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Complex Plane and Polar Form

Complex Plane and Polar Form

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Argument


 »  Angle with positive real-axis


 »  `theta = tan^(-1) (b/a)`


 »  Arguments `= 2npi+theta` where `n=0,1,2,...`


 »  Principal argument `= theta` where `0 <= theta < 2pi`

Argument of a complex Number

plain and simple summary

nub

plain and simple summary

nub

dummy

The angle made by the line segment with the real axis is the argument of the complex number.

Principal argument of a complex number is in the range `-pi` to `pi`.

simple steps to build the foundation

trek

simple steps to build the foundation

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Argument is defined for a complex number represented as a point in Argand plane.


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Starting on learning "Argument of a complex number". ;; Argument is defined for a complex number represented as a point in Argand plane.

A Complex number `a+i b` is represented on complex plane as shown in the figure.argument of complex number What is the angle the line segment `OP` makes with the real axis?

  • `tan^(-1)(b/a)`
  • `0`

The answer is '`tan^(-1)(b/a)`'

The angle between the real axis in positive direction and the line segment `OP` is called the argument of the complex number. argument of complex number The argument of a complex number is given as `text(arg) z`.

What does "argument" mean?

  • an independent element that plays a role in determining the value of something
  • complete absence and avoidance of mentioning of something

The answer is 'an independent element that plays a role in determining the value of something'.
For example the arguments of a function `f() = 2xy+3` are the variables `x` and `y`. In this case, the arguments define the value of the function.

In complex numbers, the modulus provides the absolute value. In addition to the modulus, to completely specify the complex number, the additional independent element required is the "argument".

For `z=a+i b in CC`, argument is given by `text(arg) z = tan^(-1)(b/a)`

arg(z) is called...

  • Practice Saying the Answer

The answer is 'Argument or arg'.

Find the argument of `1+i`

  • `tan^(-1)1`
  • `sqrt(2)`

The answer is '`tan^(-1)1`'

What are the possible values of `tan^(-1)1`?

  • only one possible value `pi/4`
  • Two possible values `pi/4` and `(3pi)/4`
  • `(n pi)/4` where `n = 1,3,5,cdots`

The answer is '`(n pi)/4` where `n = 1,3,5,cdots`'.

The possible values of `tan^(-1)1` are `(n pi)/4` where `n = 1,3,5,cdots`.

What are the possible values of `tan^(-1)1`, if the opposite side and adjacent sides are given as `1`?

  • Answer is same : `(n pi)/4` where `n = 1,3,5,cdots`
  • Answer is different : `(n pi)/4` where `n = 1,5,9,cdots`

'Answer is different : `(n pi)/4` where `n = 1,5,9,cdots`'

Note that: `tan^(-1)1 = (n pi)/4` where `n = 1,3,5,cdots` but

`tan^(-1)  (text(opposite-side)=1)/(text(adjacent-side)=1) = (n pi)/4` where `n = 1,5,9,cdots` and

`tan^(-1)  (text(opposite-side)=-1)/(text(adjacent-side)=-1) = (n pi)/4` where `n = 3,7,11,cdots`

The argument of a complex number can be given in the form `2n pi + theta` where `n=0,1,2...`. principal argument of complex numbers The argument corresponding to `n=0` is called the principal argument. The range of principal argument is `-pi <= theta <= pi`.

Given the complex numbers `z_1, z_2, z_3, z_4` as shown in figure.range of argument of complex numbers The value of argument of complex numbers are
 •  `0 < text(arg) z_1 < pi/2`
 •  `pi/2 < text(arg) z_2 < pi`
 •  `0< text(arg) z_3 <-pi/2`
 •  `-pi/2< text(arg) z_4 <-pi`

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Argument: For a complex number `z=a+i b in CC`, the argument of `z` is given as ` text(arg) z = tan^(-1)(b/a)`

Principal Argument: For a complex number `z=a+i b in CC`, the principal argument of `z` is given as ` text(arg) z = tan^(-1)(b/a)`, where `-pi <= text(arg) z <= pi`



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Find the principal argument of `1-i`

  • `pi/4`
  • `-pi/4`
  • `+-pi/4`

The answer is '`-pi/4`'

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Progress

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Progress

A Complex number a+i b is represented on complex plane as shown in the figure. What is the angle the line segment O P makes with the real axis?
tan;inverse;b;a
tan inverse b by a
0;zero
0
The answer is "tan inverse b by a"
The angle between the real axis in positive direction and the line segment O P is called the argument of the complex number. The argument of a complex number is given as arg of z.
What does "argument" mean?
independent;element;plays
an independent element that plays a role in determining the value of something
complete;absence;avoidance
complete absence and avoidance of mentioning of something
The answer is 'an independent element that plays a role in determining the value of something'. ;; For example the arguments of a function f = 2x y+ 3 are the variables x and y. In this case, the arguments define the value of the function.
In complex numbers, the modulus provides the absolute value. In addition to the modulus, to completely specify the complex number, the additional independent element required is the "argument". ;; For z=a+i b, argument is given by arg of z = tan inverse b by a
arg(z) is called...
argument;arg
The answer is 'Argument or arg'.
The angle made by the line segment with the real axis is the argument of the complex number.
For a complex number z = a+i b, the argument of z is tan inverse b by a.
Find the argument of 1 plus i
tan;inverse;1
tan inverse 1
square;root;2
square root 2
The answer is 'tan inverse 1'
What are the possible values of tan inverse 1?
only;one;1
only one possible value, pi by 4
two;2
Two possible values pi by 4 and 3 pi by 4
n;equals;5
n pi by 4 where n equals 1 3 5 et cetera
The answer is 'n pi by 4 where n equals 1 3 5 et cetera'
The possible values of tan inverse 1 are n pi by 4 where n equals 1, 3, 5, et cetera. ;; What are the possible values of tan inverse 1, if the opposite side and adjacent sides are given as 1?
same
Answer is same : n pi by 4 where n equals 1 3 5 et cetera
different
Answer is different: n pi by 4 where n equals 1 5 9 et cetera
Answer is different: n pi by 4 where n equals 1 5 9 et cetera
Note that: tan inverse 1 ; = n pi by 4 ; where n = 1,3,5,et cetera ;; but ; tan inverse opposite-side =1 by adjacent-side =1 ; = n pi /4 ; where n = 1,5,9,et cetera ;; and; tan inverse opposite-side =-1 by adjacent-side =-1 ; = n pi /4 ; where n = 3,7,11,et cetera
The argument of a complex number can be given in the form 2n pi + theta ; where n=0,1,2 et cetera. ;; The argument corresponding to n=0 is called the principal argument. The range of principal argument is minus pi to pi.
Principal argument of a complex number is in the range minus pi to pi
For a complex number z=a+i b, the principal argument of z is given as tan inverse b by a, where argument of z is in the range minus pi to pi.
Given the complex numbers z 1, z 2, z 3, z 4, as shown in figure. The value of argument of complex numbers are ;; argument of z 1 is, 0 to pi by 2 ;; argument of z 2 is, pi by 2 to pi ;; argument of z 3 is, 0 to minus pi by 2 ;; argument of z 4 is, minus pi by 2 to minus pi
Find the principal argument of 1 minus i.
pi by 4
pi by 4
minus
minus pi by 4
plus or minus
plus or minus pi by 4
The answer is 'minus pi by 4'

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