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Comment by ActorNightly | Hacker News Reader

Honestly, the rigid conception is the correct one. Im of the view that i as an attribute on a number rather than a number itself, in the same way a negative sign is an attribute. Its basically exists to generalize rotations through multiplication. Instead of taking an x,y vector and multiplying it by a matrix to get rotations, you can use a complex number representation, and multiply it by another complex number to rotate/scale it. If the cartesian magnitude of the second complex number is 1, then you don't get any scaling. So the idea of x/y coordinates is very much baked in to the "imaginary attribute".

I feel like the problem is that we just assume that e^(pi*i) = -1 as a given, which makes i "feel" like number, which gives some validity to other interpretations. But I would argue that that equation is not actually valid. It arises from Taylor series equivalence between e, sin and cos, but taylor series is simply an approximation of a function by matching its derivatives around a certain point, namely x=0. And just because you take 2 functions and see that their approximations around a certain point are equal, doesn't mean that the functions are equal. Even more so, that definition completely bypasses what it means to taking derivatives into the imaginary plane.

If you try to prove this any other way besides Taylor series expansion, you really cant, because the concept of taking something to the power of "imaginary value" doesn't really have any ties into other definitions.

As such, there is nothing really special about e itself either. The only reason its in there is because of a pattern artifact in math - e^x derivative is itself, while cos and sin follow cyclic patterns. If you were to replace e with any other number, note that anything you ever want to do with complex numbers would work out identically - you don't really use the value of e anywhere, all you really care about is r and theta.

So if you drop the assumption that i is a number and just treat i as an attribute of a number like a negative sign, complex numbers are basically just 2d numbers written in a special way. And of course, the rotations are easily extended into 3d space through quaternions, which use i j an k much in the same way.

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> As such, there is nothing really special about e itself either. The only reason its in there is because of a pattern artifact in math - e^x derivative is itself

Not sure I follow you here... The special thing about e is that it's self-derivative. The other exponential bases, while essentially the same in their "growth", have derivatives with an extra factor. I assume you know e is special in that sense, so I'm unclear what you're arguing?

Im saying that the definition of polar coordinates for complex numbers using e instead of any other number is irrelevant to the use of complex numbers, but its inclusion in Eulers identity makes it seem like a i is a number rather than an attribute. And if you assume i is a number, it leads to one thinking that that you can define the complex field C. But my argument is that Eulers identity is not really relevant in the sense of what the complex numbers are used for, so i is not a number but rather a tool.

We as humans had a similar argument regarding 0. The thought was that zero is not a number, just a notational trick to denote that nothing is there (in the place value system of the Mesopotamians)

But then in India we discovered that it can really participate with the the other bonafide numbers as a first class citizen of numbers.

It is not longer a place holder but can be the argument of the binary functions, PLUS, MINUS, MULTIPLY and can also be the result of these functions.

With i we have a similar observation, that it can indeed be allowed as a first class citizen as a number. Addition and multiplication can accept them as their arguments as well as their RHS. It's a number, just a different kind.

But you can define the complex field C. And it has many benefits, like making the fundamental theorem of algebra work out. I'm not seeing the issue?

On a similar note, why insist that "i" (or a negative, for that matter) is an "attribute" on a number rather than an extension of the concept of number? In one sense, this is a just a definitional choice, so I don't think either conception is right or wrong. But I'm still not getting your preference for the attribute perspective. If anything, especially in the case of negative numbers, it seems less elegant than just allowing the negatives to be numbers?

Sure, you can define any field to make your math work out. None of the interpretations are wrong per say, the question is whether or not they are useful.

The point of contention that leads to 3 interpretations is whether you assume i acts like a number. My argument is that people generally answer yes, because of Eulers identity (which is often stated as example of mathematical beauty).

My argument is that i does not act like a number, it acts more like an operator. And with i being an operator, C is not really a thing.

Rotations fell out of the structure of complex numbers. They weren't placed there on purpose. If you want to rotate things there are usually better ways.

> If you want to rotate things there are usually better ways.

Can you elaborate? If you want a representation of 2D rotations for pen-and-paper or computer calculations, unit complex numbers are to my knowledge the most common and convenient one.

For pen and paper you can hold tracing paper at an angle. Use a protractor to measure the angle. That's easier than any calculation. Or get a transparent coordinate grid, literally rotate the coordinate system and read off your new coordinates.

