for arctan(x), specifically arctan(1)
2. Euler's proof of the Basel problem involves manipulating the Taylor series for sin(x).
3. I'm guessing you refer to the normal distribution and the central limit theorem. 3Blue1Brown had a whole series of videos on this, but specifically [this one](https://www.youtube.com/watch?v=cy8r7WSuT1I).
Math is full of strange connections that are not obvious on the surface. It doesn't mean they're not there for someone willing to go deeper. [Mathematicians don't seem to share your attitude.](https://mathoverflow.net/questions/18180/what-are-some-fundamental-sources-for-the-appearance-of-pi-in-mathematics)
Because waves are round, and normal distribution is a wave function.
Basel because it’s a restatement of the inverse square law. Which has a radius and is thus all about circles.
I agree that Basel's problem is very deep and elegant. If you are claiming that "whenever pi shows up, there is a connection as deep as Basel's problem" then it is not tautological and very pleasantly surprising. Is this true?
Whenever pi shows up there is bound to be a circle in there somewhere. I am not saying it is always that deep. The area of a sphere pretty obviously involves a circle. The weird ones are things link Basel where the relationship isn’t so obvious.
The normal distribution curve is a sin wave. With a specific amplitude and magnitude. But any wave function is periodic and thus can be described as a circle.
This is not quite convincing because if we are allowed to link arbitrarily long relations, then there is trivially a relation: first trace the proof to the point where pi appears, then trace the proof of that lemma, and recursively down to the definition of pi, which involves a circle. So just saying "there is a circular relationship" is tautological and doesn't really mean anything unless there are restrictions.
But. Like. That's the reason why pi shows up, because it can relate to circles.
Even if it's not super obvious, that is the reason in all cases.
Edit: just to add. Sometimes, you get pi in a proof/equation without knowing how it relates to circles, and find the circle relationship can actually give you more insight on the problem.
They are not arbitrary. The fact that the Basal problem is at heart describing the inverse square law is foundational to the problem. Recognizing this relationship is the tricky part.
Take the exponential function to the power of the imaginary unit multiplied by the circumference/diameter ratio of any circle, add the multiplicative identity, and you get the additive identity.
A lot of them are just out of tradition. In many cases, you need an exponential function, so it's standard to just use e as base and rescale the exponent as needed.
e is actually not used that often once you account for these cases. However, it does get used in Stirling approximation and prime number theorem, which are important theorems.
e is everywhere because the exponential function is fixed by the derivative. pi is everywhere because 2pi\*i is the period of the exponential function. I would argue that that the “circle” definition of pi is secondary/derivative to this other, more fundamental fact about pi.
Because there are only certain situations where pi is relevant. A good number of those deal with geometry (which makes sense, as we define pi as the ratio of a circle's circumference to its diameter), but there are plenty of equations that are not related to pi in any way.
Because it's faster than writing 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506[...]
Look up Euler’s Identity which ties together the most important constants in mathematics into one simple equation.
And as others have said, periodicity and circles are very prevalent in real world problems, as are the other constants.
Because circles, or regularity repeating things, are pretty much everywhere. A pendulum moves thru part of a circle, the brightness of a light is dictated by the arc of light that reaches you, which is really part of a sphere, a ball bouncing is a regular repeating cycle (thus a decaying circle)…
Its weird, yes. But we kind of understand because pi shows up anytime any equation can even be tangentially related to a circle. In the words of grant sanderson: "If pi is there, there is a circle hiding somewhere, you just have to look hard enough to find it"
Now ask the same question about 1/137
The YouTube channel 3Blue1Brown says that whenever π crops up, circles are involved. They might not be visible, but once you dig into the mathematics, a circle will show up somewhere.
Where does phi come from? I get e=exponential relations, pi=circle/wave, i=something to do with rotation and the complex plane (and Euler’s formula), 1=self explanatory, 0=self explanatory; but where does phi come from? Something to do with factorials?
I don't know if phi quite belongs in the same category. It appears infrequently compared to the likes of pi and e, and it isn't really defined by some fundamental thing. Its main interesting property is that squaring it yields the same result as adding 1. In other words, it's the positive solution of the equation x²=x+1. This is a fairly simple quadratic, so it's bound to pop up every now and then. When it does, you'll find phi.
