Download Complex numbers via rigid motions

Complex numbers via rigid motions
Just a bit mathematical !
Rotations around the origin.
180 deg or pi
Y = -y
X = -x
so (1,0) goes to (-1,0)
Let us call this transformation H (for a half turn)
90 deg or pi/2
Y = x
X = -y
so (1,0) goes to (0,1) and (-1,0) goes to (0,-1)
and (0,1) goes to (-1,0) and (0,-1) goes to (1,0)
Let us call this transformation Q (for a quarter turn)
Then H(x,y) = (-x,-y)
and Q(x,y) = (-y,x)
We know that H(H(x,y)) = (x,y) and if we are bold we can write HH(x,y) = (x,y)
and then HH = I, where I is the identity or do nothing transformation.
In the same way we reach QQ = H
Now I is like multiplying the coodinates by 1
and H is like multiplying the coordinates by -1
This is not too outrageous as a dilation can be seen as a multiplication of the
coordinates
by a number <> 1
So, continuing into uncharted territory,
we have H squared = 1 (fits with (-1)*(-1) = 1
and Q squared = -1 (fits with QQ = H, at least)
So what is Q ?
Let us suppose that it is some sort of a number,
and let its value be called k.
What we can be fairly sure of is that k does not multiply each of the
coordinates.
This is most likely only for the normal numbers.
Now the "number" k describes a rotation of 90, so we would expect that the
square root of k
to describe a rotation of 45
At this point it helps if the reader is familiar with extending the rational
numbers by the
introduction of the square root of 2 (a surd, although this jargon seems to have
disappeared).
Let us assume that sqrt(k) is a simple combination of a normal number and a
multiple of k:
sqrt(k) = a + bk
Then k = sqr(a) + sqr(b)*sqr(k) + 2abk, and sqr(k) = -1
which gives k = sqr(a)-sqr(b) + 2abk and then (2ab-1)k = sqr(a) - sqr(b)
From this, since k is not a normal number, 2ab = 1 and sqr(a) = sqr(b)
which gives a = b and then a = b = 1/root(2)
Now we have a "number" representing a 45 degree rotation. namely
(1/root(2)*(1 + k)
If we plot this and the other rotation numbers as points on a coordinate axis
grid with
ordinary numbers horizontally and k numbers vertically we see that all the
points are on the
unit circle, at postions corresponding to the rotation angles they describe.
OMG there must be something in this ! ! !