What exactly is the phase of a complex number? [on hold]What exactly is a number?Is there a term to designate the “complexity” of a complex numberCan we determine if a complex number is greater than another?Re-defining the complex unit for teaching purposesEstimation with complex exponentProper Use of the Term “Witness”fourier transform - why imaginary part represents the phase shiftApplication of Complex Numbers (Electrical Engineering)What can complex numbers do that linear algebra cannot?Error propagation for complex numbers from real/imaginary to modulus/phase
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What exactly is the phase of a complex number? [on hold]
What exactly is a number?Is there a term to designate the “complexity” of a complex numberCan we determine if a complex number is greater than another?Re-defining the complex unit for teaching purposesEstimation with complex exponentProper Use of the Term “Witness”fourier transform - why imaginary part represents the phase shiftApplication of Complex Numbers (Electrical Engineering)What can complex numbers do that linear algebra cannot?Error propagation for complex numbers from real/imaginary to modulus/phase
$begingroup$
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?
complex-numbers terminology definition
asked yesterday
nbronbro
2,43363174
$endgroup$
put on hold as too broad by Learnmore, B. Goddard, Lee David Chung Lin, John Omielan, Jyrki Lahtonen yesterday
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
|
show 7 more comments
$begingroup$
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?
complex-numbers terminology definition
asked yesterday
nbronbro
2,43363174
$endgroup$
put on hold as too broad by Learnmore, B. Goddard, Lee David Chung Lin, John Omielan, Jyrki Lahtonen yesterday
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
The phase is the term $e^itheta$ in the decomposition of a complex number as $z=re^itheta$. Sometimes, one calls "phase" the real number $theta$, instead. Nothing more than that.
$endgroup$
– Giuseppe Negro
yesterday
1
$begingroup$
@GiuseppeNegro You can try to answer the other questions too and produce a more formal answer, if you want.
$endgroup$
– nbro
yesterday
$begingroup$
Seems that you're trying to do something good via self-answering, but the current formulation of the question is problematic. In particular, too many questions are being asked at the same time.
$endgroup$
– Lee David Chung Lin
yesterday
$begingroup$
@LeeDavidChungLin They are very related to each other. My answer turned out to be not so long (and I answered all the questions).
$endgroup$
– nbro
yesterday
$begingroup$
My point is that it's a bad formulation of a question post. It's indeed common to implicitly or explicitly ask several closely connected questions in a post, however, it should NOT be done in a thoughtless way (blindly listing all the questions). You should know this as you have been here long enough. I voted to close but honestly this is an edge case in my opinion, that's why I'm leaving these comments. I'll be glad to retract my vote if you edit the question.
$endgroup$
– Lee David Chung Lin
yesterday
|
show 7 more comments
$begingroup$
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?
complex-numbers terminology definition
asked yesterday
nbronbro
2,43363174
$endgroup$
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?
complex-numbers terminology definition
complex-numbers terminology definition
asked yesterday
nbronbro
2,43363174
asked yesterday
nbronbro
2,43363174
edited yesterday
nbro
asked yesterday
nbronbro
2,43363174
asked yesterday
nbronbro
2,43363174
asked yesterday
nbronbro
2,43363174
2,43363174
put on hold as too broad by Learnmore, B. Goddard, Lee David Chung Lin, John Omielan, Jyrki Lahtonen yesterday
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
put on hold as too broad by Learnmore, B. Goddard, Lee David Chung Lin, John Omielan, Jyrki Lahtonen yesterday
Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
The phase is the term $e^itheta$ in the decomposition of a complex number as $z=re^itheta$. Sometimes, one calls "phase" the real number $theta$, instead. Nothing more than that.
$endgroup$
– Giuseppe Negro
yesterday
1
$begingroup$
@GiuseppeNegro You can try to answer the other questions too and produce a more formal answer, if you want.
$endgroup$
– nbro
yesterday
$begingroup$
Seems that you're trying to do something good via self-answering, but the current formulation of the question is problematic. In particular, too many questions are being asked at the same time.
$endgroup$
– Lee David Chung Lin
yesterday
$begingroup$
@LeeDavidChungLin They are very related to each other. My answer turned out to be not so long (and I answered all the questions).
$endgroup$
– nbro
yesterday
$begingroup$
My point is that it's a bad formulation of a question post. It's indeed common to implicitly or explicitly ask several closely connected questions in a post, however, it should NOT be done in a thoughtless way (blindly listing all the questions). You should know this as you have been here long enough. I voted to close but honestly this is an edge case in my opinion, that's why I'm leaving these comments. I'll be glad to retract my vote if you edit the question.
$endgroup$
– Lee David Chung Lin
yesterday
|
show 7 more comments
2
$begingroup$
The phase is the term $e^itheta$ in the decomposition of a complex number as $z=re^itheta$. Sometimes, one calls "phase" the real number $theta$, instead. Nothing more than that.
