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The Road to Reality Study Notes: Difference between revisions
→5.4 Complex Powers
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Penrose notes an interesting curiosity for the quantity $$i^i$$. We can specify <math>logi=\frac{1}{2}πi</math> because of the general relationship <math>logw=logr+iθ</math>. If $$w=i$$, 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 $$\frac{π}{2}$$). This specification, and all rotations, amazingly achieve real number values for $$i^i$$. | Penrose notes an interesting curiosity for the quantity $$i^i$$. We can specify <math>logi=\frac{1}{2}πi</math> because of the general relationship <math>logw=logr+iθ</math>. If $$w=i$$, 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 $$\frac{π}{2}$$). This specification, and all rotations, amazingly achieve real number values for $$i^i$$. | ||
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 $$n$$ 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 Z<sub>3</sub> 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<sub>n</sub>, which contain $$n$$ 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 $$n=3$$, 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 Z<sub>3</sub> 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]] | ||