#### Activity Overview

The modulus and argument form of a complex number can be written in two ways, but how are they connected? ‘CIS’ format is a straight forward application of trigonometry, but what about *e* ? In this activity students explore the Taylor expansion of *e*^{x} and compare it with the Taylor expansion of cos(*x*) + sin(*x*) revealing a small difference relating to an alternating sequence of negative signs which can be adjusted using *i*. The activity removes the mystery and replaces it with a beautifully connected piece of mathematics

#### Objectives

- Complex numbers in polar form (modulus and argument)
- Proof of basic identities involving modulus and argument
- Proof of DeMoivre’s Theorem for integral powers

#### Vocabulary

- Taylor Series,
- Taylor Polynomial,
- Complex numbers in rectangular form,
- Complex numbers in polar form
- Modulus
- Argument

#### About the Lesson

A Taylor polynomial is a finite number of terms from a Taylor series. In this activity Taylor polynomials are used that approximate exponential and trigonometric functions. The introduction of the complex number *i* shows how the trigonometric functions can be added to produce the exponential function and therefore making the link between the two types of polar representation of a complex number.