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I've heard the term "phase" several times (e.g. in the context of waves), not just in the context of complex numbers (which are often mentioned e.g. in the context of quantum computing), but, in this post, I would like to know exactly the definition of a phase of a complex number. Why is it called a "phase"? Why do we care about phases of complex numbers? Can a complex number have different phases and still be considered the same complex number? What are examples of applications of this property of a complex number?

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    $\begingroup$ The phase is the term $e^{i\theta}$ in the decomposition of a complex number as $z=re^{i\theta}$. Sometimes, one calls "phase" the real number $\theta$, instead. Nothing more than that. $\endgroup$ – Giuseppe Negro Mar 9 '19 at 11:58
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    $\begingroup$ @GiuseppeNegro You can try to answer the other questions too and produce a more formal answer, if you want. $\endgroup$ – nbro Mar 9 '19 at 11:59
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    $\begingroup$ @LeeDavidChungLin What do you mean by "blindly listing all the questions"? What's the problem with asking a main question and then maybe related "sub-questions"? I really don't get your point. If I just asked "What is the phase of a complex number?", then a good answer would have necessarily had to answer to all sub-questions, IMHO. I just explicitly asked the sub-question (because I was more or less aware of the the answer I was expecting), and I don't see any problem with this. $\endgroup$ – nbro Mar 9 '19 at 16:08
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    $\begingroup$ @LeeDavidChungLin You're not answering my question, but, if that's your point, then my answer to that point is that this was intended to serve as a reference question. I would have answered to my own question anyway, if no one decided to answer it. $\endgroup$ – nbro Mar 9 '19 at 16:49
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    $\begingroup$ @LeeDavidChungLin I actually think my post looks nice (no need to describe the a detailed context, which I do not really have right now). I actually wrote it like this to make it as general and thus as useful as possible for people like me that in the past had the same question(s). Anyway, I added a little bit of more context. $\endgroup$ – nbro Mar 9 '19 at 16:55
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A complex number can be represented in a "rectangular" or "polar" forms.

The rectangular form of a complex number $z$ is composed of a real part and the imaginary part: $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part, and $i$ is the imaginary unity, $\sqrt{-1}$. We can visualise such number in a 2-dimensional plane , called the "complex plane", as follows:

enter image description here

where "Im" refers to the "imaginary" axis and "Re" to the real axis. Why is it called "rectangular form"? Because of the rectangle we can see in the picture above.

In the polar representation, rather than specifying the coordinates (of the complex plane) $a$ and $b$ of the complex number, we specify the angle (counterclockwise sense) between the real axis ("Re") and the vector representing the complex number, and the length of such vector. Formally, the polar representation of a complex number $z$ is $z=r(\cos \varphi +i\sin \varphi)$, where $\varphi$ is the angle between the real axis and the vector (in blue in the figures) and $r$ is the length (or magnitude or module) of such vector. We similarly visualise such polar representation as follows

enter image description here

where $y=b$ and $x=a$ (that is, we just changed letters). Why is it called "polar form"? Because the angle $\varphi$ is also often called the "polar angle".

Mathematically, the complex number $z=r(\cos \varphi +i\sin \varphi)$ is equivalent to $z=re^{i\varphi}$, because of the Euler's identity, which is $e^{i \varphi}=\cos \varphi + i \sin \varphi$, which holds for any real number $\varphi$.

enter image description here

In our case, the real number $\varphi$ is the angle.

In this context, we often call the phase either the term $e^{i\varphi}$ or simply the angle $\varphi$. Note that the angle $\varphi$ completely determines $e^{i\varphi}$, that is, given $\varphi$, we can easily retrieve $e^{i\varphi}$ (without any other information) by simply replacing this angle in the Euler's identity.

Can a complex number have different phases and still be considered the same complex number?

Suppose that we have complex numbers $z_1 = re^{i2\pi}$ (a complex number of length $r$ which lies on the real axis and points to the right) and $z_2 = re^{i4\pi}$. Clearly, $z_1 = z_2$, but they have different "phases": $z_1$ has phase $e^{i2\pi}$ (or $2\pi$), whereas $z_2$ has phase $e^{i4\pi}$ (or $4\pi$). Hence, $z_1$ and $z_2$, even though they have different phases, they represent the exact same complex number. Hence, in general, two polar representations with different phases can represent the same complex number.

Why is $\varphi$ (or $e^{i\varphi}$) called a "phase"?

Because it refers to the angle around the origin of the complex plane.

What are examples of applications of this property of a complex number?

For example, Vandermonde matrix, which represents the discrete Fourier transform (DFT), is a special matrix. If we look at the Wikipedia article on the DFT matrix, we can see the entries of such vector are complex numbers with different phases.

(Images in this answer are screenshots of the images of Wikipedia articles).

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