Product of Complex Numbers

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Use the green slider to see how the product of two complex numbers is built from a triangle of the first complex number, the origin of coordinates and point (1.0)

&bull What do the two triangles have in common and what is different?

Move the blue points (Z₁ and Z₂) to find the products of the Complex Numbers below and write their results:

(-2-2i) × (1+3i)
(2+3i) × (3-6i)
(3+i) • (-3-i)
5(-2+i)
(3+8i) • i
(-1-2i)(-1+2i)

Investigates and explains what happens when ...

... a complex number is multiplied by the number i
... a complex number is multiplied by a real number (with no imaginary part)
... a complex number is multiplied by its conjugate

And if we work with polar coordinates? Look at the figure below:

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What is the relationship between modules z₁, z₂ and z₁ ÷ z₂ ?

What will be the result of the following ratios of complex numbers? To view, right click the blue points (Z₁ and Z₂) of each complex number, select Redefine and change the coordinates.

5_30° × 1_150°
3_15° × 2_75°
8_15° • 1_90°
5_0° • 2_45°
4_60° by its conjugate.
3_150° by its opposite.

... a complex number is multiplied by the number i
... a complex number is multiplied by a real number (with no imaginary part)
... a complex number is multiplied by its conjugate

Created with GeoGebra by Manuel Sada Allo (March 2008) and adapted by Steven Lapinski