Ask Uncle Colin: A Little Complexity

Dear Uncle Colin,

I know that the imaginary part of the number $\frac{x-2i}{1+i}$ is 1, and I need to find $z$, given that $x$ is real. What do I do?

- Any Really Good Answers Now, Dear?

Hi, ARGAND, and thanks for your message!

I would start by realising the denominator: multiply top and bottom by $1-i$ to get $\frac{(x-2)+(-x-2)i}{2}$.

The imaginary part of that is $\frac{-x-2}{2}$, which equals 1, so $x=-4$.

Alternatively, you can say that $\frac{x-2i}{1+i}=a+i$, for some $a$ that you don’t really care about just now.

Multiplying across, $x-2i=(a-1)+(a+1)i$, so (comparing imaginary coefficients), $a=-3$, and (comparing real), $x=-4$.

Hope that helps!

- Uncle Colin