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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

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mathsComplex NumbersComplex Plane and Polar Form

Argument of a complex Number

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



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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)`
  • `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
  • 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)`

What is `arg(z)` called?

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is 'Argument or arg'.

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

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)`

Find the argument of `1+i`

  • `tan^(-1)1`
  • `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`
  • `(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`

'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`.

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

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`

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`

Solved Exercise Problem:

Find the principal argument of `1-i`

  • `pi/4`
  • `-pi/4`
  • `-pi/4`
  • `+-pi/4`

The answer is '`-pi/4`'

                            
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