Question

How do you divide `(-5-i)/(-10i)`?

Answer 1

I found: `0.5-0.5i`

Explanation:

Here you have a division between complex numbers that normally wouldn't be immediately possible or at least will not give a complex number in standard form: `a+ib`.

To solve the problem we need to get rid of `i` in the denominator!
To do that we multiply and divide by the complex conjugate of the denominator .

If you have a complex number, say, `3+4i` the complex conjugate is obtained changing the sign of the immaginary part as: `3color(red)(-)4i`.

In our case the denominator is:
`0-10i`

so the complex conjugate will be: `0+10i`.

Let us multiply and divide:

`(-5-5i)/(0-10i)*(0+10i)/(0+10i)=`

After this (even if it looks more difficult) we do the multiplications:
`=(-50i-50i^2)/(-100i^2)=`

remember that `i^2=-1` so:

`=(50-50i)/(100)=`

Now, the good thing is that we got rid of `i` in the denominator AND we can rearrange separating into fractions as:
`=50/100-50/100i=0.5-0.5i=1/2-1/2i`
which is in the form `a+ib`!!!