Why does $e^{i\\pi}=-1$? - Mathematics Stack Exchange
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 exponential …
Prove that $e^{i\\pi} = -1$ - Mathematics Stack Exchange
When I first found out that eiπ = −1 e i π = 1, I was blown away. Does anyone here know one of (many I'm sure) proofs of this phenomenal equation? I can perform all of the algebra to get the −1 1. But, …
How does $e^{\\pi i}$ equal $-1$ - Mathematics Stack Exchange
@ColeJohnson, eiπ = −1 e i π = 1 is a consequence of how the objects e e, π π and raising a number to the power of a complex number are defined. There is no deep truth behind the equation as some like …
Why $e^{i(π/3)} \\ne -1$? - Mathematics Stack Exchange
Mar 30, 2018 · I understand why eiπ = −1 e i π = 1 and as a result ei2π =(eiπ)2 = 1. e i 2 π = (e i π) 2 = 1. These results can be confirmed using Euler's formula But why does eiπ/3 ≠ −1 e i π / 3 ≠ 1 as we …
How to prove Euler's formula: $e^{it}=\\cos t +i\\sin t$?
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 motivation, relevant …
How can $i^i = e^{-\\pi/2}$ - Mathematics Stack Exchange
Apr 19, 2017 · I was asked a homework question: find ii i i. The solution provided was as follows: Let A = ii A = i i. log A = i log i log A = i log i. Now, log i = logeiπ/2 = iπ2 log i = log e i π / 2 = i π 2. So, log A …
How does $e^{i x}$ produce rotation around the imaginary unit circle?
Euler’s formula states that eix = cos(x) + i sin(x) 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 e to the …
What is $\sinh (x)$? And $i\pi$ - Mathematics Stack Exchange
Yes I am aware I was just stating how I got super curious about sinh but why do I get different answers for sinh (ipi) and (e^ (ipi) - e^ (-i*pi))/2?
terminology - Is there a standard word or phrase that means 'raised to ...
Mar 3, 2025 · $x^2$ is read as 'x squared' and $x^3$ is read as 'x cubed' I would have naively thought then that $x^4$ could be read as 'x tesseract-ed' or something similar.
terminology - Difference between axioms, theorems, postulates ...
Oct 25, 2010 · I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)