I walk in the x direction, if I walk twice as fast, 2x, if I walk backwards, -x. What about ix? If I say that I walk an imaginary distance ix then this means in physics and maths that I walk perpendicular to the direction of x (there must be a new dimension).

If I scale an object by 2 in it's x dimension then it doubles in size, if I scale an object by -1 then I invert it. What happens when I scale an object by i? Then I change it's size in the direction perpendicular to x.

It appears to me that imaginary numbers are used to describe things happening in the dimensions that are orthogonal to the dimension being considered. It could be error away from a line, or another effect like magnetism acting perpendicular to the observed effect, it is about the actions perpendicular to the dimension being considered?

Is this a correct way to understand it?

If there are many perpendicular dimensions can they all be described by one single complex dimension? Do we transfer the information from many orthogonal dimensions into one complex dimension for ease of seeing how all the other dimensions in the system interact with the dimension we are studying?

  • 2
    $\begingroup$ Please keep in mind that thinking about 'doubling in size' and such can be misleading since there is no ordering on complex numbers. $\endgroup$ – Ezze Nov 6 '19 at 12:43
  • $\begingroup$ Ezze: Agreed, was writing for simplicity, the concept that using imaginary numbers is talking about dimensions perpendicular to the one being considered was the important idea I was trying to convey. Would you agree with this statement? $\endgroup$ – OzOz Nov 6 '19 at 12:50
  • $\begingroup$ I do not fully understand what do you mean by perpendicular dimension. Is 'mass' perpendicular to 'frequency', for example? $\endgroup$ – Ezze Nov 6 '19 at 12:54
  • $\begingroup$ I think you mean to say "if you scale by i, you rotate it 90 degrees". That's correct. $\endgroup$ – Abhimanyu Pallavi Sudhir Nov 6 '19 at 13:02
  • $\begingroup$ Abhimanyu: Is that a correct description of the situation? It's not really about rotation, it's about things happening in a perpendicular dimension. If there are only 2 dimensions involved then you are correct, but with 3 dimensions, x, y, z, when you study the x dimension the y and z are complex, x + i(y+z) but if you study the y then y + I(x+z), which is not really a rotation by 90 degrees? It seems that the textbooks simplify it to the 2 dimension case where it is a 90 rotation? $\endgroup$ – OzOz Nov 6 '19 at 13:06

A complex number z=a+bi can be viewed as (a,b) in real numbers, hence you can think about as the dimension of complex numbers (in terms of real numbers) is 2 and the dimension of complex numbers (in terms of complex numbers) is 1. The i in a+bi means square root of -1, you cannot scale an object by square root of -1. And one cannot walk in that direction either, but you can think of it as having an orthogonal system, between real and imaginary numbers. In that case Re(z)=a and Im(z)=b. The grid between Re and Im can be sketched as orthogonal, called the Argand diagram and the vector inside it pointing at (a,b) is the complex number a+bi. Now, there are more advanced forms, like a circle in that grid is r exp(iθ), where r is the radius and θ an angle. It can also be used as exp(-iS) where S is the action. To conclude, it is a mathematical tool with great applications in physics, due to its many properties, but the most realistic visualization of it is 2dimensional space of real numbers or rotations.

  • $\begingroup$ Thank you for your answer. So, what would be the difference between them? $\endgroup$ – OzOz Nov 11 '19 at 0:31
  • $\begingroup$ Between complex and real numbers or the real and imaginary numbers? $\endgroup$ – Ash Nov 11 '19 at 11:42
  • $\begingroup$ Between complex numbers and a normal 2d vector space? $\endgroup$ – OzOz Nov 11 '19 at 12:26
  • $\begingroup$ Well as spaces, they pretty much look alike, but mostly, as algebraic structures they have big differences. So some similarities is that they are both metric spaces, groups but for example, differentiability of functions changes. Complex space allows for let's say shortcuts for proves, that real numbers won't, hence it is really popular. $\endgroup$ – Ash Nov 11 '19 at 13:00

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