Write the complex number `z=-1-i` in polar form.

We are asked to write the complex number z=-1-i in polar form.

The polar form of a complex number is 

`z=r(cos theta + i sin theta)`

where r is the distance from the origin (the modulus or absolute value of the complex number) and theta is the angle from the positive x-axis. (Note that the angle is not unique -- you can always select an angle from 0 to 2pi or 0 to 360 degrees.)

To compute r we use 

`r=sqrt(a^2+b^2)`

where a is the real part and b is the imaginary part of the complex number. So:

`r=sqrt((-1)^2+(-1)^2)=sqrt(2)`

We can find theta by

`tan theta = b/a`

In this case we can solve the angle by inspection as 225 degrees or 5pi/4.

The polar form is:

`z=sqrt(2)(cos((5pi)/4)+isin((5pi)/4))`

An alternative is to write in Euler notation:

`z=|z|e^(i theta)=re^(i theta)`

So here we have:

`-1-i=sqrt(2)e^((i(5pi)/4))`

Note again that the angle is not unique; we can add/subtract any multiples of 2pi and still have the same point. A general solution is:

`-1-i=sqrt(2)(cos((5pi)/4 + 2pi n)+isin((5pi)/4+ 2pi n))=sqrt(2)e^(i((5pi)/4+2pi n)); n in ZZ`