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# 18.2: Complex coordinates

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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.