E.g., the zeros of the function defined by the series x - x^3/3! + x^5/5! - x^7/7! + ... are nπ where n is an integer and π is our π. Another place our pi will come up is in the exponential function. It's periodic with period 2πi.
• the sum of the series 4(1 - 1/3 + 1/5 - 1/7 + …) will still be our π: https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
• the sum of the series (1 + 1/4 + 1/9 + 1/16 + 1/25 + …) will still be π²/6: https://en.wikipedia.org/wiki/Basel_problem
• (therefore) the probability that two numbers chosen uniformly at random from [1…N] are relatively prime will still approach 6/π² as N grows large
• the product 2(4/3)(16/15)(36/35)(64/63)(100/99)… will still be our π: https://en.wikipedia.org/wiki/Wallis_product
• the value of (n!/(√n (n/e)^n))²/2 as n grows large will still (very slowly) approach π: https://en.wikipedia.org/wiki/Stirling%27s_approximation (e.g. https://www.wolframalpha.com/input?i2d=true&i=N%5C%2891%29Di... )
and so on, for most of the non-geometry results listed: https://en.wikipedia.org/w/index.php?title=List_of_formulae_...
For those unwilling to click-through, it essentially posits an alternate history where infinite series were explored by mathematicians before geometry, so rather than being surprised that the 'circle constant' is found in many infinite series, we would instead be surprised that the 'infinite series constant' is found in the geometry of a circle.
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.
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.
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
exp(x) for complex x is simply defined to be the infinite sum from k = 0 to infinity of x^k/k!. That is, exp(x) = 1 + x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ...
(BTW, the motivation for this definition is that exp'(x) = exp(x), which shouldn't be too hard to see because it's already a Tailor series.)
Purely from this you can prove that exp(ix) with real x is periodic with period 6.28...
It just so happens that this number is also the circumference of the unit circle.
The Pi thing feels now less of a coincidence than the fact that exp is a power. That probably falls out of expanding the polynomials but it so ingrained as taken for granted that it is wonderous when you think about it.
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.
f(0) = 1
f'(x) = i f(x)
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.*
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.
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.
Perhaps you could share that exact value with the rest of humanity. And I mean the number value, not the nominal value.
π = 4 atan(1) = 4 (1 - 1/3 + 1/5 - 1/7 + …)
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.
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.
* Manhattan (L_1): C = 8 R
...
* Euclidean (L_2): C = 2π R
...
* Maximal Distance (L_infinity): C = 8 R