Complex numbers via rigid motions
Just a bit mathematical !
I wrote this in response to a post by Michael Pershan:
The way I have presented it is showing how mathematicians think. Get an idea, try it out, if it appears to work then attempt to produce a logical and mathematically sound derivation.
(This last part I have not included)
The idea is that wherever you have operations on things, and one operation can be followed by another of the same type, then you can consider the combinations of the operations separately from the things being operated on. The result is a new type of algebra, in this case the algebra of rotations.
Read on . . .
Rotations around the origin.
angle 180 deg or pi
Y = -y, and X = -x —> coordinate transformation
so (1,0) goes to (-1,0) and (-1,0) goes to (1,0)
Let us call this transformation H (for a half turn)
angle 90 deg or pi/2
Y = x, and 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)
Applying H twice we have 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 find 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, definitely not a normal one,
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 appears to be meaningful 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 positions corresponding to the rotation angles they describe.
OMG there must be something in this ! ! !
The continuation is left to the reader (as in some Victorian novels)
ps. root() and sqrt() are square root functions, and sqr() is the squaring function .
pps. Diagrams may be drawn at your leisure !