105
edits
The Road to Reality Study Notes: Difference between revisions
→5.3 Multiple valuedness, natural logarithms
| Line 204: | Line 204: | ||
=== 5.3 Multiple valuedness, natural logarithms === | === 5.3 Multiple valuedness, natural logarithms === | ||
We need to be careful with the above assertion of the logarithm, mainly since $$b^z$$ and logbw are ‘many valued’. Solving the equations would require a particular choice for b to isolate the solution. With this, the ‘base of natural logarithms’ is introduced as the number ‘e’, whose definition is the power series 1+1/1!+1/2!+1/3!+…If b=e, then b^z=e^z=1+z/1!+z^2/2!+z^3/3!+…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, z=logw if w=e^z. | |||
However, even with this natural logarithm we run into multi-valuedness ambiguity from above. Namely that z still has many values that lead to the same solution, z+2*pi*i*n, where n is any integer we care to choose. This represents a full rotation of 2*pi in the complex plane which achieves the same point, z. | |||
Penrose goes further in representing z with polar coordinates, z=logr+i*theta. Showing then e^z=e^logr+i*theta=re^i*theta. This formulation clearly 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 r=1, such that we recover the ‘unit circle’ in the complex plane with w=e^i*theta. We can combine trigonometry on this circle to show e^i*theta=cos(theta) + i*sin(theta). | |||
Different values can arrive at the same value. Rotation brings you back to the same place repeatedly. | Different values can arrive at the same value. Rotation brings you back to the same place repeatedly. | ||