What does this inequality mean?
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I was reading complex numbers when I came across this inequality:
$- pi ≤ arg(z) ≤ pi$
Where $z$ is a complex number, I want to know what this inequality mean?
Can't it simply be that,
$0 ≤ arg(z) ≤ 2 pi$
complex-numbers
asked Aug 28 at 16:50
Adi
134
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up vote
1
down vote
favorite
I was reading complex numbers when I came across this inequality:
$- pi ≤ arg(z) ≤ pi$
Where $z$ is a complex number, I want to know what this inequality mean?
Can't it simply be that,
$0 ≤ arg(z) ≤ 2 pi$
complex-numbers
asked Aug 28 at 16:50
Adi
134
1
A complex number $z=a+bi$ could also have been written as $z=re^itheta$, however one can notice writing it in such a way is no longer unique. You have things such as $e^icdot 0=e^icdot 2pi=1$. In order to make it unique, we sometimes refer to the principle argument of a complex number. Some authors prefer the principle argument to be in the range $-pileq arg(z)< pi$. Other authors prefer the principle argument to be in the range $0leq arg(z)<2pi$. Regardless which convention is used $z=re^itextarg(z)$ is now a unique representation.
– JMoravitz
Aug 28 at 16:56
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I was reading complex numbers when I came across this inequality:
$- pi ≤ arg(z) ≤ pi$
Where $z$ is a complex number, I want to know what this inequality mean?
Can't it simply be that,
$0 ≤ arg(z) ≤ 2 pi$
complex-numbers
asked Aug 28 at 16:50
Adi
134
I was reading complex numbers when I came across this inequality:
$- pi ≤ arg(z) ≤ pi$
Where $z$ is a complex number, I want to know what this inequality mean?
Can't it simply be that,
$0 ≤ arg(z) ≤ 2 pi$
complex-numbers
asked Aug 28 at 16:50
Adi
134
asked Aug 28 at 16:50
Adi
134
asked Aug 28 at 16:50
Adi
134
asked Aug 28 at 16:50
Adi
134
134
1
A complex number $z=a+bi$ could also have been written as $z=re^itheta$, however one can notice writing it in such a way is no longer unique. You have things such as $e^icdot 0=e^icdot 2pi=1$. In order to make it unique, we sometimes refer to the principle argument of a complex number. Some authors prefer the principle argument to be in the range $-pileq arg(z)< pi$. Other authors prefer the principle argument to be in the range $0leq arg(z)<2pi$. Regardless which convention is used $z=re^itextarg(z)$ is now a unique representation.
– JMoravitz
Aug 28 at 16:56
add a comment |Â
1
A complex number $z=a+bi$ could also have been written as $z=re^itheta$, however one can notice writing it in such a way is no longer unique. You have things such as $e^icdot 0=e^icdot 2pi=1$. In order to make it unique, we sometimes refer to the principle argument of a complex number. Some authors prefer the principle argument to be in the range $-pileq arg(z)< pi$. Other authors prefer the principle argument to be in the range $0leq arg(z)<2pi$. Regardless which convention is used $z=re^itextarg(z)$ is now a unique representation.
– JMoravitz
Aug 28 at 16:56
1
1
A complex number $z=a+bi$ could also have been written as $z=re^itheta$, however one can notice writing it in such a way is no longer unique. You have things such as $e^icdot 0=e^icdot 2pi=1$. In order to make it unique, we sometimes refer to the principle argument of a complex number. Some authors prefer the principle argument to be in the range $-pileq arg(z)< pi$. Other authors prefer the principle argument to be in the range $0leq arg(z)<2pi$. Regardless which convention is used $z=re^itextarg(z)$ is now a unique representation.
– JMoravitz
Aug 28 at 16:56
A complex number $z=a+bi$ could also have been written as $z=re^itheta$, however one can notice writing it in such a way is no longer unique. You have things such as $e^icdot 0=e^icdot 2pi=1$. In order to make it unique, we sometimes refer to the principle argument of a complex number. Some authors prefer the principle argument to be in the range $-pileq arg(z)< pi$. Other authors prefer the principle argument to be in the range $0leq arg(z)<2pi$. Regardless which convention is used $z=re^itextarg(z)$ is now a unique representation.
– JMoravitz
Aug 28 at 16:56
add a comment |Â
2 Answers
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You probably want strict inequality at one end of the interval, otherwise the argument is not unambiguously defined.
Any interval of length $2pi$ which includes one endpoint but not the other will work as the set of possible values for the argument of any nonzero complex number. That's because it represents exactly one traversal of the unit circle.
answered Aug 28 at 16:56
MPW
28.6k11853
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0
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It is called Argument.
The argument of $z$ is the angle between the positive real axis and the line joining the point to the origin, the argument is a multi-valued function operating on the nonzero complex numbers.
$$z=x+iy$$
$$arg(z)=tan^-1(fracyx)$$
The $tan^-1$ function usually returns a value in the range (−À, À].
When a well-defined function is required than the usual choice, known as the principal value, is the value in the open-closed interval $(−À,À]$, that is from −À to À radians, excluding −À rad itself.