For computers, you could use a complex number since it's effectively a cache of sin(a) and cos(a), but you often want general affine transformations and not just rotations, so you use a matrix instead.

> For computers, you could use a complex number since it's effectively a cache of sin(a) and cos(a), but you often want general affine transformations and not just rotations, so you use a matrix instead.

That makes sense in some contexts but in, say, 2D physics simulations, you don't want general homogeneous matrices or affine transformations to represent the position/orientation of a rigid body, because you want to be able to easily update it over time without breaking the orthogonality constraint.

I guess you could say that your tuple (c, s) is a matrix [ c -s ; s c ] instead of a complex number c + si, or that it's some abstract element of SO(2), or indeed that it's "a cache of sin(a) and cos(a)", but it's simplest to just say it's a unit complex number.

Why use a unit complex number (2 numbers) instead of an angle (1 number)? Maybe it optimizes out the sins and cosses better — I don't know — but a cache is not a new type of number.

There's a significant advantage in using a tuple over a scalar to represent angles.

For many operations you can get rid of calls to trigonometric functions, or reduce the number of calls necessary. These calls may not be supported by standard libraries in minimalistic hardware. Even if it were, avoiding calls to transcendental can be useful.

Because rotations with complex numbers is not just rotations, its rotations+scaling.

The advantage of complex numbers is to rotate+scale something (or more generally move somewhere in a complex plane), is a one step multiplication operation.

Do you find yourself scaling things a lot in 2D physics, but not translating them? I'd think translation and rotation are more common than scaling.

If you need to support zoom, scaling shows up very frequently.

I can give an example from real life. A piece of code one of my colleagues was working on required finding a point on the angular bisector. The code became a tangle of trigonometry calls both the forward and inverse functions. The code base was Python, so there was native support of complex numbers.

So you need angular bisector of two points p and q ? just take their geometric mean and you are done. At the Python code base level you only have a call to sqrt. That simplifies things.

No.

The whole idea of imaginary number is its basically an extension of negative numbers in concept. When you have a negative number, you essentially have scaling + attribute which defines direction. When you encounter two negative attributes and multiply them, you get a positive number, which is a rotation by 180 degrees. Imaginary numbers extend this concept to continuous rotation that is not limited to 180 degrees.

With just i, you get rotations in the x/y plane. When you multiply by 1i you get 90 degree rotation to 1i. Multiply by i again, you get another 90 degree rotation to -1 . And so on. You can do this in xyz with i and j, and you can do this in 4dimentions with i j and k, like quaternions do, using the extra dimension to get rid of gimbal lock computation for vehicle control (where pointed straight up, yaw and roll are identicall)

The fact that i maps to sqrt of -1 is basically just part of this definition - you are using multiplication to express rotations, so when you ask what is the sqrt of -1 you are asking which 2 identical number create a rotation of 180 degrees, and the answer is 1i and 1i.

Note that the definition also very much assumes that you are only using i, i.e analogous to having the x/y plane. If you are working within x y z plane and have i and j, to get to -1 you can rotate through x/y plane or x/z plane. So sqrt of -1 can either mean "sqrt for i" or "sqrt for j" and the answer would be either i or j, both would be valid. So you pretty much have to specify the rotation aspect when you ask for a square root.

Note also that you can you can define i to be <90 degree rotation, like say 60 degrees and everything would still be consistent. In which case cube root of -1 would be i, but square root of -1 would not be i, it would be a complex number with real and imaginary parts.

The thing to understand about math is under the hood, its pretty much objects and operations. A lot of times you will have conflicts where doing an operation on a particular object is undefined - for example there are functions that assymptotically approach zero but are never equal to it. So instead, you have to form other rules or append other systems to existing systems, which all just means you start with a definition. Anything that arises from that definition is not a universal truth of the world, but simply tools that help you deal with the inconsistencies.

The whole idea of an imaginary number is that it squares to a negative number. Everything else is accidental. Nobody expected that exp(i*a)=cos(a)+i*sin(a). Totally wacky discovery.

Imaginary numbers don't work in 3D, by the way. The most natural representation of a 3D rotation is a normalized 4D quaternion, and it's still pretty weird.

This is the most confident Ive seen someone in being so very wrong lol.

