Complex numbers and Euler's formula

Idea

Multiplying by $e^{i\theta }$ turns an arrow counterclockwise by $\theta$. On this page that same arrow is also a sum, a difference, a product, or a quotient of two complex numbers, and the power in De Moivre’s formula. The horizontal projection is $\cos \theta$ and the vertical projection is $\sin \theta$. This is not the separate page that traces only the imaginary part as a sine wave.

Interaction

• Complex arithmetic: switch Operation among addition, subtraction, multiplication, and division, and see where $z_1$ and $z_2$ land. Drag the tips of $z_1$ and $z_2$
• Euler’s formula: drag Angle θ and watch $e^{i\theta }$ on the unit circle, together with the cosine and sine projections
• De Moivre’s formula: compare $(\cos \theta +i\sin \theta {)}^n$ with $\cos (n\theta )+i\sin (n\theta )$. Power n and Angle θ move the point at angle $n\theta$

Suggested order: Complex arithmetic, then Euler’s formula, then De Moivre’s formula.

What to notice

• Multiplying by a complex number rotates and scales one arrow. Rectangular coordinates are the components of that arrow.
• $e^{i\theta }$ reads the same rotation as a point on the unit circle and as $\cos \theta +i\sin \theta$.
• Addition, multiplication, and rotation share one complex plane. The difference is a shift, or a turn about the origin, not three unrelated operations.

Related

• Geometric significance of complex arithmetic
• Polar form of a complex number
• Euler’s formula, rotating
• Julia set

Further reading

• Euler’s formula (Wikipedia)
• De Moivre’s formula (Wikipedia)