Editing The Road to Reality Study Notes (section)

=== 5.3 Multiple valuedness, natural logarithms ===
We need to be careful with the above assertion of the logarithm, mainly since <math>b^z</math> and <math>log_bw</math> are ‘many valued’.  Solving the equations would require a particular choice for <math>b</math> to isolate the solution.  With this, the ‘base of natural logarithms’ is introduced as the [https://en.wikipedia.org/wiki/E_(mathematical_constant) number e], whose definition is the power series <math>1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+…</math>. This power series converges for all values of z which then makes for an interesting choice to solve the ambiguity problem above.  Thus we can rephrase the problem above with the natural logarithm, <math>z=logw</math> if <math>w=e^z</math>.

However, even with this natural logarithm we run into multi-valuedness ambiguity from above.  Namely that <math>z</math> still has many values that lead to the same solution with <math>z+2πin</math>, where <math>n</math> is any integer we care to choose.  This represents a full rotation of <math>2π</math> in the complex plane with all multiples of <math>n</math> achieving the same point, <math>z</math>.

Penrose goes further in representing <math>z</math> with polar coordinates showing <math>z=logr+iθ</math>, then <math>e^z=re^{iθ}</math>.  This formulation shows us that when we multiply two complex numbers, we take the product of their moduli and the sum of their arguments (using the addition to multiplication formula introduced in 5.2).

Rounding out the chapter, Penrose gives us another further representation of assuming <math>r=1</math>, such that we recover the ‘unit circle’ in the complex plane with <math>w=e^{iθ}</math>.  We can 'encapsulate the essentials of trigonometry in the much simpler properties of complex exponential functions' on this circle by showing <math>e^{iθ}=cos(θ) + isin(θ)</math>.  

* <math>e^{i\theta}</math> is helpful notation for understanding rotating 
* <math>e^{i\theta} = cos \theta + i sin \theta</math>
* (Worth looking into [https://en.wikipedia.org/wiki/Taylor_series Taylor Series], which is related.)