Nope. (Just to imitate your style)

There's more to it than rotation by 180 degrees. More pedagogically ...

Define a tuple (a,b) and define addition as pointwise addition. (a, b) + (c, d) = (a+c, b+d). Apples to apples, oranges to oranges. Fair enough.

How shall I define multiplication, so that multiplication so defined is a group by itself and interacts with the addition defined earlier in a distributive way. Just the way addition and multiplication behave for reals.

Ah! I have to define it this way. OK that's interesting.

But wait, then the algebra works out as if (0, 1) * (0, 1) = (-1, 0) but right hand side is isomorphic to -1. The (x, 0)s behave with each other just the way the real numbers behave with each other.

All this writing of tuples is cumbersome, so let me write (0,1) as i.

Addition looks like the all too familiar vector addition. What does this multiplication look like? Let me plot in the coordinate axes.

Ah! It's just scaled rotation, These numbers are just the 2x2 scaled rotation matrices that are parameterized not by 4 real numbers but just by two. One controls degree of rotation the other the amount of scaling.

If I multiply two such matrices together I get back a scaled rotation matrix. OK, understandable and expected, rotation composed is a rotation after all. But if I add two of them I get back another scaled rotation matrix, wow neato!

Because there are really only two independent parameters one isomorphic to the reals, let's call the other one "imaginary" and the tupled one "complex".

What if I negate the i in a tuple? Oh! it's reflection along the x axis. I got translation, rotation and reflection using these tuples.

What more can I do? I can surely do polynomials because I can add and multiply. Can I do calculus by falling back to Taylor expansions ? Hmm let me define a metric and see ...

I think we are agreeing.

You made it seem like rotations are an emergent property of complex numbers, where the original definition relies on defining the sqrt of -1.

Im saying that the origin of complex numbers is the ability to do arbitrary rotations and scaling through multiplication, and that i being the sqrt of -1 is the emergent property.

> Im saying that the origin of complex numbers is the ability to do arbitrary rotations and scaling through multiplication, and that i being the sqrt of -1 is the emergent property.

Not true historically -- the origin goes back to Cardano solving cubic equations.

But that point aside, it seems like you are trying to find something like "the true meaning of complex numbers," basing your judgement on some mix of practical application and what seems most intuitive to you. I think that's fruitless. The essence lies precisely in the equivalence of the various conceptions by means of proof. "i" as a way "to do arbitrary rotations and scaling through multiplication", or as a way give the solution space of polynomials closure, or as the equivalence of Taylor series, etc -- these are all structurally the same mathematical "i".

So "i" is all of these things, and all of these things are useful depending on what you're doing. Again, by what principle do you give priority to some uses over others?

>he origin goes back to Cardano solving cubic equations.

Whether or not mathematicians realized this at the time, there is no functional difference in assuming some imaginary number that when multiplied with another imaginary number gives a negative number, and essentially moving in more than 1 dimension on the number line.

Because it was the same way with negative numbers. By creating the "space" of negative numbers allows you do operations like 3-5+6 which has an answer in positive numbers, but if you are restricted to positive only, you can't compute that.

In the same way like I mentioned, Quaternions allow movement through 4 dimentions to arrive at a solution that is not possible to achieve with operations in 3 when you have gimbal lock.

So my argument is that complex numbers are fundamental to this, and any field or topological construction on that is secondary.

Maybe.

You disagreed with the parent comment that said

"Rotations fell out of the structure of complex numbers. They weren't placed there on purpose. If you want to rotate things there are usually better ways."

I see Complex numbers in the light of doing addition and multiplication on pairs. If one does that, rotation naturally falls out of that. So I would agree with the parent comment especially if we follow the historical development. The structure is identical to that of scaled rotation matrices parameterized by two real numbers, although historically they were discovered through a different route.

I think all of us agree with the properties of complex numbers, it's just that we may be splitting hairs differently.

>"Rotations fell out of the structure of complex numbers. They weren't placed there on purpose. If you want to rotate things there are usually better ways."

I mean, the derivation to rotate things with complex numbers is pretty simple to prove.

