18.2: Complex coordinates
Recall that one can think of the Euclidean plane as the set of all pairs of real numbers \((x,y)\) equipped with the metric
\(AB=\sqrt{(x_A-x_B)^2+(y_A-y_B)^2},\)
where \(A=(x_A,y_A)\) and \(B=(x_B,y_B)\) .
One can pack the coordinates \((x,y)\) of a point in one complex number \(z=x+i\cdot y\) . This way we get a one-to-one correspondence between points of the Euclidean plane and \(\mathbb{C}\) . Given a point \(Z=(x,y)\) , the complex number \(z=x+ i\cdot y\) is called the complex coordinate of \(Z\) .
Note that if \(O\) , \(E\) , and \(I\) are points in the plane with complex coordinates \(0\) , \(1\) , and \(i\) , then \(\measuredangle EOI=\pm\dfrac{\pi}{2}\) . Further, we assume that \(\measuredangle EOI=\dfrac{\pi}{2}\) ; if not, one has to change the direction of the \(y\) -coordinate.