Some authors define the range of the principal value as being in the closed-open interval [0, 2À).
answered Aug 28 at 16:54
Deepesh Meena
3,0982824
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
You probably want strict inequality at one end of the interval, otherwise the argument is not unambiguously defined.
Any interval of length $2pi$ which includes one endpoint but not the other will work as the set of possible values for the argument of any nonzero complex number. That's because it represents exactly one traversal of the unit circle.
answered Aug 28 at 16:56
MPW
28.6k11853
add a comment |Â
up vote
0
down vote
You probably want strict inequality at one end of the interval, otherwise the argument is not unambiguously defined.
Any interval of length $2pi$ which includes one endpoint but not the other will work as the set of possible values for the argument of any nonzero complex number. That's because it represents exactly one traversal of the unit circle.
answered Aug 28 at 16:56
MPW
28.6k11853
add a comment |Â
up vote
0
down vote
up vote
0
down vote
You probably want strict inequality at one end of the interval, otherwise the argument is not unambiguously defined.
Any interval of length $2pi$ which includes one endpoint but not the other will work as the set of possible values for the argument of any nonzero complex number. That's because it represents exactly one traversal of the unit circle.
answered Aug 28 at 16:56
MPW
28.6k11853
You probably want strict inequality at one end of the interval, otherwise the argument is not unambiguously defined.
Any interval of length $2pi$ which includes one endpoint but not the other will work as the set of possible values for the argument of any nonzero complex number. That's because it represents exactly one traversal of the unit circle.
answered Aug 28 at 16:56
MPW
28.6k11853
answered Aug 28 at 16:56
MPW
28.6k11853
answered Aug 28 at 16:56
MPW
28.6k11853
answered Aug 28 at 16:56
MPW
28.6k11853
28.6k11853
add a comment |Â
add a comment |Â
up vote
0
down vote
It is called Argument.
The argument of $z$ is the angle between the positive real axis and the line joining the point to the origin, the argument is a multi-valued function operating on the nonzero complex numbers.
$$z=x+iy$$
$$arg(z)=tan^-1(fracyx)$$
The $tan^-1$ function usually returns a value in the range (−À, À].
When a well-defined function is required than the usual choice, known as the principal value, is the value in the open-closed interval $(−À,À]$, that is from −À to À radians, excluding −À rad itself.
Some authors define the range of the principal value as being in the closed-open interval [0, 2À).
answered Aug 28 at 16:54
Deepesh Meena
3,0982824
add a comment |Â
up vote
0
down vote
It is called Argument.
The argument of $z$ is the angle between the positive real axis and the line joining the point to the origin, the argument is a multi-valued function operating on the nonzero complex numbers.
$$z=x+iy$$
$$arg(z)=tan^-1(fracyx)$$
The $tan^-1$ function usually returns a value in the range (−À, À].
When a well-defined function is required than the usual choice, known as the principal value, is the value in the open-closed interval $(−À,À]$, that is from −À to À radians, excluding −À rad itself.
Some authors define the range of the principal value as being in the closed-open interval [0, 2À).
answered Aug 28 at 16:54
Deepesh Meena
3,0982824
add a comment |Â
up vote
0
down vote
up vote
0
down vote
It is called Argument.
The argument of $z$ is the angle between the positive real axis and the line joining the point to the origin, the argument is a multi-valued function operating on the nonzero complex numbers.
$$z=x+iy$$
$$arg(z)=tan^-1(fracyx)$$
The $tan^-1$ function usually returns a value in the range (−À, À].
When a well-defined function is required than the usual choice, known as the principal value, is the value in the open-closed interval $(−À,À]$, that is from −À to À radians, excluding −À rad itself.
Some authors define the range of the principal value as being in the closed-open interval [0, 2À).
answered Aug 28 at 16:54
Deepesh Meena
3,0982824
It is called Argument.
The argument of $z$ is the angle between the positive real axis and the line joining the point to the origin, the argument is a multi-valued function operating on the nonzero complex numbers.
$$z=x+iy$$
$$arg(z)=tan^-1(fracyx)$$
The $tan^-1$ function usually returns a value in the range (−À, À].
When a well-defined function is required than the usual choice, known as the principal value, is the value in the open-closed interval $(−À,À]$, that is from −À to À radians, excluding −À rad itself.
Some authors define the range of the principal value as being in the closed-open interval [0, 2À).
answered Aug 28 at 16:54
Deepesh Meena
3,0982824
edited Aug 28 at 16:59
answered Aug 28 at 16:54
Deepesh Meena
3,0982824
answered Aug 28 at 16:54
Deepesh Meena
3,0982824
answered Aug 28 at 16:54
Deepesh Meena
3,0982824
3,0982824
add a comment |Â
add a comment |Â
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1
A complex number $z=a+bi$ could also have been written as $z=re^itheta$, however one can notice writing it in such a way is no longer unique. You have things such as $e^icdot 0=e^icdot 2pi=1$. In order to make it unique, we sometimes refer to the principle argument of a complex number. Some authors prefer the principle argument to be in the range $-pileq arg(z)< pi$. Other authors prefer the principle argument to be in the range $0leq arg(z)<2pi$. Regardless which convention is used $z=re^itextarg(z)$ is now a unique representation.
– JMoravitz
Aug 28 at 16:56