If you convert to cartesian, the rotation is a scaling operation by a matrix, which you have to compute from r and theta. And Im sure you know that for x and y, the rotation matrix to the new vector x' and y' is

x' = cos(theta)*x - sin(theta)*y

y' = sin(theta)*x + cos(theta)*y

However, like you said, say you want to have some representation of rotation using only 2 parameters instead of 4, and simplify the math. You can define (xr,yr) in the same coordinates as the original vector. To compute theta, you would need ArcTan(yr/xr), which then plugged back into Sin and Cos in original rotation matrix give you back xr and yr. Assuming unit vectors:

x'= xr*x - yr*y

y'= yr*x + xr*y

the only trick you need is to take care negative sign on the upper right corner term. So you notice that if you just mark the y components as i, and when you see i*i you take that to be -1, everything works out.

So overall, all of this is just construction, not emergence.

Yes it's simple and I agree with almost everything except that arctan bit (it loses information, but that's aside story).

But all that you said is not about the point that I was trying to convey.

What I showed was you if you define addition of tuples a certain, fairly natural way. And then define multiplication on the same tuples in such a way that multiplication and addition follow the distributive law (so that you can do polynomials with them). Then your hands are forced to define multiplication in very specific way, just to ensure distributivity. [To be honest their is another sneaky way to do it if the rules are changed a bit, by using reflection matrices]

Rotation so far is nowhere in the picture in our desiderata, we just want the distributive law to apply to the multiplication of tuples. That's it.

But once I do that, lo and behold this multiplication has exactly the same structure as multiplication by rotation matrices (emergence? or equivalently, recognition of the consequences of our desire)

In other words, these tuples have secretly been the (scaled) cos theta, sin theta tuples all along, although when I had invited them to my party I had not put a restriction on them that they have to be related to theta via these trig functions.

Or in other words, the only tuples that have distributive addition and multiplication are the (scaled) cos theta sin theta tuples, but when we were constructing them there was no notion of theta just the desire to satisfy few algebraic relations (distributivity of add and multiply).

I just don't like this characterization of

> "How shall I define multiplication, so that multiplication so defined is a group by itself and interacts with the addition defined earlier in a distributive way. Just the way addition and multiplication behave for reals."

which eventually becomes

> "Ah! It's just scaled rotation"

and the implication is that emergent.

Its like you have a set of objects, and defining operations on those objects that have properties of rotations baked in ( because that is the the only way that (0, 1) * (0, 1) = (-1, 0) ever works out in your definition), and then you are surprised that you get something that behaves like rotation.

Meanwhile, when you define other "multiplicative" like operations on tuples, namely dot and cross product, you don't get rotations.

> I just don't like this characterization

That's ok. It's a personal value judgement.

However, the fact remains that rotations can "emerge" just from the desire to do additions and multiplications on tuples to be able to do polynomials with them ... which is more directly tied to its historical path of discovery, to solve polynomial equations, starting with cubic.

>historical path of discovery, to solve polynomial equations, starting with cubic.

Even with polynomial equations that have complex roots, the idea of a rotation is baked in in solving them. Rotation+scaling with complex numbers is basically an arbitrary translation through the complex plane. So when you are faced with a*x*x + b*x + c = 0, where a b and c all lie on the real number line, and you are trying to basically get to 0, often you can't do it by having x on a number line, so you have to start with more dimentions and then rotate+scale so you end up at zero.

Its the same reason for negative numbers existing. When you have positive numbers only, and you define addition and subtraction, things like 5-6+10 become impossible to compute, even though all the values are positive. But when you introduce the space of negative numbers, even though they don't represent anything in reality, that operation becomes possible.

Yes but it was a fundamental mathematical achievement to see this equivalence. That knowledge had to emerge, be discovered. This eventually led to the theory of Galois fields.

The connection with rotation emerged naturally from a line of thought that initially had nothing to do with rotations. It was a consequence of a desire to satisfy distributive laws and maintain vector addition.

Connection between seemingly unrelated mathematical fields happen from time to time and those are considered events of surprise, understanding and celebration.

You can define that, but (if you don't already know about complex numbers) it's not obvious that it does anything mathematically interesting. It's just a cache for sin and cos, not a new type of anything. I could say that when evaluating 4th degree polynomials it's useful to have x, x^2 and x^3 immediately at hand, but the combination of those three isn't a new type of number, just a cache.

>it does anything mathematically interesting

You are right - its not interesting. You already know that rotation can be done through multiplication (i.e rotation matrix), and you are just simplifying it further.