$endgroup$
– Giuseppe Negro
yesterday
1
$begingroup$
@GiuseppeNegro You can try to answer the other questions too and produce a more formal answer, if you want.
$endgroup$
– nbro
yesterday
$begingroup$
Seems that you're trying to do something good via self-answering, but the current formulation of the question is problematic. In particular, too many questions are being asked at the same time.
$endgroup$
– Lee David Chung Lin
yesterday
$begingroup$
@LeeDavidChungLin They are very related to each other. My answer turned out to be not so long (and I answered all the questions).
$endgroup$
– nbro
yesterday
$begingroup$
My point is that it's a bad formulation of a question post. It's indeed common to implicitly or explicitly ask several closely connected questions in a post, however, it should NOT be done in a thoughtless way (blindly listing all the questions). You should know this as you have been here long enough. I voted to close but honestly this is an edge case in my opinion, that's why I'm leaving these comments. I'll be glad to retract my vote if you edit the question.
$endgroup$
– Lee David Chung Lin
yesterday
2
2
$begingroup$
The phase is the term $e^itheta$ in the decomposition of a complex number as $z=re^itheta$. Sometimes, one calls "phase" the real number $theta$, instead. Nothing more than that.
$endgroup$
– Giuseppe Negro
yesterday
$begingroup$
The phase is the term $e^itheta$ in the decomposition of a complex number as $z=re^itheta$. Sometimes, one calls "phase" the real number $theta$, instead. Nothing more than that.
$endgroup$
– Giuseppe Negro
yesterday
1
1
$begingroup$
@GiuseppeNegro You can try to answer the other questions too and produce a more formal answer, if you want.
$endgroup$
– nbro
yesterday
$begingroup$
@GiuseppeNegro You can try to answer the other questions too and produce a more formal answer, if you want.
$endgroup$
– nbro
yesterday
$begingroup$
Seems that you're trying to do something good via self-answering, but the current formulation of the question is problematic. In particular, too many questions are being asked at the same time.
$endgroup$
– Lee David Chung Lin
yesterday
$begingroup$
Seems that you're trying to do something good via self-answering, but the current formulation of the question is problematic. In particular, too many questions are being asked at the same time.
$endgroup$
– Lee David Chung Lin
yesterday
$begingroup$
@LeeDavidChungLin They are very related to each other. My answer turned out to be not so long (and I answered all the questions).
$endgroup$
– nbro
yesterday
$begingroup$
@LeeDavidChungLin They are very related to each other. My answer turned out to be not so long (and I answered all the questions).
$endgroup$
– nbro
yesterday
$begingroup$
My point is that it's a bad formulation of a question post. It's indeed common to implicitly or explicitly ask several closely connected questions in a post, however, it should NOT be done in a thoughtless way (blindly listing all the questions). You should know this as you have been here long enough. I voted to close but honestly this is an edge case in my opinion, that's why I'm leaving these comments. I'll be glad to retract my vote if you edit the question.
$endgroup$
– Lee David Chung Lin
yesterday
$begingroup$
My point is that it's a bad formulation of a question post. It's indeed common to implicitly or explicitly ask several closely connected questions in a post, however, it should NOT be done in a thoughtless way (blindly listing all the questions). You should know this as you have been here long enough. I voted to close but honestly this is an edge case in my opinion, that's why I'm leaving these comments. I'll be glad to retract my vote if you edit the question.
$endgroup$
– Lee David Chung Lin
yesterday
|
show 7 more comments
1 Answer
1
active
oldest
votes
$begingroup$
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:
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 $z=r(cos varphi +isin varphi)$, 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
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 $z=r(cos varphi +isin varphi)$ is equivalent to $z=re^ivarphi$, because of the Euler's identity, which is $e^i varphi=cos varphi + i sin varphi$, which holds for any real number $varphi$.
In our case, the real number $varphi$ is the angle.
In this context, we often call the phase either the term $e^ivarphi$ or simply the angle $varphi$. Note that the angle $varphi$ completely determines $e^ivarphi$, that is, given $varphi$, we can easily retrieve $e^ivarphi$ (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 $z_1 = re^i2pi$ (a complex number of length $r$ which lies on the real axis and points to the right) and $z_2 = re^i4pi$. Clearly, $z_1 = z_2$, but they have different "phases": $z_1$ has phase $e^i2pi$ (or $2pi$), whereas $z_2$ has phase $e^i4pi$ (or $4pi$). 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 $e^ivarphi$) 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).
answered yesterday
nbronbro
2,43363174
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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:
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 $z=r(cos varphi +isin varphi)$, 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
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 $z=r(cos varphi +isin varphi)$ is equivalent to $z=re^ivarphi$, because of the Euler's identity, which is $e^i varphi=cos varphi + i sin varphi$, which holds for any real number $varphi$.
In our case, the real number $varphi$ is the angle.
