Complex numbers are used widely in quantum mechanics and the waveform, is there a physical interpretation of what this means about the structure of the universe? Why is it not used in macro physics?   Do physicists really think that calling imaginary numbers a 90 degree rotation is a good enough answer? It seems to be used in many areas to mean similar things.  Is there an explanation to do with dimensions as I have tried in this conversation a better way to understand them?
18930
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?
24570
Although this question is going to seem completely trivial to anyone with any exposure to path integrals, I'm looking to answer this precisely and haven't been able to find any materials after looking for about 40 minutes, which leads me to believe that it makes sense to ask the question here. In particular I'm looking for an answer wherin any mathematical claims are phrased as precisely as possible, with detailed proofs either provided or referenced. Also my search for a solution has led me to think I'm actually looking for a good explenation of Wick rotation, which I can't really claim to understand in detail. Any good references about this would be very welcome as well.  I'm looking to make sense of the following integral identity:  $$\int_{-\infty}^{\infty} dx \ \exp\left(i\frac{a}{2}x^2+iJx\right)=\left(\frac{2\pi i}{a}\right)^{1/2}\exp\left(\frac{-iJ^2}{2a}\right), \qquad a,J\in\mathbb{R}$$   (and various other sources) say that "This result is valid as an integration in the complex plane as long as a has a positive imaginary part." Clearly the left hand side does not exist in Lebesgue sense for real $a, J$. An answer to the question "" claims:      "it is convergent as a Riemann integral, thanks to some rather delicate cancellations. To make the integral well defined -- equivalently to see how these cancellations occur -- we need to supply some additional information.   Wick rotation provides a way of doing this. You observe that the left hand side is analytic in t   , and that the right hand side is well-defined if Im(t)&lt;0. Then you can define the integral for real t by saying that it's analytic continued from complex t with negative imaginary part."   I want to see the gory details and all known motivation for the validity of this procedure for the kinds of applications where such integrals occur. Suggestions such as "include an $i\epsilon$ in order to make it finite" seem arbitrary. In that case one would have to motivate that prescription from the very start, that is within the modeling procedure that ends up giving that integral expression (which is likely the correct way to approach this). I'm also not sure how to interpret the right hand side, since it involves the square root of an imaginary number, which should involve some choice of branch cut, which I have never seen specified in connection to this formula.
34461
Suppose now we are trying to explain to students who do not know complex numbers, how do we distinguish $i$ and $-i$ to them? They will object that they both squared to $-1$ and thus they are indistinguishable. Is there a way of explaining this in an elementary way without go into introductory things in complex analysis?
89315
Since the  makes a statement about domains defined in terms of real numbers, it would appear to invalidate statements about nature that we derived by applying real analysis. My reasoning is this:  If you can "duplicate" an abstract 3-dimensional ball defined, in the usual way, using the domain of real numbers, then clearly the domain of real numbers must be unsuited to describing physical objects (for instance The Earth), because duplicating them would double their mass (and thereby energy). But we use real numbers to reason about nature and derive new laws (or basic results) all the time in physics.  In the same way as we could integrate over the earth's volume to determine its mass from its density (for instance), we could apply the Banach-Tarski theorem to show that we could generate as many earths as we liked, right?  Does this mean that the mathematical foundation of physics is flawed? (I hope this question is not too provocative!)
54040
I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this.  In the domain of natural numbers, addition and multiplication always generate natural numbers, staying in the same domain.  However subtraction of a large number from a smaller one needs to "escape" into the domain of integers, and division may result in escape to the real domain (like 3 / 5 -&gt; 0.6).  It was a simple step from there to taking the square root of a negative number, hence requiring the escape into the complex domain, such as 4+7i.  He quite easily picked up that each of these domains was a superset of another, natural -&gt; integer -&gt; real -&gt; complex.  However, he then asked if an operation on a complex number would require yet another escape, a question I had to investigate. Now, it turns out that the square root of a complex number is simply another complex number along the lines of mathematical distribution: (a+bi)2 -> a2 + 2abi - b2, from memory.  But I'm wondering if there are other mathematical operations performed on complex numbers (or any of its subset domains) that can't be represented within the complex domain.  Apologies if I've used the wrong terms, it's been about 30 years since I did University level math.