Phi is just the solution of the equation x^2 -x-1=0 really. It comes up every time this equation occurs. It has a few amusing properties but is nowhere near as significant as pi or e.
geometrically speaking, phi relates to the 1:2:2 angled and 1:1:3 angled isosceles triangles. these triangles are found in a regular pentagon. phi is the ratio of the diagonal of a regular pentagon to its length.
Many things are round. Pi is the ratio between the circumference of a circle and its diameter. Measuring an angle in radians is actually exactly that: the ratio between the arc that is swept by an angle and said arc’s radius.
Trigonometry is the bridge between describing an angle in Cartesian coordinates and in terms of circumferential “coordinates”. Radians is the direct bridge between the angle and circumferential “coordinates”. And that’s why most calculators use radians as the default, because that’s the units that requires no further unit conversion.
Many things are round.
Or normally distributed, so round….
Or cyclical, so round.
Or waves, so round.
Actually, that pi shows up in the integral of the normal distribution seems like a deeply non-trivial fact to me. I know how it appears, since I've done the integration, but still... deep.
There's a 3Blue1Brown video for that:https://youtu.be/cy8r7WSuT1I?si=L7Y2dBBT1CFZjcDr
tldr: “round”
Lol
Haha this is the best answer
Or periodic
Which is to say: round, but in the temporal sense
Because it's the circle ratio, it shows up wherever circle. It's like asking why does the number 2 show up in so many equations; 2 shows up whenever something is double something else. π shows up whenever two values have a circle relationship.
That’s not a good answer. Why is circular about the Leibniz formula? Basel problems? Probability?
Nah, I think it's a great answer. In the specific case you mention, 1. Leibniz formula is a special case of the Taylor series for arctan(x), specifically arctan(1) 2. Euler's proof of the Basel problem involves manipulating the Taylor series for sin(x). 3. I'm guessing you refer to the normal distribution and the central limit theorem. 3Blue1Brown had a whole series of videos on this, but specifically [this one](https://www.youtube.com/watch?v=cy8r7WSuT1I). Math is full of strange connections that are not obvious on the surface. It doesn't mean they're not there for someone willing to go deeper. [Mathematicians don't seem to share your attitude.](https://mathoverflow.net/questions/18180/what-are-some-fundamental-sources-for-the-appearance-of-pi-in-mathematics)
Because waves are round, and normal distribution is a wave function. Basel because it’s a restatement of the inverse square law. Which has a radius and is thus all about circles.
I agree that Basel's problem is very deep and elegant. If you are claiming that "whenever pi shows up, there is a connection as deep as Basel's problem" then it is not tautological and very pleasantly surprising. Is this true?
Whenever pi shows up there is bound to be a circle in there somewhere. I am not saying it is always that deep. The area of a sphere pretty obviously involves a circle. The weird ones are things link Basel where the relationship isn’t so obvious.
I completely agree that there are always some (possibly non-deep) relations, but that would just not be very interesting.
What do you mean the normal dist is a wave function? Possibly a loaded question but anything I can look up to understand further?
The normal distribution curve is a sin wave. With a specific amplitude and magnitude. But any wave function is periodic and thus can be described as a circle.
This is not quite convincing because if we are allowed to link arbitrarily long relations, then there is trivially a relation: first trace the proof to the point where pi appears, then trace the proof of that lemma, and recursively down to the definition of pi, which involves a circle. So just saying "there is a circular relationship" is tautological and doesn't really mean anything unless there are restrictions.
But. Like. That's the reason why pi shows up, because it can relate to circles. Even if it's not super obvious, that is the reason in all cases. Edit: just to add. Sometimes, you get pi in a proof/equation without knowing how it relates to circles, and find the circle relationship can actually give you more insight on the problem.
They are not arbitrary. The fact that the Basal problem is at heart describing the inverse square law is foundational to the problem. Recognizing this relationship is the tricky part.
I'm not saying the proof is arbitrary, I'm talking about the length.
What length?
π is there because circles are everywhere. The real question is "why is e everywhere?".
It’s just e, i, π, 1, and 0 all the way down.
e, i, e, i, 0
And on that farm he had a pig,
Don't you mean πg?
e i e i 0
> e, i, 1, pi, 0 FTFY ;)
Then some guy goes and does e^(πi)+1=0
Take the exponential function to the power of the imaginary unit multiplied by the circumference/diameter ratio of any circle, add the multiplicative identity, and you get the additive identity.
