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sciencesolve eNotes educator| Certified Educator

To find the argument of the complex number 1-i you need to transform the given rectangular form into the polar form.

You need to remember how to transform the rectangular form of a complex number z = x+iy into the polar form`z = (sqrt(x^2+y^2))*(cos theta + isin theta).`

`theta`  denotes the argument of the complex number.

`theta = arctan (y/x)`

Comparing the standard form of a comples number with the given complex number yields: x = 1; y = -1

`theta = arctan(-1/1) = arctan (-1) = -pi/4`

`sqrt(x^2+y^2) = sqrt(1+1) = sqrt2`

The polar form of the given complex number is `z=sqrt2*(cos(pi/4) - i*sin(pi/4)).`

The argument  of the given complex number is `theta = -pi/4` .

neela | Student

Shall I edit the question like this?

Find the argument(1-i) or in short form arg(1-i):

 

1-i  is a complex number, which can be written in polar coordinates, explaied below.

The complex number, z is written in the form :

z = x+iy, where x is  the real part and y is the imaginary part of the complex number .

We nomalise the complex number and write it as below in order to express its real and imaginary parts  as cosine of an angle a and sine of an angle a times the modulus of the complex number to conver it into polar form:

z = sqrt(x^2+y^2){x/sqrt(x^2+y^2) +i* y/sqrt(x^2+y^2)}

=r(cos a  + i*sin a), where  r=sqrt(x^2+y^2) and sina =x/r

cosa = y/r

Then,  tan a = x/y. or

arg(z) =a = arctg(x/y).

In the given problem:

Let z = 1- i = sqrt(2){ 1/sqrt2+i(-1/sqrt2)}

x=1 and y=-1.

arg(z) = arg(1 -i) =  arctg [(-1/(+1)] = (-pi/4) = -pi/4 = -45 degree.

Hope this helps you.

 

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