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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 the definition of a phase of a complex number. So, what is it?

Here are a few sub-questions to help you answer (more thoroughly) the main question above (so you don't have to answer all these sub-questions, but, if you answer them, the main question will probably be answered more satisfactorily).

  • 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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  • $\begingroup$ Let me compose a hypothetical example in the next comment. The main issue (as I see it) is that currently the post looks exactly the same as a "gimme teh codez" question by someone who puts no effort in their homework or no effort in even reading their textbook. $\endgroup$ Mar 9, 2019 at 16:09
  • $\begingroup$ Here is a variation of your question post. Title: (the same) "What exactly is the phase of a complex number?" Body: "I've encountered the term "phase" under various contexts. and I'd like to know exactly (with some formatting to highlight) what is the definition of a phase of a complex number. (new paragraph) Why is it called a "phase"? In engineering or physics, ... insert a couple of sentences for that context, therefore the notion of phase is nice and useful. However, I don't see the point of such a concept for the complex numbers. Can someone provide some examples?" $\endgroup$ Mar 9, 2019 at 16:49
  • $\begingroup$ My point is that you should "compose" the question instead of just listing consecutive short question marks. Again, like I said, this is an edge case for me. $\endgroup$ Mar 9, 2019 at 16:51
  • $\begingroup$ Many users of this site don't want to be bothered with formality, and that's fine. However, in this case where the question doesn't stem from a person currently needing help, formality becomes very important. If you want the post to be a useful entry in the database, then efforts should be made in having the question post "looks nice". $\endgroup$ Mar 9, 2019 at 16:54
  • $\begingroup$ @LeeDavidChungLin This is the third Google result for "complex phase" for me. I can't say I understand your argument here. Please forgive me for finding this post useful $\endgroup$ Sep 5, 2020 at 21:16

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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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