I had been solving an equation with complex numbers: $z = \frac{x - iy}{x + iy}$ I solved it up to the point where I get: $z = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}$. But I have no idea how to simplify it! Could anyone help me? I would be glad to learn how to simplify $\frac{x^{2} + y^{2}}{x^{2} - y^{2}}$ as well...
2026-04-17 21:21:54.1776460914
How could I simplify this algebra expresion?
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$$z=\frac{x-iy}{x+iy}=\frac{(x-iy)^2}{(x+iy)(x-iy)}\\=\frac{x^2-y^2-2ixy}{x^2+y^2}$$ So the real part is your answer $\frac{x^2-y^2}{x^2+y^2}$, but it also has an imaginary part: $\frac{-2xy}{x^2+y^2}$.
It doesn't get simpler than that; but at least the real and imaginary parts are untangled.