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:

\(\displaystyle \left\vert{z}\right\vert\) \(=\) \(\displaystyle \left\vert{x + y i}\right\vert\)
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {x^2 + y^2}\) Definition of Modulus


and its distance from the origin on the complex plane:

\(\displaystyle d \left({z, O}\right)\) \(=\) \(\displaystyle d \left({\left({x, y}\right), \left({0, 0}\right)}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sqrt{\left({x - 0}\right)^2 + \left({y - 0}\right)^2}\) Distance Formula
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {x^2 + y^2}\)


The two are seen to be equal.

$\blacksquare$