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> Another place our pi will come up is in the exponential function. It's periodic with period 2πi. Isn't it the opposite? As I understand, we (European civiliz

> Another place our pi will come up is in the exponential function. It's periodic with period 2πi. Isn't it the opposite? As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our previously defined sin and cos functions. We could have defined it to have another period — for example, if we define "360° angle" to be equal to 1 instead of 2*Pi, and define sin0=0, sin0.25=1, sin0.5=0, sin0.75=-1, sin1=0, we'd also define periodicity of e^ix to be 1. UPD: Same idea as for why we use base-ten numbers. The only reason is that we have ten fingers on two hands, and historically we've been using base-ten numbers for the past few hundred years. But there's no reason to expect that "aliens" would be having ten digits also.

moring 3y ago
Where in the definition of the complex exponential function is π used? IIRC its: the exponential function is defined to be its own derivative (note 1), i is the square root of -1, and exp(ix) is observed to have a period of 2π. There isn't any arbitrary choices in there that could be said to be defined in such a way that π results. note 1: exp(x) can alternatively be defined by the exponential series, but that series does contain arbitrary numbers that could be said to be selected in such a way that π results.
- nlitened 3y ago
  As I understand, "complex exponential" function f(x) = e^ix must satisfy only two equalities: f(0) = 1 f'(x) = i f(x) So any function that satisfies these equalities can work as a "complex exponential" function which we denote as e^ix. So we can define a function with period of 1, and use it everywhere — then "2π" vanishes from most equations, and the complex math still works and all equalities hold.*
  - messe 3y ago
    The only function that satisfies those two equalities is e^ix which has period 2π.
    - nlitened 3y ago
      Hey, you're actually right! My bad
  - jojobas 3y ago
    Consider simultaneous functional equations: f(x) = dg(x)/dx g(x) = df(x)/dx Its only linearly independent solutions are sin(x+c) and cos(x+c) with, x being in radians, periodic by 2pi.
    - nlitened 3y ago
      Yep, you’re right, my bad.
    - xigoi 3y ago
      You forgot a minus before one of the equations, otherwise you get sinh and cosh.
      - jojobas 3y ago
        Yep, silly me.
jlokier 3y ago
> As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our previously defined sin and cos functions. We could have defined it to have another period — for example, if we define "360° angle" to be equal to 1 instead of 2Pi, and define sin0=0, sin0.25=1, sin0.5=0, sin0.75=-1, sin1=0, we'd also define periodicity of e^ix to be 1.* No, it doesn't work in degrees. The definition of e isn't that arbitrary. 2π is the unique period which satisfies the definition of e using derivatives and the extension of real number algebraic laws to complex numbers. This shows up as a real world physical measurement, which I describe below. The (natural) exponential function eˣ is defined as the unique function which equals its own derivative and satisfies e⁰ = 1 (like other exponentials). The value of e comes from this. Combine that with the definition i² = -1 and using basic rules of algebra which are observed on real numbers with exponentials and derivatives (such as (xª)ᵇ = xªᵇ) and you find the function eˣ must be periodic with period 2πi. This comes from sin(x) and cos(x) and their derivatives. The derivative of sin(x) is cos(x), and of cos(x) it is -sin(x), but only if sin(x) and cos(x) are defined in the usual math way with period 2π. Those sin/cos derivatives and that little negative sign are enough to make them components of the unique solution to the derivative definition of eˣ applied to a complex argument, and thereby fix its period in the complex plane and prove Euler's famous identity (without needing the Taylor expansion). That in turn has.a more physical basis. Asin(x+B) with constants A, B are the family of functions whose second derivative equal themselves negated. Physically, it means an object whose acceleration is proportional to its displacement from a fixed position and in the opposite direction will oscillate with a period of exactly 2π seconds, if the acceleration is -1m/s² per 1m displacement. This setup is called a harmonic oscillator. In this way, 2π arises (and is measurable!) from physical properties of time, force and inertia, of things moving in straight lines. No circles required.
- nlitened 3y ago
  I agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π. If we defined a function sin to take not an angle in radians, but in degrees (with a period of 360.0), and used that definition of sin in our math, then our complex e^ix would have a period of exactly 360, and the entire complex math would still work — for example, Euler's formula below would still hold: e^ix = cos x + i sin x And people in comments would rave about how magic number 360 is, and its magic properties were discovered by Romans two thousand years ago.
zoomablemind 3y ago
Fact that pi is irrational may point at something fundamental missing in our knowledge system. It appears that we cannot precisely measure circle length/area in units of radius and vice versa. Basically, the unity as such does not exist in our knowledge, nor can we truly comprehend infinity. Perhaps, unity and infinity are just our abstractions for something else.
- xigoi 3y ago
  The fact that π is irrational has absolutely nothing to do with physically measuring circles or with infinity. We know the value of π exactly.
  - zoomablemind 3y ago
    > ...We know the value of π exactly. Perhaps you could share that exact value with the rest of humanity. And I mean the number value, not the nominal value.
    - xigoi 3y ago
      What do you mean by “number value”? There are many different ways to calculate π, the most famous (but converging very slowly) being π = 4 atan(1) = 4 (1 - 1/3 + 1/5 - 1/7 + …)
      - zoomablemind 3y ago
        There are many ways to obtain inexact value of pi. Being irrational number, there are no finite number of digits (e.g. in decimal form or other base) to represent the pi value exactly. Nor can such value be expressed as ratio of integers. Likewise, in your representation the ellipsis hides away the infinity.
        xigoi 3y ago
        π has a finite amount of digits in base π. Why arbitrarily limit yourself to integer bases? If you can compute a number to any desired precision, then you know its exact value. And this still has nothing to do with measurement of the physical world. We cannot measure anything exactly.