After all, the only application of imaginary numbers outside their definition is roots of a polynomial. And if you think of rotation+scaling as simple movement through the complex plane to get back to the real one, it makes perfect sense.

You can apply this principle generically as well. Say you have an operation on some ordered set S that produces elements in a smaller subset of S called S' It then follows that the inverse operation of elements of the complement of S' with respect to the original set S is undefined.

But you can create a system where you enhance the dimension of the original set with another set, giving the definition of that inverse operation for compliment of S'. And if that extra set also has ordering, then you are by definition doing something analogous to rotation+scaling.

It seems obvious now only because of significant mathematical discoveries of prominent mathematicians.

If one is taught what those discoveries revealed then of course they would seem obvious.

Arguing as you are, it would appear one can call all and every theorem in mathematics that connects to different fields as something obvious. They weren't, till someone proved the connection and that knowledge percolated down to how maths is taught, to text books.

That the integral of

        exp(-x*x)

over the entire real line is sqrt pi can be surprising or obvious depending on how you were taught. At face value it has nothing to do with circles, unless you are taught the connection or you are a mathematician of high calibre who can see it without being taught the background information.

This completely misses the point of why the complex numbers were even invented. i is a number: it is one of the 2 solutions to the equation x^2 = -1 (the other being -i, of course). The whole point of inventing the complex numbers was to have a set of numbers for which any polynomial has a root. And sure, you can call this number (0,1) if you want to, but it's important to remember that C is not the same as R².

Your whole point about Taylor series is also wrong, as Taylor series are not approximations, they are actually equal to the original function if you take their infinite limit for the relevant functions here (e^x, sin x, cos x). So there is no approximation to be talked about, and no problem in identifying these functions with their Taylor series expansions.

I'd also note that there is no need to use Taylor series to prove Euler's formula. Other series that converge to e^x,cos x, sin x can also get you there.

>The whole point of inventing the complex numbers was to have a set of numbers for which any polynomial has a root

Think about what this implies.

You have an operation, like exponentiation, that has limits. Something squared can never be negative if you are talking about any real number.

In terms of Sets, you essentially have an operation that produces results only in a finite subset of the overall set. And so the inverse of that operation, when applied to the complement of that finite subset, is undefined.

However you can introduce another (ordered) set in complement to your original set and combine them to form a new set, with operations that define how you move around the values of those sets. So in the case of imaginary numbers, you basically redefine all your reals as "real number + 0 i". And now you have a way to apply that inverse operation to the complement of the finite subset, which means you can get answers to the roots of the polynomial.

And in defining the operation of multiplication, you essentially define a way to move around the 2 dimensional set now. And moving around 2 dimensions is exactly the same thing as rotation+scaling. And note that when you say sqrt(-1) = i, you basically assume that the complex plane is 2d. There is nothing that is stopping you from making a complex plane 3d or 4d or nd. So sqrt(-1) can also be j, or it can be k. To know what it is, you have to specify the axis of the plane when you specify the sqrt operation, which again, brings it back to the concept of rotations.

And thats my whole point, there is nothing special about i, its simply just a construct that bakes in rotations through any way you wanna define it.

>our whole point about Taylor series is also wrong, as Taylor series are not approximations, they are actually equal to the original function if you take their infinite limit

Looking back at what I wrote, I worded it very poorly.

I don't have a problem with any math involved, not trying to say that Eulers identity is not valid.

What Im trying to say is that all the definitions sort of assume that if you have some operation that you can do on real numbers, and have a result, if you just plug in a complex value, it all works out, so people think that i behaves like a number. I personally don't think that this is the case, specifically about i behaving like a number just because those results work out.

For example, even without Taylor series, you can prove Eulers identity using the limit formula for e(x). The idea is that you have (1+xi/n)^n as n goes to infinity, but because you baked in the rotation as a multiplication in the definition, all you are doing is starting at 1+0i and doing smaller and smaller rotations to get to some value, and the limit of that value is essentially the unit vector rotated by a certain angle. So naturally the cos and sin equivalence arises.

My issue is that the limit equation for e, in the case of the reals, take e x times in multiplication and then compute the limit equation, and you get equivalence. But in the case of the complex, you don't really have any idea what it takes something to ith power, but you can compute the limit equation, and so you end up with a definition of what it means to take something to the ith power.

