Visualizing Division of Complex Numbers on the Complex Plane

To revise, multiplication involves adding the arguments or angles of two numbers and multiplying their moduli together to create the new transformed number. Then we should also remember that the angle of a multiplicative inverse of a complex number has its modulus the reciprocal of the original and its angle is the negative (or coterminal) of the original angle.

Once we understand the graphical interpretation of multiplication, the graphical interpretation of division can be theorized. In the case of $\dfrac{z_1}{z_2}$ , this can be written as $z_1 \times z_2^{-1}$ . Based on this, we can make the simple claims.

$\arg\left(\dfrac{z_1}{z_2}\right) = \arg(z_1 \times z_2^{-1}) = \arg(z_1) - \arg(z_2)$

$\left|\dfrac{z_1}{z_2}\right| = |z_1 \times z_2^{-1}| = |z_1| \times |z_2^{-1}| = |z_1| \times \dfrac{1}{|z_2|} = \dfrac{|z_1|}{|z_2|}$

The green and red points represent $z_1$ and $z_2$ respectively while the red point represents $\dfrac{z_1}{z_2}$ . Move the two points around to try and draw some conclusions about the behaviour.

Visualizing Transformation by Division

Insights

We should see that our two claims are vindicated. The angle of the result involves subtracting the angle of the denominator from the numerator. Also, the numerator's modulus is directly proportional to the modulus of the result while the modulus of the denominator is inversely proportional to the modulus of the result.

End of Lesson