Here Goes...

This is as far as I got in Road when I was initially working through it in March and April of last year.  I was stumped and remain stumped by the second part of the challenge Penrose sets in Exercise 5.4, which is to show without detailed calculation or trigonometry that multiplication of complex number z by another complex number w corresponds to two geometric transformations of the representation of z on the complex plane: (a) a uniform expansion/contraction and (b) a non-reflective rotation.  Penrose hints that this is due to two properties of complex numbers z and w:

w(z₁+z₂) = wz₁+wz₂

(1)

w(iz) = i(wz)

(2)

    Penrose also tells us that (1) implies the preservation of linearity in the multiplication of complex numbers; and that (2) implies the preservation of right angles in the multiplication of complex numbers.  Which leaves me confused as to whether we are allowed to use the fact that multiplication of a complex number by i corresponds to a rotation of z through a right angle in the complex plane, or whether we still have to prove that.  My guess is that this is simply "guilty knowledge" that will have to be shown logically, but that ends up sucking for me because while (a) is fairly easy to show in a way that I think is consistent, (b) has proven very difficult to show without making use of trigonometry and/or "detailed calculation," whatever the hell "detailed calculation" actually means. Anyhoo, I'll talk more about this when I get around to talking about possible approaches to showing (b). 

     In general, Penrose seems to want for us to be able to think about what's happening with these complex number in terms of geometry, not algebra, and to use "mostly" geometric arguments to demonstrate these properties of complex number multiplication.  Since I've always been a more rote-calculation person it's a good exercise for my brain to try the harder, or at least different, geometric approach.

Setup

      Just to make sure I know what I'm talking about throughout, here are some basic definitions.  For the rest of the time we'll be talking about two arbitrary complex numbers z and w defined as follows:

z = a+bi

w = c+di

  

      Where a, b, c, and d are all real numbers.  Also we will be talking about representing z and w as vectors on the "complex plane" which is defined by a horizontal "real" axis and a vertical "imaginary" axis, as shown in the following diagram:

    We are also going to need to assume some givens about how these vectors can be "added" geometrically.  For example, z can be represented in the complex plane as the "resultant" vector z of vectors a and b, whose magnitudes correspond to the real numbers a and b respectively.  Finding the resultant vector from component vectors on the complex plane works according to the same parallelogram rule as works for vectors on the real plane:

Uniform Expansion/Contraction

     Show that multiplying z by w involves a uniform expansion/contraction.  Not sure what Penrose means by "uniform, " but I assume that he means an expansion/contraction by a constant factor.  In any case he seems to be asking about how multiplication affects the geometric length of the vector z in the complex plane  Let's represent this length as |z|.  Penrose is asking us to look at |wz|, i.e. the magnitude of the length of the geometric representation of the complex product wz, and see whether it differs from |z| by a constant factor.

      We begin by looking at a bit of algebra (which hopefully doesn't count as the sort of "detailed calculation" which Penrose is disallowing):

|wz| = |z(c+di)| = |cz| + z(di)| = c|z| + id|z|

(3)

     This assumes some things about two complex numbers z and w that seem to me to be true:

|zw| = |z||w|

|z+w| = |z| + |w|

|i| = i

     The first of these facts is the most important for us.  In particular, we need to show in a simpler case that |bz| = b|z| where b is a real number.  

     We can show this quite easily by interpreting bz as z added to itself b times, which can be represented in the complex plane as a simple geometric addition of the vector z to itself b times (b is a scalar value):

     Inspecting this we can see that |bz| = b|z|, which makes me feel better about saying |bz|=b|z|.

   

     Now according to (3) |zw| can be rewritten as c|z| + id|z|.  Since |z| is a scalar value, we can interpret this sum as a third complex number whose geometric interpretation is::

     Which means that the magnitude |z(c+di)| depends on the magnitude |z| multiplied by constant factors c and d.  Given (3), this means that |wz| also depends on these constant factors.  Which means that the multiplication of z by w results in a third complex number, whose magnitude is related to the magnitude of z by a constant factor.  Which is (I think) what Penrose wanted us to prove.

Rotation

      Now we have to show that multiplying z by w also involves a "non-reflective" rotation.  Since (as will be clear shortly) multiplying z by the factor i² = -1 is analogous to a geometric reflection of z through the origin, I guess that Penrose did not mean for us to consider "degenerate" cases such as when w equals i, -1, 1, or 0.  Even so, we will need to start out by considering what happens, geometrically in the complex plane, when you multiply z by the "special" value of i.  Algebraic multiplication by i of z = a+bi results in a new number z' = -b+ai.  A relationship between z and z' becomes evident upon examinations of their analagous vectors z and z' in the complex plane:

      We are most interested in the angle between z and z'.  If you imagine the two vectors defining triangles with one side along the real and imaginary axes respectively you will see that the triangles are similar.  Therefore the angle γ between z and the real axis is equal to the angle between z' and the imaginary axis.  Define β as the angle between z and the imaginary axis.  We know that γ + β equals 90 degrees.  Then note that angle between z and z' is equal to β plus the angle between z' and the imaginary axis.  But this is equal to β+γ as well, therefore the angle between z and z' is equal to 90 degrees.  And so multiplication of the complex number z by i is analogous to a counterclockwise 90-degree rotation of  z in the complex plane.  

      Now consider multiplication of z  by another arbitrary complex number w.  We can represent w by the corresponding vector w on the complex plane.  Suppose that w has components w_R and w_I:

     Examining the above, and using (1), we see that:

zw = z(w_R+w_I) = zw_R+zw_I

     Or algebraically:

zw = zw_R+zw_I

(4)

     However again by examination we know that w_I is nothing more than i multiplied by some real constant C, while w_R is nothing more than a real constant D.  Therefore (4) can be rewritten as:

zw = z(iC)+zD

     Using (2) we slightly rewrite this as:

zw = iCz+Dz

zw = iCz+Dz

(5)

     We have shown that multiplication of a vector such as z by a constant such as C or D results in  a dilation or contraction.  We have also shown that a multiplication of a vector in the complex plane by i results in a 90-degree counterclockwise rotation.  Therefore, the vector product zw is equivalent to combining a counterclockwise 90-degree rotation and dilation/contraction of z with another, not necessarily identical contraction/dilation of z:

     One component of the resultant vector equal to zw will be a stretched out/compressed copy of z rotated ninety degrees, and the other will be a non-rotated copy of z stretched out/compressed by a different amount.  Assuming that w is not one of various "degenerate" cases, the product of the two will therefore be a contraction/expansion combined with a rotation which is what we were supposed to show.