11 ms·

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 sig

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.

direwolf20 8mo ago
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.
- ActorNightly 8mo ago
  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.
  - deleted 8mo ago
    [deleted]
  - srean 8mo ago
ttoinou 8mo ago
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
- chongli 8mo ago
  If you take this tack, then 0 and 1 are not numbers either.
  - ttoinou 8mo ago
    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
    - chongli 8mo ago
      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
      - ttoinou 8mo ago
        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
        chongli 8mo ago
tsimionescu 8mo ago
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.
- ActorNightly 8mo ago
  >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.
jonahx 8mo ago
> 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?
- ActorNightly 8mo ago
  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.
  - srean 8mo ago
    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.
  - jonahx 8mo ago
    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?

direwolf20 8mo ago

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.

ActorNightly 8mo ago
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.
- deleted 8mo ago
  [deleted]
- srean 8mo ago

ttoinou 8mo ago

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

chongli 8mo ago
If you take this tack, then 0 and 1 are not numbers either.
- ttoinou 8mo ago
  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
  - chongli 8mo ago
    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
    - ttoinou 8mo ago
      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
      - chongli 8mo ago

tsimionescu 8mo ago

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.

ActorNightly 8mo ago
>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.

jonahx 8mo ago

> 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?

ActorNightly 8mo ago
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.
- srean 8mo ago
  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.
- jonahx 8mo ago
  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?