Euler's formula describes two equivalent ways to move in a circle. Starting at the number $1$, see multiplication as a transformation that changes the number $1 \cdot e^ {i\pi}$. Regular …

Oct 13, 2021 · Prove Euler's identity $e^ {i\theta} = \cos \theta + i \sin \theta$ using Taylor series. Then plug in $\theta = \pi$.

Mar 22, 2014 · Can somebody explain to me why the absolute value of a complex exponential is 1? (Or at least that's what my textbook says.) For example: $$|e^ {-2i}|=1, i=\sqrt {-1}$$

I love how the OP said, "I put it into the calculator and it works". Love this, my favorite example of how non-intuitive math can be.

Apr 20, 2015 · @Jamie: FWIW, I first saw this fact outside of class in about 7th grade, saw it "formally" for the first time in pre-calculus, got a first non-rigorous proof in Calculus II, and …

Aug 28, 2010 · Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and …

Apr 19, 2017 · Then go build some other numbers which behave the way you want instead.

Mar 30, 2018 · The community reviewed whether to reopen this question 4 years ago and left it closed: Original close reason (s) were not resolved

Jul 24, 2013 · Complex conjugation is an automorphism of order 2, meaning $\,\overline {\overline z}=z\,\,,\,\,\forall\,z\in\Bbb C\,$ , so if the conjugate of $\,e^ {-iwt}\,$ is ...

Oct 25, 2021 · How can I find all roots of logz=ipi/2? I tried to use the formula Logz=lnr+itheta, but I cannot do the next step.