Comment by ActorNightly
1 day ago
The things is, there are no accidents in math. If you end up with formulas that look like something else, that something else was a defining part of the original definition, whether it was obvious or not.
You would agree that in the quest to solve x^2+1=0, we had to introduce another dimension, with a multiply operator that lets us move through that dimension. All im saying is that introducing another dimension and ability to move in that dimension is the same thing as rotation and scaling.
As for e, the point im trying to make is that if you look at what it means to take something to the power of something when it comes to reals, there is a clear definition. But taking something to imaginary power is meaningless it iself. For reals, the power operator has a strict definition with multiplication and division for rationals, and generically extended to reals through limits. And to compute a limit means that you have to have continuity and smoothness. So by extension, to compute exponentiation, you have to have continuity and smoothness, and vice versa.
My argument is that there is no guarantee of continuity/smoothness on the complex plane - one can define it as such (i.e the analytic view) but in my stricter philosophical view, to make something with similar properties like real numbers, you have to have all the analogous operations work within the numbers itself. I.e you have to be able to define what 2i^3i is without ever referencing anything from the real numbers.
This is not done for complex plane - to define something to the i power you need to borrow definitions from the real plane.
As such, you can't define exponentiation to the i, because you can't compute limits. Computing the taylor series expansion equivalence or limit formulas in the way me and you presented them are "holograms" - i.e meaningless results.
And it seems that way. Say that you ignore taylor series formulation, skip it completely, and define the polar form to be r(10)^ix = cos(x)+isin(x), where r is radius and x is angle. JUST BECAUSE YOU CAN.
Would you lose anything? Not really. You would still have a way to compute 2^i - it would just be 10^(ilog(2)) -> r = 1, angle=log(2). I.e the map still exists in quite a good shape.
Basically it doesn't matter if r(10)^ix = cos(x)+isin(x) or r(e)^ix = cos(x)+isin(x), as counterintuitive as it may seem, which furthers my point that exponentiation operation to imaginary numbers is not defined.
So without exponentiation, all you are left with is basically a more strict rigid construction of another set with an ordering property, a multiplication operation definition, (i^2 = -1), and the resultant orthogonality that represents a cartesian plane.
And if you think thats silly, consider the fact that the algebraic view of complex numbers doesn't even consider calculus on them to be valid.
I forgot to respond to this:
> And it seems that way. Say that you ignore taylor series formulation, skip it completely, and define the polar form to be r(10)^ix = cos(x)+isin(x), where r is radius and x is angle. JUST BECAUSE YOU CAN.
The thing is, you can't do this. Your numbers will not work out correctly. For example, (e^(pi*i))^i = 1/e^pi is a direct consequence of how exponentiation works and the definition of i. If you define 10^(xi) = cos x + i sin x, you will get:
I very much doubt that sin(cos (log_10 e * pi) + i sin (log_10 e * pi)) is 0, so this number can't possibly be equal to the real number 1/e^pi. So, with your definition, the property (a^x)^y = a^(x*y) doesn't hold for all a, x, y. So, your definition doesn't represent the exponential function at all.
> a direct consequence of how exponentiation works and the definition of i
I dunno if Im just not coming across clearly or my thinking is too abstract, but when you say
"how exponentiation works"
you are referring to how exponentiation works for real exponents
You would have to prove that it works the same way for i exponents before you can go further.
I think the point is to look at it the other way: we invented/chose i so that exponentiation (and all other functions) maintain the same properties for the complex numbers as they have for the reals. The whole goal of defining the complex numbers is to extend R in a way that maintains all of the properties of functions on R.
> You would agree that in the quest to solve x^2+1=0, we had to introduce another dimension, with a multiply operator that lets us move through that dimension.
I don't really agree, no. In the analytic definition, there is no second dimension. Sure, C is isomorphic to R×R, but that is not how you construct it. C is not R×R, it's R+{a + b * i | a,b in R, b≠0} in this view. Just like Z is N+{a * -1 | a in N, a≠0}. You introduce one new number that you need to solve the equation, and then all of the numbers needed to make the new construction a field again. You don't introduce a new notion of multiplication, C uses the exact same multiplication operation as R does, or at least as polynomials over R do, as I showed.
The fact that C is isomorphic to R×R, and that multiplication of numbers in C is isomorphic to scaling+rotation of vectors in R×R, is not part of the construction.
I do agree that there are no accidents in math, so I imagine this isomorphism is related to some more fundamental relationship between polynomials, 2d vectors, and rotation matrices - because our construction of C is strictly motivated by polynomials.
> For reals, the power operator has a strict definition with multiplication and division.