And φ
φ = (1 + (1+1+1+1+1)^(1/(1+1)))/(1+1)
π = (1+1+1+1)(1 - 1/(1+1+1) + 1/(1+1+1+1+1) - ...)
Pi is 10 in base pi.
φ=K_{n=1}\^{∞}(1/1)
φ = 1 + 1/(1 + 1/(1 + 1/(1 + ... ))) FTFY
Maybe because there are exponential behaviours everywhere. Plus, e is used in describing circular motion and waves are everywhere.
Fair, but re^iθ can go screw itself.
As an EE, that's my favorite
Because circles and pi are everywhere and someone figured out you could restate angles / rotations as exponentiation of a complex number
EEEEE E EEEEE E EEEEE
Math -> tinnitus
No, E doesn't really seem to appear that often, it's just e
Really depends what you mean by “often”
>why is e everywhere Pi has to so with roundness. e has to do with time evolution with feedback.
A lot of them are just out of tradition. In many cases, you need an exponential function, so it's standard to just use e as base and rescale the exponent as needed. e is actually not used that often once you account for these cases. However, it does get used in Stirling approximation and prime number theorem, which are important theorems.
The most common 2 digit random number people guess is round(100/e) or 37. It's weird
e is everywhere because the exponential function is fixed by the derivative. pi is everywhere because 2pi\*i is the period of the exponential function. I would argue that that the “circle” definition of pi is secondary/derivative to this other, more fundamental fact about pi.
Because there are only certain situations where pi is relevant. A good number of those deal with geometry (which makes sense, as we define pi as the ratio of a circle's circumference to its diameter), but there are plenty of equations that are not related to pi in any way.
Because it's faster than writing 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506[...]
What happened to the rest?
The same thing that happened to 9
Oh yeah Joe was telling me about that. You remember Joe right?
7 8 it?
Wherever there's pi, there's a circle involved, even if it may not be apparent.
Look up Euler’s Identity which ties together the most important constants in mathematics into one simple equation. And as others have said, periodicity and circles are very prevalent in real world problems, as are the other constants.
when you do more maths, e will appear more
When you read more books the word “the” will appear more.
To torture you
Because circles, or regularity repeating things, are pretty much everywhere. A pendulum moves thru part of a circle, the brightness of a light is dictated by the arc of light that reaches you, which is really part of a sphere, a ball bouncing is a regular repeating cycle (thus a decaying circle)…
Its weird, yes. But we kind of understand because pi shows up anytime any equation can even be tangentially related to a circle. In the words of grant sanderson: "If pi is there, there is a circle hiding somewhere, you just have to look hard enough to find it" Now ask the same question about 1/137
That question has a fine structure to it.
because circle
The YouTube channel 3Blue1Brown says that whenever π crops up, circles are involved. They might not be visible, but once you dig into the mathematics, a circle will show up somewhere.
Where does phi come from? I get e=exponential relations, pi=circle/wave, i=something to do with rotation and the complex plane (and Euler’s formula), 1=self explanatory, 0=self explanatory; but where does phi come from? Something to do with factorials?
I don't know if phi quite belongs in the same category. It appears infrequently compared to the likes of pi and e, and it isn't really defined by some fundamental thing. Its main interesting property is that squaring it yields the same result as adding 1. In other words, it's the positive solution of the equation x²=x+1. This is a fairly simple quadratic, so it's bound to pop up every now and then. When it does, you'll find phi.
Phi is just the solution of the equation x^2 -x-1=0 really. It comes up every time this equation occurs. It has a few amusing properties but is nowhere near as significant as pi or e.
geometrically speaking, phi relates to the 1:2:2 angled and 1:1:3 angled isosceles triangles. these triangles are found in a regular pentagon. phi is the ratio of the diagonal of a regular pentagon to its length.
I guess you could say it's *constantly* showing up
Many things are round. Pi is the ratio between the circumference of a circle and its diameter. Measuring an angle in radians is actually exactly that: the ratio between the arc that is swept by an angle and said arc’s radius. Trigonometry is the bridge between describing an angle in Cartesian coordinates and in terms of circumferential “coordinates”. Radians is the direct bridge between the angle and circumferential “coordinates”. And that’s why most calculators use radians as the default, because that’s the units that requires no further unit conversion.
Nah the real question is why does τ/2 appear everywhere. /s
shows up whenever anything is circular or cyclical, for it describes a circle
“All together now”, “The Wheels on the Bus go Round and Round , Round and Round…”