Editing The Road to Reality Study Notes (section)

=== 5.4 Complex Powers ===
Returning to the ambiguity problem of multi-valuedness, it seems the best way to avoid issues is when a particular choice of <math>logw</math> has been specified.  As an example, <math>w^z</math> with <math>z=\frac{1}{2}</math>.  We can specify a rotation for <math>logw</math> to achieve <math>+w^\frac{1}{2}</math>, then another rotation of <math>logw</math> to achieve <math>-w^\frac{1}{2}</math>.  The sign change is achieved because of the Euler formula <math>e^{πi}=-1</math>. Note the process:
<math>w^z=e^{zlogw}=e^{zre^{iθ}}=e^{ze^{iθ}}</math>, then specifying rotations for theta allows us to achieve either <math>+w^\frac{1}{2}</math> or <math>-w^\frac{1}{2}</math>.

Penrose notes an interesting curiosity for the quantity <math>i^i</math>.  We can specify <math>logi=\frac{1}{2}πi</math> because of the general relationship <math>logw=logr+iθ</math>.  If <math>w=i</math>, then its easy to see <math>logi=\frac{1}{2}πi</math> from noting that y is on the vertical axis in the complex plane (rotation of <math>\frac{π}{2}</math>).  This specification, and all rotations, amazingly achieve real number values for <math>i^i</math>.

We end the section with the idea of finite multiplicative groups, or [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups] Z<sub>n</sub>, which contain <math>n</math> quantities ([https://en.wikipedia.org/wiki/Root_of_unity#:~:text=The%20nth%20roots%20of%20unity%20are%2C%20by%20definition%2C%20the,and%20often%20denoted%20%CE%A6n. nth roots of unity if around the unit circle]) with the property that any two can be multiplied together to get another member of the group.  As an example, Penrose gives us <math>w^z=e^{ze^{i(θ+2πin)}}</math> with <math>n=3</math>, leading to three elements <math>1, ω, ω^2</math> with <math>ω=e^\frac{2πi}{3}</math>. Note <math>ω^3=1</math> and <math>ω^-1=ω^2</math>.  These form a cyclic group Z<sub>3</sub> and in the complex plane, represent vertices of an equilateral triangle.  Multiplication by ω rotates the triangle through <math>\frac{2}{3}π</math> anticlockwise and multiplication by <math>ω^2</math> turns it through <math>\frac{2}{3}π</math> clockwise. The cyclic group is graphically shown below:
[[File:Fig5p11.png|thumb|center]]