From a real world perspective each dimension in the 3-D Cartesian System can be represented by an axis that is perpendicular to 2 other axes. I read somewhere else that the effect of ${i}$ is to reorient data 90 degrees on the "imaginary" axis. I guess my question is this: What role ( if any)   does $i$ serve in everyday or even quantum physics?  ${Edit}$  To clarify, I am more concerned with the dimensional aspects of $i$. If for instance you were to consider a 3-D location to be a "point" with length width and height in a complex graph representation with three additional mutually perpendicular axes?   PS. I know this thread is getting a bit hair brained. But in my defense, I did look for a "specualation" tag before I posted this thread here.  $EDIT$  I changed $\sqrt{-1}$ back to $i$, because I am concerned with the hypothetical real world effect of the entity. Not the mathematical representation.
The use of imaginary and complex values comes up in many physics and engineering derivations. I have a question about that: Is the use of complex numbers simply to make the process of derivation easier, or is it an essential ingredient, without which it would be impossible to derive some results?   I can identify two different settings where the answer may be different:   It doesn't look like it is mandatory for results in Newtonian mechanics, General Relativity and classical electrodynamics. Can we say the same thing about quantum mechanics either way for sure?   Could this be a difference in quantum mechanics over the classical picture?
Proving that $AB-BA=cI$ for nontrivial $c \in \mathbb{C}$ I have a homework question I can`t solve:  Let $X$ be a normed linear space, $A,B \in B(X)$. Show that there exists no nontrivial $c \in \mathbb{C} $ such that $AB-BA=cI$.  Thanks alot already guys! I seem to have a professor who gives impossible homework problems.  The hint he gave: $A^{n}B-BA^{n}=ncA^{n-1}$, $n \in \mathbb{N}$ How do I gunzip all files recursively in a target directory? I'd like to know what is the command to use to gunzip all files in a target directory recursively? I tried to use the unzip command but it didn't work.  I tried the command from Is it possible to detect 32 bit vs 64 bit in a bash script? I am writing a bash script to deal with some installations in an automated way... I have the possibility of getting one such program in 32 or 64 bit binary... is it possible to detect the machine architecture from bash so I can select the correct binary?  This will be for Ubuntu machines. When installing an application, the application lists permissions that it needs to perform its  functions.    I am creating this list of the the system defined permissions and a description of what they mean. It is a community wiki so if new permissions are added in the future they can be added to this list. Extract all numbers from a single string in R How can I tell whether beetroot is still edible? I have some raw beetroot I bought about a week ago, having stored them at room temperature since. It's the first time I bought it raw, and I'm unsure whether it's still edible.   It is already soft, and when I cut it, it smells... Well, different than I expected. It's not a rotten smell as such, but it's nothing like the sweet beet-rooty smell I know from the pre-packaged, pre-boiled supermarket variety.  How can I tell whether it's still safe to eat? remote action function encountering CPU time out limit exception How can I remove all duplicates so that NONE are left in a data frame? In the 70's I had a stack of old Amateur Radio magazines (50s-60s), and for a long time I saved an article about using the  to combine a number of resistors to achieve a specific value. Does anyone recall and have a copy of this article, or know how the Euclidean algorithm is applied to solve this problem? What does the 'A' trailing letter in Atmel's ATtiny devices mean? I have both on stock: ATtiny861A-SU and ATtiny861-20SU.  But I can not find out any difference between them except 'In-System Programmable via SPI Port' available for 'A' version.  Could anyone tell me what is the difference between those two devices?  Thanks! Is product of two closed sets closed? If $A$ is closed in $X$ and $B$ is closed in $Y$,   does it follow that $A \times B$ is closed in $X \times Y$ ? I have this problem I'm working on. Hints are much appreciated (I don't want complete proof):  In a normed vector space, if $ x_n \longrightarrow x $ then  $ z_n = \frac{x_1 + \dots +x_n}{n} \longrightarrow x $   I've been trying adding and subtracting inside the norm... but I don't seem to get anywhere.  Thanks! Why won't this Path.Combine work? How can I invert a 'clip' selection within TikZ? I wish to draw an arc for a reflex angle on the outside of a triangle with coordinates (A), (B) and (C).  If I wanted to draw the arc inside the triangle I could use something like  \begin{scope} \path[clip] (A) -- (B) -- (C) -- cycle; \draw (B) circle (2mm);    % this is the little angle marker \end{scope}   Is there a straightforward way to exclude the part of the circle that's in the triangle, as in the pseudo-code below?  \begin{scope} \path[magicalinverseclipcommand] (A) -- (B) -- (C) -- cycle; \draw (B) circle (2mm);    % this is the little angle marker \end{scope}   EDIT: Although I included the code above as a motivating example of what an inverse clip would do, I would prefer a general solution that works systematically with any closed path, and not just in the particular example of a triangle with a circle clipped out of it. Looking at my question, I realise this isn't clear, and I will be happy with the best solution to my particular problem, but wonder if there is a less ad hoc solution than those listed so far. What is a good algorithm for estimating the median of a huge read-once data set? With regard to the theory that each time a wave function collapses the universe  so that each possible outcome really exists - where does all the energy required to create all the new universes come from? What is the closed form of $\int e^{x^2} \, dx$? How can I do a closed form expansion of $\int e^{x^2} \, dx$? Please be specific as to which method I must use. I am trying to show $$\lim_{n\rightarrow \infty} \int_{[0,1]^n} \frac{|x|}{\sqrt{n}}dx=\frac{1}{\sqrt{3}}$$ where $[0,1]^n$ is the unit cube in $\mathbb{R}^n$. I would appreciate any hint. How many syllables are in the word 'hour'? Does the word 'hour' have 1 or 2 syllables? Brace expansion not working in a script