So, I'm trying to do some calculations for my Electrical engineering homework. This requires a bit of algebra with complex numbers. I have been finding that some of the calculations that my calculator has been making don't really make sense. This is especially apparent if I try to enter complex numbers into my calculator in their polar form. Correct me if I'm wrong, but it my understanding of switching back and forth between rectangular and polar forms of complex numbers is the following : A*e^(i*x) => A cos(x) + i sin(x)
However, if I input (while the calculator is in degrees mode, and it is set up to output complex quantities in rectangular form) e^(i*45) and press enter the output is : .525+.851 * i. My understanding of complex numbers leads me to believe that the output should have been: cos(45)+sin(45)*i = .707 + .707*i. Also, if I put the calculator in polar mode and input e^(i*45) the output is: 1e^(58.31). Why is it giving me a different output? What is going wrong here? Also, it is saying that e^(i*100)/e^(i*50)= e^(i*-15.211). Also an input of 1+i yields an output of 1.414*e^(i*45)…which is correct. Any idea what's going on here? Is the ti-84 just garbage when it comes to complex algebra in polar form?
Best Answer
Your calculator won't work pleasantly with degrees in this case. It likely defines $e^{ix}$ as the power series $$e^{z} := \sum_{n=0}^\infty \frac {z^n}{n!}$$ In the case of $z=ix$, this simplifies to $\cos(x) + i\sin(x)$ by looking at the respective power series. These latter terms are, of course, in terms of radians; but changing 'radians' to 'degrees' doesn't change the above power series (or else your calculator would screw up, say, $e^2$ and instead output $e^{\pi/90}$ - trying to convert the entire exponent from 'radians' to 'degrees', even though they don't apply here!) So if you're doing things in degrees, you'll either need to plug into $\cos(x) + i\sin(x)$ directly, or first convert into radians.