Express the following complex quantity in the form $a+bi$ I'm currently working on a problem and I need to simplify the following complex quantity in the form of $a+bi$. Assuming that $a$ and $b$ are real numbers, here is the quantity: $${(a+bi)\over (a-bi)}- {(a-bi)\over (a+bi)}$$ I've rationalized the left and the right terms and arrived at an answer of $\dfrac{4abi}{a^2 + b^2}$, but do not know where to go from here. I double checked my math and everything should be correct up to this point. What's the problem here?

You have $z=a+bi$ and want to compute $$\frac{z}{\bar z}-\frac{\bar z } {z}$$ Where $\bar z= a-bi$. Combine denominators to get $$\frac{z^2-\bar z^2}{z\bar z}$$ From which you get $$\frac{4iab}{a^2+b^2}$$ So if you wanted to put this in the shape of a complex number $A+Bi$ you get that $A=0$ and $B=\frac{4ab}{a^2+b^2}$.