De Moivre’s Theorem

Martin McBride
Technological Singularity
9 min readApr 22, 2024

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De Moivre’s theorem can be simply stated as:

Notice that the sine function on both sides is multiplied by the imaginary unit i. This isn’t a pure trigonometrical identity, instead it is a formula that applies to complex numbers.

This formula is an early precursor to Euler’s identity. It is easier to prove — it derives directly from the definition of complex multiplication — but it is not as versatile.

Complex multiplication

When we first learn about complex multiplication, we simply apply the FOIL rule. For example, if we multiply 1 + i (which we will call a) and 1 + 4i (which we will call b), we get the result c, which is -3 + 5i:

This is shown here on an Argand diagram:

Modulus argument form

A complex number can be represented in the form x + iy, but it can also be represented in modulus argument form. In that case, we define the complex number using the angle Θ that the number makes with the real axis, and the length r from the number to the origin. This diagram shows the two alternative ways of representing the same complex number:

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