A complex number represents a rotation and a scaling and translation of another complex number.
That is about as physical as it gets in mathematics.
Given two complex numbers,
a+b is translating
a*b is rotating
a by the angle of
b, then scaling the result by the magnitude of
A strange thing happens because the same value
b represents both a scaling/rotation operation, and a translation operation, on another complex number.
So how do we inspire that physically?
If we have:
(a + b)*c
this is obviously "start with
a. Move by
b. Then rotate/scale based on
c. The algebra lets us break this apart:
a*c + b*c
which is really neat. There is a whole pile of things you can align this algebraic manipulation to physical operations here.
The next bit is a bit strange:
a * (b + c)
what happens when you add complex rotate-and-scale operations? Well, algebra tells us this is:
a*b + a*c
b+c becomes "what if you took something, rotated it scaled by
b and then by
c, then added the result".
This is strange operation. But you know what isn't a strange operation?
(b * lambda + c * (1-lambda))
This is called an affine combination of
So we have two different rotation/scales,
c. And we want to interpolate between them smoothly.
a* (b * lambda + c * (1-lambda))
as lambda goes from 0 to 1 gives us the result of transforming
a first by
b, then finally by
c, and having a smooth transformation in between.
a * (b * c)
b, then by
b*c is the combination of the two rotations/scales in one value.
So applying the same rotation/scale twice would be:
a * (b*b)
a * b^2
which means that if c^2 = b, then
a*c^2 = a*b
c is the operation that if you do twice, you get
On the real line there are two different operations such that if you do either one twice, you get 4. They are
2. The same holds in the complex numbers.
Of interest is
a * c^3
because on the real line, there is only one scale factor that can get
8 if you do it 3 times. But in the complex numbers there are 3.
To see this, look at scaling by 1. On the real line, there are two scale/rotates that reach 1 by applying twice -- "-1" and "1". These correspond to the complex numbers
1 e^0 and
1 e^(pi i) -- no scale, no rotation, and no-scale, half rotation.
If you do half rotation twice, you get a full rotation, ie nothing.
How about the cube root of 1? Something you do 3 times that is a scale or rotation, and afterwards you end up being back where you started.