Difference between revisions of "The Road to Reality Study Notes"

We end the section with the idea of finite multiplicative groups, or [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups] Z_n, which contain $$n$$ quantities 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 $$n=3$$, leading to three elements $$1, ω, ω^2$$ with <math>ω=e^\frac{2πi}{3}</math>. Note $$ω^3=1$$ and $$ω^-1=ω^2$$.  These form a cyclic group Z₃ and in the complex plane, represent vertices of an equilateral triangle.  Multiplication by ω rotates the triangle through $$\frac{2}{3}π$$ anticlockwise and multiplication by $$ω^2$$ turns it through $$\frac{2}{3}π$$ clockwise. The cyclic group is graphically shown below:

We end the section with the idea of finite multiplicative groups, or [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups] Z_n, which contain $$n$$ quantities 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 $$n=3$$, leading to three elements $$1, ω, ω^2$$ with <math>ω=e^\frac{2πi}{3}</math>. Note $$ω^3=1$$ and $$ω^-1=ω^2$$.  These form a cyclic group Z₃ and in the complex plane, represent vertices of an equilateral triangle.  Multiplication by ω rotates the triangle through $$\frac{2}{3}π$$ anticlockwise and multiplication by $$ω^2$$ turns it through $$\frac{2}{3}π$$ clockwise. The cyclic group is graphically shown below:

[[File:Fig5p11.png|thumb|center]]

[[File:Fig5p11.png|thumb|center]]

== Chapter 6 Real-number calculus ==

== Chapter 6 Real-number calculus ==