moring 3y ago

Where in the definition of the complex exponential function is π used? IIRC its: the exponential function is defined to be its own derivative (note 1), i is the square root of -1, and exp(ix) is observed to have a period of 2π. There isn't any arbitrary choices in there that could be said to be defined in such a way that π results. note 1: exp(x) can alternatively be defined by the exponential series, but that series does contain arbitrary numbers that could be said to be selected in such a way that π results.

nlitened 3y ago
As I understand, "complex exponential" function f(x) = e^ix must satisfy only two equalities: f(0) = 1 f'(x) = i f(x) So any function that satisfies these equalities can work as a "complex exponential" function which we denote as e^ix. So we can define a function with period of 1, and use it everywhere — then "2π" vanishes from most equations, and the complex math still works and all equalities hold.*
- messe 3y ago
  The only function that satisfies those two equalities is e^ix which has period 2π.
  - nlitened 3y ago
    Hey, you're actually right! My bad
- jojobas 3y ago
  Consider simultaneous functional equations: f(x) = dg(x)/dx g(x) = df(x)/dx Its only linearly independent solutions are sin(x+c) and cos(x+c) with, x being in radians, periodic by 2pi.
  - nlitened 3y ago
    Yep, you’re right, my bad.
  - xigoi 3y ago
    You forgot a minus before one of the equations, otherwise you get sinh and cosh.
    - jojobas 3y ago
      Yep, silly me.

jlokier 3y ago

> As I understand, we (European civilization humans) historically _define_ our complex exponential function to have a period of 2πi to match the period of our previously defined sin and cos functions. We could have defined it to have another period — for example, if we define "360° angle" to be equal to 1 instead of 2Pi, and define sin0=0, sin0.25=1, sin0.5=0, sin0.75=-1, sin1=0, we'd also define periodicity of e^ix to be 1.* No, it doesn't work in degrees. The definition of e isn't that arbitrary. 2π is the unique period which satisfies the definition of e using derivatives and the extension of real number algebraic laws to complex numbers. This shows up as a real world physical measurement, which I describe below. The (natural) exponential function eˣ is defined as the unique function which equals its own derivative and satisfies e⁰ = 1 (like other exponentials). The value of e comes from this. Combine that with the definition i² = -1 and using basic rules of algebra which are observed on real numbers with exponentials and derivatives (such as (xª)ᵇ = xªᵇ) and you find the function eˣ must be periodic with period 2πi. This comes from sin(x) and cos(x) and their derivatives. The derivative of sin(x) is cos(x), and of cos(x) it is -sin(x), but only if sin(x) and cos(x) are defined in the usual math way with period 2π. Those sin/cos derivatives and that little negative sign are enough to make them components of the unique solution to the derivative definition of eˣ applied to a complex argument, and thereby fix its period in the complex plane and prove Euler's famous identity (without needing the Taylor expansion). That in turn has.a more physical basis. Asin(x+B) with constants A, B are the family of functions whose second derivative equal themselves negated. Physically, it means an object whose acceleration is proportional to its displacement from a fixed position and in the opposite direction will oscillate with a period of exactly 2π seconds, if the acceleration is -1m/s² per 1m displacement. This setup is called a harmonic oscillator. In this way, 2π arises (and is measurable!) from physical properties of time, force and inertia, of things moving in straight lines. No circles required.

nlitened 3y ago
I agree with you that e is not arbitrary. I say that the period of 2π for e^ix is arbitrary, because we've arbitrarily defined periods of sin and cos as 2π. If we defined a function sin to take not an angle in radians, but in degrees (with a period of 360.0), and used that definition of sin in our math, then our complex e^ix would have a period of exactly 360, and the entire complex math would still work — for example, Euler's formula below would still hold: e^ix = cos x + i sin x And people in comments would rave about how magic number 360 is, and its magic properties were discovered by Romans two thousand years ago.

zoomablemind 3y ago

Fact that pi is irrational may point at something fundamental missing in our knowledge system. It appears that we cannot precisely measure circle length/area in units of radius and vice versa. Basically, the unity as such does not exist in our knowledge, nor can we truly comprehend infinity. Perhaps, unity and infinity are just our abstractions for something else.

xigoi 3y ago
The fact that π is irrational has absolutely nothing to do with physically measuring circles or with infinity. We know the value of π exactly.
- zoomablemind 3y ago
  > ...We know the value of π exactly. Perhaps you could share that exact value with the rest of humanity. And I mean the number value, not the nominal value.
  - xigoi 3y ago
    What do you mean by “number value”? There are many different ways to calculate π, the most famous (but converging very slowly) being π = 4 atan(1) = 4 (1 - 1/3 + 1/5 - 1/7 + …)
    - zoomablemind 3y ago
      There are many ways to obtain inexact value of pi. Being irrational number, there are no finite number of digits (e.g. in decimal form or other base) to represent the pi value exactly. Nor can such value be expressed as ratio of integers. Likewise, in your representation the ellipsis hides away the infinity.
      - xigoi 3y ago
        π has a finite amount of digits in base π. Why arbitrarily limit yourself to integer bases? If you can compute a number to any desired precision, then you know its exact value. And this still has nothing to do with measurement of the physical world. We cannot measure anything exactly.