Geometric significance of complex arithmetic

Parametric equations

Adding complex numbers translates and combines vectors in the plane. Multiplying them adds the arguments and multiplies the moduli. The moving arrows show the algebra and the geometry at the same time.

Let $z_1=r_1{e}^{i{\theta }_1}$ and $z_2=r_2{e}^{i{\theta }_2}$:

Interaction

• z₁ modulus r₁ and z₁ argument θ₁: set the modulus and argument of $z_1$
• z₂ modulus r₂ and z₂ argument θ₂: set the modulus and argument of $z_2$
• The sum $z_1+z_2$ lies on the diagonal of the parallelogram, and the product $z_1{z}_2$ multiplies the moduli and adds the arguments

What to notice

• Adding complex numbers is adding vectors in the plane.
• In polar form, multiplying complex numbers is a scaling and a rotation composed together.
• The unit circle is a reference scale of modulus 1.

Related

• Polar form of a complex number
• Euler’s formula, rotating
• Complex numbers and Euler’s formula

Further reading

• Complex number (Wikipedia)
• Complex plane (Wikipedia)