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The Road to Reality Study Notes: Difference between revisions
→5.4 Complex Powers
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=== 5.4 Complex Powers === | === 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 $$logw$$ has been specified. As an example, $$w^z$$ with $$z=frac{1}{2}$$. We can specify a rotation for $$logw$$ to achieve $$+w^frac{1}{2}$$, then another rotation of $$logw$$ to achieve $$-w^frac{1}{2}$$. The sign change is achieved because of the Euler formula <math>e^{π*i}=-1</math>. Note the process: | Returning to the ambiguity problem of multi-valuedness, it seems the best way to avoid issues is when a particular choice of $$logw$$ has been specified. As an example, $$w^z$$ with $$z=\frac{1}{2}$$. We can specify a rotation for $$logw$$ to achieve $$+w^\frac{1}{2}$$, then another rotation of $$logw$$ to achieve $$-w^frac{1}{2}$$. The sign change is achieved because of the Euler formula <math>e^{π*i}=-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 $$+w^frac{1}{2}$$ or $$-w^{frac{1}{2}}$$. | <math>w^z=e^{zlogw}=e^{zre^{iθ}}=e^{ze^{iθ}}</math>, then specifying rotations for theta allows us to achieve either $$+w^\frac{1}{2}$$ or $$-w^{\frac{1}{2}}$$. | ||
Penrose notes an interesting curiosity for the quantity $$i^i$$. We can specify <math>logi=(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=(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] \BbbZ, 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 w^z= e^ze^i(theta+2*pi*i*n) with n=3, leading to three elements 1, XX, XX^2 with XX=e^2*pi*i/3. Note XX^3=1 and XX^-1=XX^2. These form a cyclic group Z3 and in the complex plane, represent vertices of an equilateral triangle. Multiplication by XX rotates the triangle through 2/3*pi anticlockwise and multiplication by XX^2 turns it through 2/3*pi clockwise. | We end the section with the idea of finite multiplicative groups, or [https://en.wikipedia.org/wiki/Cyclic_group cyclic groups] \BbbZ, 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 w^z= e^ze^i(theta+2*pi*i*n) with n=3, leading to three elements 1, XX, XX^2 with XX=e^2*pi*i/3. Note XX^3=1 and XX^-1=XX^2. These form a cyclic group Z3 and in the complex plane, represent vertices of an equilateral triangle. Multiplication by XX rotates the triangle through 2/3*pi anticlockwise and multiplication by XX^2 turns it through 2/3*pi clockwise. | ||