122940
Why is  quaternion algebra 4D and not 3D?  Complex algebra is 2D and what is known as quaternion algebra jumps to 4D.    $ i^2 = j^2 = k^2 = ijk = -1 $  Using $1, i, j,$ and $k$ as the base (where complex uses $1$ and $i$ (or $j$ if you are an EE)) which results in a 4-axis space.  Why is there no 3D algebra?
95751
The SR invariance formula makes space-like intervals imaginary (e.g., the distance $x$ in a given frame has interval $ix$). Yet modern physicists consider it bad form to define the distance itself as $ix$, as Minkowski did. It seems to me that this distinction is trivial—what’s wrong with defining the distance as the value of its interval? An answer to a previous question of mine on this topic referred to a paper by Visser showing that there is a way to do this that can be generalized to GR.  So what’s the fuss about? Even if there weren’t such a way, I don’t see how that would invalidate Minkowski’s definition for SR. Since the interval measure is what’s important, what fundamental difference can it make whether one defines the distance as x or as the imaginary interval $ix$?  The whole idea of a LT being an imaginary rotation (important, because phi is additive) depends on the use of the invariant measure, which is the complex length between two points in C2, the space in which complex rotations exist, and so the space in which the x-t plane is being considered to be embedded when we talk of complex rotations. In C2 it can easily be shown that the angle between any two axes, real or imaginary, is a real right angle, and this explains the invariance formula—it is simply the Pythagorean theorem for that complex triangle. So all this ‘invariant interval geometry’ is an integral part of special relativity. If you insist that distances are real and use a Pseudo-Euclidean metric, which would imply entirely different values for these angles, you are throwing out an enormous portion of SR. It is hard to see how that can be billed as an advance in our understanding.
119603
I recently read that a quantum state is actually defined by a ray and not a vector. That is it is possible to multiply a state $\psi$ by any complex number $c\in \mathbb{C}$ and you won't be changing the physics in any way. I understand this mathematically, but I don't understand what the physical meaning of such an "equivalent state" would be since the new state need not be normalised if $c$ is not of the form $e^{i\phi}$.
57240
I am confused as to what numbers are. Numbers are defined to be what they are, so numbers aren't real? But numbers are found in nature, right? So if we invented them, how can they be found in nature? How can everything be based on something that is not real. Sorry if what I am saying don't make sense, it hard to put it into words. I feel like I've been known numbers all my life yet I don't know what they are. Also why don't complex numbers have an order? Real numbers are an invention just as complex numbers are, so how are imaginary numbers any more imaginary than real numbers? I mean an order was given to the real numbers, so why can't complex numbers be given a definite order?  Thanks
13270
While teaching complex numbers for those who are encounter for the first time (usually 10th grader and 11th grader), I get the question like      "Can even squared number give negative results? How come there be square root of negative numbers? How much length does $i \; \mathrm{cm}$ represent in meter scale?"     in every year and in every section. I have been answering      "Well, this is just a construction such that it gives solution to equation $x^2 + 1 = 0$ which must have two solution by use of Fundamental theorem of algebra but since we do not have that in Real numbers, we are forced to construct $i$ that way." Then I go to history accepting the difficulty of some numbers like $\sqrt 2 $ by ancient  (just a red herring). Then I continue "Natural number is used to represent Money while Real number is used to measure length, complex number, while not representing length or weight, it my represent some other things than money or length that has applications in physics, math and engineering and so it is valid to take construct a symbol $i^2 = -1$."   I don't know if I am speaking too much out of my limited knowledge. What is the best way this knucklehead to explain those knucklehead who have been brainwashed in secondary and lower-secondary school that you cannot take square root of negative numbers, while still being correct and make them accept it easily?  EDIT: I have last question. Are some problems that is impossible to solve without use of Complex Numbers or all problems involving complex numbers have other workarounds?  EDIT-ADDED:: What would mathematics be like without the use of complex numbers? Perhaps, if I mention this clearly, maybe it would be helpful for students. One example is given by @Semiclassical on the comments in the first answer. What other things cannot be absolutely solved without the use of complex numbers?