I don't agree with this statement. It's true for the integers, but already for the the rationals it loses any direct relationship to multiplication and division (how do you get 4^(1/2) = 2 by repeated multiplication?). And for the reals, it's completely gone. We can't even define many properties of real-valued exponentials - we don't even know if e^pi is an irrational number, for example.
> When you search for something like taylor series or limit form of e and you see a way to compute e^i what you are doing is basically using operators designed for real numbers and extending them to the complex numbers (when you substitute i for where normally a real number would be in the power term.
We've already agreed that there are no accidents in math. So just as the multiplication of polynomials is fundamentally linked with rotations of 2d real-valued vectors, we must accept that real exponentiation is fundamentally linked to complex exponentiation, otherwise the formulas wouldn't work the way they do.
Note also that calculus is not limited to operations on the number line - you can take the derivatives or integrate or calculate limits of n-dimensional curves, and even differentiate over surfaces and n-dimensional manifolds more generally. Smoothness and continuity are anyway part of the structure of C, regardless what definition of it you use.
> And if you think thats silly, consider the fact that the algebraic view of complex numbers doesn't even consider calculus on them to be valid.
I don't understand what you mean by this. Calculus is well defined for functions over any field of size continuum, and that is exactly what <C,+,*,0,1> is in the algebraic view.
>C uses the exact same multiplication operation as R does
Not quite. If it did, the -i*-i would be i^2, not -1. And yes, I totally agree that C is R+, not RxR. The point is that you still introducing something extra with some rules, where you introduce the concept of geometric orthogonality into i^2 = -1, whether that is your intention or not.
>For reals, the power operator has a strict definition with multiplication and division for rationals, and generically extended to reals through limits.
I accidentally swapped reals and rationals there. The whole point was to highlight that exponentiation for real exponents relies on limits which relies on continutity.
>So just as the multiplication of polynomials is fundamentally linked with rotations of 2d real-valued vectors, we must accept that real exponentiation is fundamentally linked to complex exponentiation, otherwise the formulas wouldn't work the way they do.
Don't agree.
Multiplication of polynomials involves operations that are clearly defined. When you do (a+bi)^2, you have defined what it means to multiply complex numbers in their construction, without needing to use any such formula from real numbers.
Exponentiation where you have i exponent however, is not defined solely in the complex field.
>I don't understand what you mean by this. Calculus is well defined for functions over any field of size continuum, and that is exactly what <C,+,*,0,1> is in the algebraic view.
Algebraic view is basically the idea of that numbers are only defined if there is some relation that expresses their definition.
For example, 1+2=3, locks all 3 number down. 1 is 3-2, 2 is 3-1 and 3 is 1+2.
pi or e on the other hand, are "something else", because there is no algebraic formula that defines them. To do so you have to invoke the computation of limits, which is an analytic view, not algebraic.*
> Not quite. If it did, the -i-i would be i^2, not -1.
It is i², though, but that is equal to -1. Just like 22 = 2² = 4. I also maintain that the historical view is that this something extra comes from the properties of polynomials and their roots, not from geometric orthogonality.
> Exponentiation where you have i exponent however, is not defined solely in the complex field.
We can take another tack for defining the complex exponential function, if you'd prefer. One of the definitions of the exponential function is that e^x is the only function that respects f'(x) = f(x) (well, up to constant multiplication).
So, we need to look for a function f(z) such that f'(z) = f(z). There are various ways to do this (for example, using the Taylor series expansion and noting that all of the f derived n times factors are equal to f(z), which yields the power series definition). You don't need to appeal to limits of e^x to get there this way.
> Algebraic view is basically the idea of that numbers are only defined if there is some relation that expresses their definition.
Understood, you are using a different sense of "algebraic" than what I was - I was thinking more of the abstract algebraic definition of C.
Still, the sense you are using seems to be the concept of algebraic numbers, which is more formalized - the algebraic numbers are all those that represent the root of a polynomial with integer or rational coefficients. Interestingly, while pi and e are not algebraic numbers, i or i+7 are still algebraic.
However, I'm not sure what the point of bringing this up is. Exponentiation is simply not defined over the algebraic numbers, especially not e^x where x is algebraic - so if we restrict ourselves to algebraic numbers, e^i is not defined, true, but neither is e^1. And while 2^2 is defined, of course, I'm not even sure you can define 2^sqrt(2), so I wouldn't be surprised if 2^i doesn't make sense either.
Either way, the algebraic numbers are not "a way of thinking about numbers", they are a restricted subset of what is generally meant by "number", and many famous and useful results from many branches of math do not work over this subset (for example, you can't even use the same set of algebraic numbers to refer to the length of a circle and the length of a square).
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