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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.


    This page titled 18.2: Complex coordinates is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.