Geometrical Interpretation of Complex Modulus
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Theorem
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.
Proof
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 $z$.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups