Modulus of Complex Number equals its Distance from Origin
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Theorem
The modulus of a complex number equals its distance from the origin on the complex plane.
Proof
Let $z = x + y i$ be a complex number and $O = 0 + 0 i$ be the origin on the complex plane.
We have its modulus:
| \(\ds \cmod z\) | \(=\) | \(\ds \cmod {x + y i}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {x^2 + y^2}\) | Definition of Complex Modulus |
and its distance from the origin on the complex plane:
| \(\ds \map d {z, O}\) | \(=\) | \(\ds \map d {\tuple {x, y}, \tuple {0, 0} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {\paren {x - 0}^2 + \paren {y - 0}^2}\) | Distance Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {x^2 + y^2}\) |
The two are seen to be equal.
$\blacksquare$
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