I know that it’s true but why? I’ve never grasped this topic.
In: Mathematics
First, complex numbers are strange
Second, e^(i Pi)=-1 is bests understood by thinking of representing your complex numbers on a graph. The X axis is real numbers, and the Y axis is imaginary numbers. e^(i Pi) is better generalized as R e^(i theta) where R is the distance of your point from (0,0) (aka the magnitude of the complex number) and Theta is the angle(in radians) that you rotate it through on that graph. A circle has 2 Pi radians in it so if you rotate your point around the origin by Pi radians then it is 1 unit (our R) directly to the left of the origin so its -1 on the x axis
If you were to set theta to Pi/2 then the point would be 1 unit up on Y axis and 0 units over on the X axis so it’d be i. If theta is 3Pi/2 then you get -i, same as if its 7 Pi/2 because the circle resets every 2 Pi radians.
You can represent any complex number this way. 17-6i becomes 18.0278 e^(i 1.892 Pi). [Wolfram alpha even shows you the complex plane to help visualize the answer](https://www.wolframalpha.com/input/?i=17+-+6i)
Latest Answers