# Homework Help: Argument of a complex number

1. Dec 30, 2011

### sara_87

1. The problem statement, all variables and given/known data

I have a complex number
z=1-i

I want to find the argument of this complex number

2. Relevant equations

The angle it makes with the positive real axis is arctan(1/1)=pi/4

3. The attempt at a solution

This point lies in the fourth quadrant of the argand diagram.

Arg(z)= angle the z makes anticlockwise with the positive real axis = 2*pi - pi/4

In matlab, the argument of this number is -pi/4 so I don't understand, doesn't the argument have to always be measured anticlockwise from the positive real axis?

2. Dec 30, 2011

### Staff: Mentor

That might be so for positive angles, but negative angles are measured in the clockwise direction. I can't imagine that matlab would require all angles to be positive.

The angles 7$\pi$/4 and -$\pi$/4 are of course different, but they have the same reference point.

3. Dec 30, 2011

### espen180

You can freely add multiples of 2pi to the arument of a complex number without changing its value. For different multiples of 2pi, you obtain different representations of the same argument. After all, the radian mesure is periodic in 2pi. However, there exists the notion of the principal argument (or the principal value of the argument) of a complex number. Of all the possible representations of the argument, this is the one lying between 0 and 2pi.

Therefore -pi/4 is a valid representation of arg(1-i), but it is not the principal argument. The principal argument is 7pi/4, as you stated.