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

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Charlie Hooper eNotes educator| Certified Educator

The complex number `z` has the form `z=x+iy` . In the complex plane where `x` is the real axis and `y` is the imaginary axis we have `x=-1` and `y=-1` .

Find the hypotenuse.

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

There are infinite values for `theta` that can make this triangle in the third quadrant.

`theta=(5pi)/4+2pi n, n=0,+-1,+-2...`

Now use the polar relations for `x` and `y` .

`x+iy=[rcos(theta)]+i[r sin(theta)]`

`-1-i=sqrt(2)cos((5pi)/4+2pi n)+i sqrt(2) sin((5pi)/4+2pi n)`

`-1-i=sqrt(2)[cos((5pi)/4+2pi n)+i sin((5pi)/4+2pi n)]`

Use Euler's relation

`rcos(theta)]+i r sin(theta)=r*e^(i*theta)`

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

Therefore, `z` is a point on the circle `sqrt(2)e^(i*theta)` where theta has the angle (5pi)/4 which also works for every revolution of `2pi` around the circle.

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embizze eNotes educator| Certified Educator

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`