alt Hacker NewsRotations 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.
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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.
> 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.