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Mathematics LibreTexts

18.2: Complex coordinates

  • Page ID
    23697
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    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.