Euler's identity and exponential form

Comparing the trigonometric form with the expansion of the exponential reveals a deep link between the exponential function and the trigonometric functions.

Property — Euler's identity and exponential form

Anticipating a result that will be justified rigorously in Year Five with series: $e^{i\theta} = \cos\theta + i\sin\theta.$ As a consequence, every complex number is written in exponential form as $z = \rho\,e^{i\theta}.$

The exponential form is the most compact: it encloses in a single symbol the information about modulus ($\rho$) and argument ($\theta$). It is also the most convenient for multiplications, divisions and powers, as will be seen with De Moivre’s formula.

Links

Topics: Complex numbers
Concepts: Exponential form · Trigonometric form · Euler’s identity
Skills: Use formulae
People: Leonhard Euler (Eulero)