Linked Questions

11
votes
5answers
6k views

How does $e^{i x}$ produce rotation around the imaginary unit circle?

Euler’s formula states that $e^{i x} = \cos(x) + i \sin(x)$. I can see from the MacLaurin Expansion that this is indeed true; however, I don’t intuitively understand how raising $e$ to the power of $...
2
votes
3answers
81 views

The proof of the formula $z = |z|e^{i \phi}$ [duplicate]

I would like to proof that: $$z = |z|\big(\cos(\phi) + i \sin(\phi)\big)$$ of course the second part can be shown using Euler's formula. That's why I would like to prove that: $$z = |z|e^{i \phi}$$ I ...
422
votes
23answers
125k views

How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?

How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my math ...
27
votes
5answers
9k views

Difference between $\mathbb C$ and $\mathbb R^2$ [duplicate]

What are the basic differences between $\mathbb C$ and $\mathbb R^2$? The points in these two sets are written as ordered pairs, I mean the structure looks similar to me. So what is the reason to ...
16
votes
6answers
3k views

Simple Proof of the Euler Identity $\exp{i\theta}=\cos{\theta}+i\sin{\theta}$

My question is too simple. We know all that if we define the exponential function on $\mathbb{C}$ then we define the real part and imaginary part of $\exp{it}$ as $\cos{t}$ and $\sin{t}$. So if we ...
2
votes
4answers
293 views

Philosophical question about Pi and connections in maths

Pi is the ratio of circumference of a circle to its diameter. Okay. Got that, easy enough. Now, why does the following equality hold true? $$ \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{...
1
vote
2answers
141 views

Complex numbers and conjugates. [closed]

Given that $|z|=√3$, solve the equation $$2\overline{z}+\frac3{iz}=\sqrt{15}.$$ How to solve this question without a calculator?
0
votes
4answers
142 views

How can $e^{i\pi}+1$ be zero? [duplicate]

So, most of us are familliar with Euler’s equation stating that $e^{i\pi}+1=0$. But I was wondering: how can an irrational number to the power of another irrational number equal a whole integer? And ...
-6
votes
2answers
157 views

Is $e^{i\theta}$ a circle? [duplicate]

$e^{i\theta}=\cos\theta +i\sin\theta$. By that statement, we conclude that e^iθ is a circle on the complex plane for real values of $\theta$ because the second side of the equation $\cos\theta +i\sin\...
1
vote
1answer
73 views

Using the fifth root of unity to show the cosine equation

Consider the equation expressed the fifth root of unity: $z^5-1=0$ To show that: $$2(\cos(\frac{2\pi}{5})+\cos(\frac{4\pi}{5}))=-1\\4\cos(\frac{2\pi}{5})\cos(\frac{4\pi}{5})=-1$$ I have already ...