Geometrical Interpretation of Complex Addition

Theorem

Let $a, b \in \C$ be complex numbers expressed as vectors $\mathbf a$ and $\mathbf b$ respectively.

Let $OA$ and $OB$ be two adjacent sides of the parallelogram $OACB$ such that $OA$ corresponds to $\mathbf a$ and $OB$ corresponds to $\mathbf b$.


Then the diagonal $OC$ of $OACB$ corresponds to $\mathbf a + \mathbf b$, the sum of $a$ and $b$ expressed as a vector.


Proof

Let $a = a_x + i a_y$ and $b = b_x + i b_y$.

Then by definition of complex addition:

$a + b = \paren {a_x + b_x} + i \paren {a_y + b_y}$

Thus $\mathbf a + \mathbf b$ is the vector whose components are $a_x + b_x$ and $a_y + b_y$.


Similarly, we have:

$b + a = \paren {b_x + a_x} + i \paren {b_y + a_y}$

Thus $\mathbf b + \mathbf a$ is the vector whose components are $b_x + a_x$ and $b_y + a_y$.

It follows that both $\mathbf a + \mathbf b$ and $\mathbf b + \mathbf a$ both correspond to the diagonal $OC$ of $OACB$.

$\blacksquare$


Sources

  • 1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 3$. Geometric Representation of Complex Numbers
  • 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations
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