the function of i

30 04 2011

Interested development of thought, tying together multiplication by i on the complex plane, and the standard model as represented below:

If we take the dimensions to be unconscious to conscious to be our x axis and incompetence to competence to be our y axis, we get a similar mapping from bottom left quadrant to bottom right and so on, anti-clockwise, which is equivalent to multiplying a complex number by powers of i.

We can abstract this system further, if we consider any two dimensions of “opposites”. The establishment of zero on the number line as a continuous and contiguous part, gives rise to the extension of the number line into the negative direction. It allows the notion of reflection, how 6 can be converted into -6 by the multiplication of -1. That is, the negative identity in multiplication, -1, transforms any number x to -x, and vice versa. I mentally refer to this as reflection, whereas it can be conceived as being rotation.

When we superimpose one dimension with another at right angles centreed at the mutual zero, the usual x and y axes, the multiplication by i (ie the root of -1) rotates a point through 90˚. Note that multiplication by i^2 is the same as multiplying it by -1, the negative identity of multiplication, and hence performs the same function: rotation by 180˚ (the notion of reflection breaks down in two dimensions in this case).

How can this function occur if we do not have zero in our number line? When we consider simply A and not A as “opposites”?




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