My argument is that its not really applicable - not that its wrong, but the fact that its not defining exponentiation to the ith power in the sense that i has "number like" qualities like real numbers do. You would have to prove that an equivalence

What is really happening is that you never really escape the real numbers, and your complex numbers are just simplified operations that rotate/scale a number, like rotation matricies do through multiplication, and that in the nature of the definition of those rotations, you get stuff like Eulers identity, which is somewhat pointless because like I mentioned - the value of e (i.e 2.7) is never really used to compute anything in regards to complex numbers in polar form of re^ix, all you care about is r and the x which is the angle.

And for this reason, I don't consider i a number, so the analytic/smooth interpretations to me are meaningless.

> And in defining the operation of multiplication, you essentially define a way to move around the 2 dimensional set now. And moving around 2 dimensions is exactly the same thing as rotation+scaling.

Again, this is not how complex numbers were defined. The only original goal was to come up with a number that can solve the equation x^2 + 1 = 0, so that (R + {this number}, + , *), becomes an algebraically closed field. Once you've set this goal, there is really a single simple choice for how the operations will work, because everything else is already constrained. If x^2 + 1 = 0, we already know that:

  (a + bx) (c + dx) =                         [polynomial multiplication]
     = ac + (bc + ad)x + (bd)x^2 =
     = ac + (bc + ad)x + bd(x^2 + 1 - 1) =  
     = ac + (bc + ad)x + bd(x^2 + 1) - bd =   [x^2 + 1 = 0 by definition]
     = ac + (bc + ad)x + bd * 0 - bd =
     = (ac - bd) + x(bc + ad).

So the formula for complex number multiplication comes out of the arithmetic of real numbers, extended with this extra entity defined simply by being a root of x^2 + 1. The fact that this operation happens to represent a rotation in the RxR plane is "an accident" (I'm sure there are deep ties that make this necessary, probably related to the structure of polynomials themselves).

And while you can define other algebraically closed fields that include the reals as a subfield, the complex numbers are the simplest such set. R^n for n>2 is clearly more complex, for example. So there is a clear reason to prefer sqrt(-1) = i, and thus ending up with a 2d vector space.

> What Im trying to say is that all the definitions sort of assume that if you have some operation that you can do on real numbers, and have a result, if you just plug in a complex value, it all works out, so people think that i behaves like a number. I personally don't think that this is the case, specifically about i behaving like a number just because those results work out.

Again, complex multiplication is not some intentional construction, it is baked into how we defined the complex numbers the first time we did. Again, our goal was to find (well, define) the solution(s) of the equation x^2 + 1 = 0. The fact that we can plug in this x to any formula that involves other numbers just falls out of this goal, it's not an additional assumption. In the case of e(x), this is simpler to see with the power series formula:

  e(nx) = sum(1 + nx + (nx)^2/2! + (nx)^3/3! + (nx)^4/4! + ...)
  but, by definition, x^2 + 1 = 0, so x^2 = -1, so the formula becomes:
  e(nx) = sum(1 + nx - n^2/2! - xn^3/3! + n^4/4! + ...) = 
       = sum(1 - n^2/2! + n^4/4! ...) + sum(xn - xn^3/3! + xn^5/5! ...) = 
       = sum(1 - n^2/2! + n^4/4! ...) + x sum(n - n^3/3! + n^5/5! ...) = 
       = cos n + x sin n

Note that this falls out of the properties of e(x), cos(x), and sin(x) for real numbers, and the single property of i that it is a solution of x^2 + 1 = 0.

I also think that the definition that e(x) is "take e x times in multiplication and then compute the limit" is any more intuitive. I certainly don't think that `e^2 = e(2) = lim (1 + 2/n) ^ n, with n-> infinity` is any more intuitive than the definition of `e(2i) = lim (1 + 2i/n)^n, with n -> infinity`.

> which is somewhat pointless because like I mentioned - the value of e (i.e 2.7) is never really used to compute anything in regards to complex numbers in polar form of re^ix, all you care about is r and the x which is the angle

This is also not really true, because the value of e is deeply tied to the values of cos x and sin x. This also becomes visible if you want to compute 2^(ix). 2^(ix) = e ^ ix (log_e 2) = cos (x log_e 2) + i sin (x log_e 2), using log_e to denote the natural logarithm to make it clearer that it is related to e. So the value of e itself is still there in the formula, even if we discount the relationship between e and cos and sin.

