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.
> 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.
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.
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.
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.
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.
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.
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
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 anglesThere'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