I dreamed this up so I'll write it down and see what happens!
I asked myself this question:
Why is it that a 2 dimensional number is represented by the square root of minus one as its one-dimensional representitive?
I draw a square with sides "1 unit".
This is a 2-D unit.
Imagine the square is standing vertically on the desk. Suppose I look down at it so I only see the top edge. Now it looks 1-dimensional; just a 1-unit line.
What is this 1 visible Dimension, this visible top edge to the square; as a 2D viewpoint?
It is a view of the missing side of the unit square! It is the square root of the missing one!
So the square root of minus one is: an alternative 2-D view of one of the alternative perspectives in a 2-D view.
A view of a square that is different from the usual view, by looking at just the top edge and seeing that edge as being a 2-D view so that edge is the square root of the missing remainder of the square.
So an alternative view that something has of itself via a different view of itself (in this case, via looking at itself as a top edge of a vertical square).
So the square root of minus one involves seeing a different dimension of itself via self-reference?
Complex number: a + ib
An ordinary number 6 can be seen as:
a vertical rectangle with vertical side 3 units and top side 2 units.
Look at this from above: you see 2 units.
A 2-D view of that above-view:
(1D) x2 + (square root minus 1D x 3units)
2 + i3
a - ib so 2 - i3
Real number: (a + ib)(a - ib) =
(2 + i3) (2 - i3) = 4 - - 9 = 4 + 9 = 13.
So the complex number gives a 2-D view of 13? A 2x2 square plus a 3x3 square; a view of 13 as a sum of squares? A sum of a 2x2 square and a 3x3 square?
A 1-D view take a side from each square and get 2x3 = 6?