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:

  (e^(pi\*i))^i = 
    = (10^(log_10 e \* pi \* i))^i
    = (cos (log_10 e \* pi) + i sin (log_10 e \* pi)) ^ i
    = 10^(log_10 $exp)i
    = cos $exp + i sin $exp

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).

I still think you misunderstanding what im saying. i don't have a problem with the math working out, its that i have a problem with the math being able to be applied in the first place.

Let me try a different way.

Suppose you start with just the i number line with the rules that exist. You have zero, integer i's, rational is, and even irrational i's. All seems good. Then you start to define operations. i*i is undefined (because it goes to -1, and the reals are outside of your domain that you are working with currently). And this means you can't effectively do any sort of futher work in defining exponentiation, or limits, because you can have purely complex polynomials with just undefined terms.

So like you said, complex is R+, not something like RxR. The definition of complex numbers is intrinsic to real numbers - its an enhancement on real numbers. And by extension, all the math works out when you do taylor series with e^i and such.

But this pretty much means its a rigid definition, i.e you are defining something in a certain construction to supplement reals.

And as for geometric claim, my argument with that is that just like when you have x, and then you add <x,y> in some form and way, you are defining geometry. So in defining a+bi, you are defining geometry.*

Yeah i is not a number. Once you define complex numbers from reals and i, i becomes a complex numbers but that's a trick

i is not a "trick" or a conceit to shortcut certain calculations like, say, the small angle approximation. i is a number and this must be true because of the fundamental theorem of algebra. Disbelieving in the imaginary numbers is no different from disbelieving in negative numbers.

"Imaginary" is an unfortunate name which gives makes this misunderstanding intuitive.

https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...

However, what's true about what you and GP have suggested is that both i and -1 are used as units. Writing -10 or 10i is similar to writing 10kg (more clearly, 10 × i, 10 × -1, 10 × 1kg). Units are not normally numbers, but they are for certain dimensionless quantities like % (1/100) or moles (6.02214076 × 10^23) and i and -1. That is another wrinkle which is genuinely confusing.

https://en.wikipedia.org/wiki/Dimensionless_quantity

Yes I agree, I was just arguing against "i=0+1*i so it's by definition a complex number" which is a self referential definition

If you take this tack, then 0 and 1 are not numbers either.

i is not a real number, is not an integer, is not a rational etc.

You need a base to define complex numbers, in that new space i=0+1*i and you could call that a complex number

0 and 1 help define integers, without {Empty, Something} (or empty, set of the empty, or whatever else base axioms you are using) there is no integers

The simple fact you wanted to write this:

i=0+1*i

Makes i a number. Since * is a binary operator in your space, i needs to be a number for 1*i to make any sense.

Similarly, if = is to be a binary relation in your space, i needs to be a number for i={anything} to make sense.

Comparing i with a unary operator like - shows the difference:

i*i=-1 makes perfect sense

-*-=???? does not make sense

i is a complex number, complex numbers are of the form real + i*real... Don't you see the recursive definition ? Same with 0 and 1 they are not numbers until you can actually define numbers, using 0 and 1

  i*i=-1 makes perfect sense

This is one definition of i. Or you could geometrically say i is the orthogonal unit vector in the (real,real) plane where you define multiplication as multiplying length and adding angles

There's no issue with recursive definitions. That's how arithmetic was original formalized by Peano's axioms [1].

[1] https://en.wikipedia.org/wiki/Peano_axioms

i is also a quaternion. So by this logic we could say complex numbers are made up of quaternions. But we don’t say such things because they wouldn’t be a good mental model of what we want to talk about.

> i is also a quaternion

Yeah, so ?

The fraction 1/2 is a rational and also a real and also a complex numbers and also a quaternion also an octonion ....

We use a number along with the minimal abstraction that is sufficient for our purpose.

i=0+1*i =0+1*(0+1*(0+1*(0+1*(0+1*(0+1*( ...*

... = -2 - (-3) = -1 - (-2) = 0 - (-1) = 1 - 0 = 1 = 2 - 1 = 3 - 2 = 4 - 3 = 5 - 4 = ...

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