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Questions tagged [complex-numbers]

Numbers of the form ${z= x+ i,y:;x,, y\in\mathbb{R}}$ where $i^2 = -1$. Useful especially as quantum mechanics, where system states take complex vector values.

Complex numbers - together with multiplication and addition - are a field of numbers of the form $\{z= x+ i\,y:\;x,\, y\in\mathbb{R}\}$ where $i^2 = -1$. In physics they are a useful representation of quantities that have magnitude and phase such as quantities that vary sinuosoidally with time. System states in quantum mechanics live in a complex Hilbert space usually with a countably infinite basis but sometimes of finite dimension.

The complex numbers with addition and multiplication are the smallest algebraically closed field containing the ring of integers with addition and multiplication, i.e. they are the smallest field needed to solve any polynomial equation $p(x)=0$. They are also the largest connected, locally compact, topological field: intuitively - the biggest field with "everyday" continuous arithmetical operations.

NB: The tag includes quaternion, octonions,...

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Can one do the maths of physics without using $\sqrt{-1}$?

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…
Ajay
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What are functions of a complex variable used for in physics?

Whenever someone asks "Why are complex numbers important?" the answer, at least in the context of physics, usually includes things like quantum mechanics, oscillators and AC circuits. This is all very fine, but I've never seen anyone talk about…
Javier
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How are complex numbers, being imaginary (non real), able to represent reality (physics)?

Complex numbers came from solving $x^2 = -a$, taking $i\equiv\sqrt{-1}$. It isn't representing anything real. Then how are we able to represent lots of physics using them? How can something non-existing (not real/imaginary) represent reality?
Sherbaj Thind
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In classical mechanics, are complex numbers unphysical?

Possible Duplicate: Physics math without $\sqrt{-1}$ When I produce a complex final solution to a problem that began without complex coefficients at all, I have so far (with my limited expertise) tended to discard it as unphysical, unless I had…
Meow
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What is the physical significance of $i=\sqrt{-1}$?

What is the physical significance of $i=\sqrt{-1}$?
Shen
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Converting magnitude ratio to complex form

I want to inquire as to how we can convert a magnitude ratio, given in dB, to complex form, in real and imaginary form, provided of course that we have the phase given as well. Eg. Suppose I have an S-parameter reading as -11dB with a phase of 1.4…
C Vith
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Orientation Quaternion

I am looking to work out the orientation between two quaternions to establish if they are parallel face-face orientation, side-side orientation or perpendicular orientation. At the moment I am taking the absolute of the dot product of two…
raby
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Physical Applications of Complex Analysis

I'm writing an essay about Physical Applications of Complex Analysis. Which ones do you think are more interesting and appropriate for an undergraduate-level course? All suggestions are welcome!
I.Ald
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Without resorting to QM how do I understand the real world uses of imaginary numbers

I am trying to understand the real world uses of imaginary numbers. I have been told many times that it is not just a convenient tool mathematicians invented but has its places in some fundamental part of physical reality. But when I research its…
Qiulang 邱朗
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Meaning of cross product of real and complex part of a complex eigenvector

Let's say we have a complex eigenvector (v) as a result of a natural frequency calculation of multi DoF system. I'm wondering what kind of physical meaning has the cross product of Re(v) $\times$ Im(v). Thank's for the answer in advance
G.David
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Can we get rid of complex number?

Complex number is very weird to me, especially so when they appear in engineering and physics equations. It is possible to represent complex number as real matrix where $i = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ and $1= I$ However, the…
somebody4
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Use of Imaginary Angles in Physics

I am studying higher algebra. I have learnt that as well as for reals, trigonometric functions can also be defined for complex numbers, by means of power series. Such as: $$\sin(z)=z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+...(to \infty…
Ayan Biswas
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Physical Interpretation of the Residue

In my complex analysis course (taught by the physics department no less) we've obviously paid close attention to the residue for solving our problems, but there has been no attempt to try and attach any kind of concrete way to think of the residue,…
Skyler
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dv/dt = iv . what can be concluded from it?

Suppose there is a vector $V = e^{it}$ , and taking $dV/dt$ would just give me $iV$. What can be concluded from this... So what I am asking is that what would the presence of $i$ next to $V$ after differentiation mean? what other operators return…
dumpy
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