In this context, we often call the phase either the term $e^ivarphi$ or simply the angle $varphi$. Note that the angle $varphi$ completely determines $e^ivarphi$, that is, given $varphi$, we can easily retrieve $e^ivarphi$ (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 $z_1 = re^i2pi$ (a complex number of length $r$ which lies on the real axis and points to the right) and $z_2 = re^i4pi$. Clearly, $z_1 = z_2$, but they have different "phases": $z_1$ has phase $e^i2pi$ (or $2pi$), whereas $z_2$ has phase $e^i4pi$ (or $4pi$). 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 $e^ivarphi$) 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).
answered yesterday
nbronbro
2,43363174
$endgroup$
add a comment |
$begingroup$
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:
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 $z=r(cos varphi +isin varphi)$, 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
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 $z=r(cos varphi +isin varphi)$ is equivalent to $z=re^ivarphi$, because of the Euler's identity, which is $e^i varphi=cos varphi + i sin varphi$, which holds for any real number $varphi$.
In our case, the real number $varphi$ is the angle.
In this context, we often call the phase either the term $e^ivarphi$ or simply the angle $varphi$. Note that the angle $varphi$ completely determines $e^ivarphi$, that is, given $varphi$, we can easily retrieve $e^ivarphi$ (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 $z_1 = re^i2pi$ (a complex number of length $r$ which lies on the real axis and points to the right) and $z_2 = re^i4pi$. Clearly, $z_1 = z_2$, but they have different "phases": $z_1$ has phase $e^i2pi$ (or $2pi$), whereas $z_2$ has phase $e^i4pi$ (or $4pi$). 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 $e^ivarphi$) 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).
answered yesterday
nbronbro
2,43363174
$endgroup$
add a comment |
$begingroup$
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:
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 $z=r(cos varphi +isin varphi)$, 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
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 $z=r(cos varphi +isin varphi)$ is equivalent to $z=re^ivarphi$, because of the Euler's identity, which is $e^i varphi=cos varphi + i sin varphi$, which holds for any real number $varphi$.
In our case, the real number $varphi$ is the angle.
In this context, we often call the phase either the term $e^ivarphi$ or simply the angle $varphi$. Note that the angle $varphi$ completely determines $e^ivarphi$, that is, given $varphi$, we can easily retrieve $e^ivarphi$ (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 $z_1 = re^i2pi$ (a complex number of length $r$ which lies on the real axis and points to the right) and $z_2 = re^i4pi$. Clearly, $z_1 = z_2$, but they have different "phases": $z_1$ has phase $e^i2pi$ (or $2pi$), whereas $z_2$ has phase $e^i4pi$ (or $4pi$). 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 $e^ivarphi$) 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).
answered yesterday
nbronbro
2,43363174
$endgroup$
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:
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 $z=r(cos varphi +isin varphi)$, 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
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 $z=r(cos varphi +isin varphi)$ is equivalent to $z=re^ivarphi$, because of the Euler's identity, which is $e^i varphi=cos varphi + i sin varphi$, which holds for any real number $varphi$.
In our case, the real number $varphi$ is the angle.
In this context, we often call the phase either the term $e^ivarphi$ or simply the angle $varphi$. Note that the angle $varphi$ completely determines $e^ivarphi$, that is, given $varphi$, we can easily retrieve $e^ivarphi$ (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 $z_1 = re^i2pi$ (a complex number of length $r$ which lies on the real axis and points to the right) and $z_2 = re^i4pi$. Clearly, $z_1 = z_2$, but they have different "phases": $z_1$ has phase $e^i2pi$ (or $2pi$), whereas $z_2$ has phase $e^i4pi$ (or $4pi$). 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 $e^ivarphi$) 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).
answered yesterday
nbronbro
2,43363174
edited yesterday
answered yesterday
nbronbro
2,43363174
answered yesterday
nbronbro
2,43363174
answered yesterday
nbronbro
2,43363174
2,43363174
add a comment |
add a comment |
2
$begingroup$
The phase is the term $e^itheta$ in the decomposition of a complex number as $z=re^itheta$. Sometimes, one calls "phase" the real number $theta$, instead. Nothing more than that.
$endgroup$
– Giuseppe Negro
yesterday
1
$begingroup$
@GiuseppeNegro You can try to answer the other questions too and produce a more formal answer, if you want.
$endgroup$
– nbro
yesterday
$begingroup$
Seems that you're trying to do something good via self-answering, but the current formulation of the question is problematic. In particular, too many questions are being asked at the same time.
$endgroup$
– Lee David Chung Lin
yesterday
$begingroup$
@LeeDavidChungLin They are very related to each other. My answer turned out to be not so long (and I answered all the questions).
$endgroup$
– nbro
yesterday
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My point is that it's a bad formulation of a question post. It's indeed common to implicitly or explicitly ask several closely connected questions in a post, however, it should NOT be done in a thoughtless way (blindly listing all the questions). You should know this as you have been here long enough. I voted to close but honestly this is an edge case in my opinion, that's why I'm leaving these comments. I'll be glad to retract my vote if you edit the question.
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– Lee David Chung Lin
yesterday