14964
Should I write complex numbers as $a+ib$ or $a+bi$? Maybe there are certain cases where I should write it one way and other cases where I should write it the other way. What do you think?
86002
Introduction:  Recently I found out that $i^i \approx 0.20788$ has no imaginary part. I got interested and then wanted to know whether there are other $n$ for which $\underbrace {i^{i^{i^{.^{.^{.^{i}}}}}}}_{n \ times}$ has no inaginary part.  So I wrote  (I'm a beginner at python, could be very bad code :) which plots what I call the $i-Tower\ up\ to\ n = 100$. It looks like this:   Question:  Let's use this convention: ${}^ni = \underbrace{i^{i^{i^{.^{.^{.^{i}}}}}}}_{n \ times}$.   Why is the sequence ${}^ni\ |\ n \in \mathbb{N}$ converging?   ${}^{20}i \approx 0.48770 + 0.41217i$   ${}^{60}i \approx 0.437584 + 0.360535i$   ${}^{100}i \approx 0.43829+ 0.36059i$  What I already noticed is that the angle between the lines you can draw from ${}^ni$ over ${}^{n+1}i$ to ${}^{n+2}i$ is $&lt;90°$.   Is that angle equal for all $n$? Is there a way to give the exact value of $x = {}^ni \ $with$\ {n \rightarrow \infty}$? What is the connection between $i$ and $x$. Is $x$ a special complex number that is maybe already known or comes up in other places?
58413
As far as I've read online, there isn't a good explanation for the . Is this the case? Why does taking the square of the wave function give you the Probability? Naturally it removes negatives and imaginary numbers, but why is it the square, not the fourth or some higher power?
79557
I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$).  I say this is practically the case, so in every calculation you just assume that "$a + 0i$" is equal to the real number "$a$" and you always get the right results (this might not even be true, but I am no expert and as far as I can see this is the case). But then again, I believe that just cutting away "$+ 0i$" makes this number is not the same kind of number anymore, it's changed its structure and is not totally equivalent anymore. This is where he says I am absolutely wrong.  So, is it mathematically (strictly speaking!) perfectly fine to say that every complex number with an imaginary part of $0$ is just a real number? Or does this change the mathematical structure so much that it cannot be mathematically-valid but just something that happens to work (which is what I believe)?  I must say that I believe a complex number is one entity, whilst he believes that it is just a conglomerate of other entitites. So, as one said here, "$(a + bi) - bi = a$", I believe is barely a good argument. But then again: is it correct to view $a + bi$ as a single entity (like "$5$" or "$2$", just consisting of multiple symbols)? I'm (as a non-mathematician) not sure there either.
57804
I've been struggling with this problem, actually I was doing a program in python and did   1j ** 1j(complex numbers) (In python a**b = $a^b$ )   and found out the answer to be a real number with value $0.2079$, How to calculate this value of $i^i$?
83507
I'm currently doing some experiments on fractals and in one of my equation I need to flip the real and imaginary components of a complex number, such as :  $$ z = a + bi $$  Becomes :  $$ z = b + ai $$  What would be the correct mathematical term or notation for this ? flip ?
3089
As a Graduate Mathematics student, my interest lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications of Number theory to physics. I have heard Applications of linear algebra and analysis to many branches of physics, but not number theory.  Waiting forward in receiving interesting answers!