Geometrical Interpretation of Complex Modulus

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Let $z \in \C$ be a complex number expressed in the complex plane.

Then the modulus of $z$ can be interpreted as the distance of $z$ from the origin.


Let $z = x + i y$.

By definition of the complex plane, it can be represented by the point $\tuple {x, y}$.

By the Distance Formula, the distance $d$ of $z$ from the origin is:

\(\ds d\) \(=\) \(\ds \sqrt {\paren {x - 0}^2 + \paren {y - 0}^2}\)
\(\ds \) \(=\) \(\ds \sqrt {x^2 + y^2}\)

which is precisely the modulus of $z$.