Editing The Road to Reality Study Notes (section)

== Chapter 5 Geometry of logarithms, powers, and roots ==

=== 5.1 Geometry of complex algebra ===
Penrose asks us to view complex addition and multiplication as transformations from the complex plane to itself, rather than just as simple addition and multiplication.  The visual representations of these operations are given as the parallelogram and similar-triangle laws for addition and multiplication respectively.  
[[File:Fig 5p1.png|thumb|center]]
Rather than just ‘adding’ and ‘multiplying’ these can be viewed as ‘translation’ and ‘rotation’ within the complex plane. For example, multiply a real number by the complex number <math>i</math> rotates the point in the complex plane π/2 and viewing the parallelogram and similar-triangle laws as translation and rotation:
[[File:Fig 5p2.png|thumb|center]]
Penrose further introduces the concept of polar coordinates where <math>r</math> is the distance from the origin and <math>θ</math> is the angle made from the real axis in an anticlockwise direction.
[[File:Fig 5p4.png|thumb|center]]

=== 5.2 The idea of the complex logarithm ===

Armed with both the cartesian and polar representations of complex numbers, it is now possible to show that the multiplication of two complex numbers leads to adding their arguments and multiplying the moduli.  This, for the moduli, converts multiplication into addition.

This idea is fundamental in the use of logarithms.  We first start with the expression <math>b^{m+n} = b^m \times b^n</math>, which represents the idea of [https://en.wikipedia.org/wiki/Exponentiation exponentiation]. This is easy to grasp for <math>m</math> and <math>n</math> being positive integers, as each side just represents <math>m+n</math> instances of the number <math>b</math>, all multiplied together.  If <math>b</math> is positive, this law is then showed to hold for exponents that are positive integers, values of 0, negative, and fractions.  If <math>b</math> is negative, we require further expansion into the complex plane. 

We would need a definition of <math>b^p</math> for all complex numbers <math>p,q,b</math> such that <math>b^{p+q} = b^p \times b^q</math>.  If we then define the [https://en.wikipedia.org/wiki/Logarithm logarithm to the base b] as the inverse of the function <math>f(z) = b^z</math> such that <math>z=log_bw</math> for <math>w=b^z</math> then we should expect <math>z=log_b(p \times q) = log_bp + log_bq</math>.  This would then convert multiplication into addition and allow for exponentiation in the complex plane.

=== 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.)

=== 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]]

=== 5.5 Some Relations To Modern Particle Physics ===
Penrose rounds out the chapter with some examples of complex concepts in the world of particle physics.  Additive quantum numbers were briefly introduced in section 3.5, and here we are introduced to multiplicative quantum numbers, which are quantified in terms of nth roots of unity.

The notion of [https://en.wikipedia.org/wiki/Parity_(physics) parity] is introduced as approximately a multiplicative quantum number with n=2, and an example is the family of particles called [https://en.wikipedia.org/wiki/Boson bosons].  Penrose notes that [https://en.wikipedia.org/wiki/Fermion fermions] could also be considered a parity group but it is not the normal convention.   The distinction between these two particles are that bosons are completely restored to their original states under a <math>2π</math> rotation, whereas fermions require <math>4π</math> (two rotations).  Thus a multiplicative quantum number of <math>-1</math> can be assigned to a fermion and <math>+1</math> to a boson.

An example of a multiplicative quantum number with <math>n=3</math> relates to quarks, which have values for electric charge that are not integer multiples of the electron’s charge, but in fact <math>\frac{1}{3}</math> multiples.  If <math>q</math> is the value of electric charge with respect to an electron (<math>q=-1</math> for electron charge), then quarks have q=<math>\frac{2}{3}</math> or <math>-\frac{1}{3}</math> and antiquarks q=<math>\frac{1}{3}</math> or <math>-\frac{2}{3}</math>.  If we take the multiplicative quantum number <math>e^{-2qπi}</math>, then we find the values <math>1,ω,ω^2</math> from section 5p4, which constitute the cyclic group Z<sub>3</sub>.