Linked Questions

2
votes
2answers
1k views

Prove $e^{i \pi} = -1$ [duplicate]

Possible Duplicate: How to prove Euler's formula: $\exp(i t)=\cos(t)+i\sin(t)$ ? I recently heard that $e^{i \pi} = -1$. WolframAlpha confirmed this for me, however, I don't see how ...
5
votes
2answers
820 views

Intuition behind euler's formula [duplicate]

Possible Duplicate: How to prove Euler's formula: $\\exp(i t)=\\cos(t)+i\\sin(t)$ ? Hi, I've been curious for quite a long time whether it is actually possible to have an intuitive ...
0
votes
1answer
230 views

Why Euler's formula is true? [duplicate]

Possible Duplicate: How to prove Euler’s formula: $\exp(i t)=\cos(t)+i\sin(t)$? I need to know why Euler's formula is true? I mean why is the following true: $$ e^{ix} = \cos(x) + i\sin(x) ...
2
votes
4answers
175 views

Why does $e^{i\pi}=-1$? [duplicate]

I will first say that I fully understand how to prove this equation from the use of power series, what I am interested in though is why $e$ and $\pi$ should be linked like they are. As far as I know ...
-1
votes
2answers
86 views

Why is $sinx$ the imaginary part of $e^{ix}$? [duplicate]

Most of us who are studying mathematics are familiar with the famous $e^{ix}=cos(x)+isin(x)$. Why is it that we have $e^{ix}=cos(x)+isin(x)$ and not $e^{ix}=sin(x)+icos(x)$? I haven't studied Complex ...
40
votes
14answers
11k views

Why negative times negative = positive?

Someone recently asked me why a negative * a negative is positive, and why a negative * a positive is negative, etc. I went ahead and gave them a proof by contradiction like so: Assume $(-x) * (-y) ...
44
votes
13answers
3k views

Pseudo Proofs that are intuitively reasonable

What are nice "proofs" of true facts that are not really rigorous but give the right answer and still make sense on some level? Personally, I consider them to be guilty pleasures. Here are examples ...
12
votes
6answers
1k views

How does $e^{\pi i}$ equal $-1$

I was on xkcd a while back and there was the equation $e^{\pi i}$ which somehow miraculously equals $-1$. So, I put it into Google and it works. So, I tried solving it on my own: $\begin{align} ...
10
votes
3answers
11k views

How do I divide a function into even and odd sections?

While working on a proof showing that all functions limited to the domain of real numbers can be expressed as a sum of their odd and even components, I stumbled into a troublesome roadblock; namely, I ...
12
votes
6answers
1k views

Natural derivation of the complex exponential function?

Bourbaki shows in a very natural way that every continuous group isomorphism of the additive reals to the positive multiplicative reals is determined by its value at $1$, and in fact, that every such ...
15
votes
3answers
2k views

How does e, or the exponential function, relate to rotation?

$e^{i \pi} = -1$. I get why this works from a sum-of-series perspective and from an integration perspective, as in I can evaluate the integrals and find this result. However, I don't understand it ...
2
votes
5answers
564 views

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

Euler' 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 $e^{i x}$ power ...
6
votes
3answers
472 views

Given an exponential generating function, is it possible to isolate only the even terms?

Suppose you have an exponential generating function $$ F(x)=F_0+F_1x+F_2\frac{x^2}{2!}+\cdots+F_n\frac{x^n}{n!}+\cdots $$ and you want to get only the even terms $$ ...
6
votes
3answers
265 views

Coloring of an $1\times n$ board using 4 colors?

How can I find the number of ways to color an $1\times n$ board using the colors red, blue, green and orange if: # of red squares is even # of green squares is even We did the tilings of a $1\times ...

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