Complex numbers as arrows

Drag z and w on the Argand plane: add them tip to tail, or multiply the lengths and add the angles.

Interactive

z1.3 + 0.75i

w0.6 + 1.04i

z·w0 + 1.8i

length, angle1.8, 90°

Show

Length of z, |z|1.5Angle of z30 °Length of w, |w|1.2Angle of w60 °

The math, with your numbers

|zw| = |z|\,|w| = 1.5 \times 1.2 = 1.8

\arg(zw) = \arg z + \arg w = 30^\circ + 60^\circ = 90^\circ

Set w to length 1 at 90°: multiplying by it is multiplying by i, a quarter turn.

Try this

→Multiply: set w = i (length 1, angle 90°). What happens to z? Do it twice in your head: what is i²?
→Add: how is adding complex numbers like adding forces?
→Multiply: can you make z·w land exactly on the real axis?

Want to know why it behaves like this?

The lesson “Complex Numbers: Numbers as Arrows” explains it step by step: How can a number squared be negative, and why would an engineer want one?

Complex numbers as arrows

Drag z and w on the Argand plane: add them tip to tail, or multiply the lengths and add the angles.

Complex numbers as arrows

Interactive

z1.3 + 0.75i

w0.6 + 1.04i

z·w0 + 1.8i

length, angle1.8, 90°

Show

Length of z, |z|1.5Angle of z30 °Length of w, |w|1.2Angle of w60 °

The math, with your numbers

|zw| = |z|\,|w| = 1.5 \times 1.2 = 1.8

\arg(zw) = \arg z + \arg w = 30^\circ + 60^\circ = 90^\circ

Set w to length 1 at 90°: multiplying by it is multiplying by i, a quarter turn.

Try this

→Multiply: set w = i (length 1, angle 90°). What happens to z? Do it twice in your head: what is i²?
→Add: how is adding complex numbers like adding forces?
→Multiply: can you make z·w land exactly on the real axis?

Want to know why it behaves like this?

The lesson “Complex Numbers: Numbers as Arrows” explains it step by step: How can a number squared be negative, and why would an engineer want one?