Complex Numbers For Beginners

In the first chapter, students will learn the cartesian(algebraic) form of complex number and its complex conjugate. They will learn the four basic operations involving complex numbers. We will look at how to solve simple two simultaneous equations involving complex numbers. Then finally they will learn how to use Fundamental Theorem of Algebra to solve polynomial equations of real coefficients up to degree 4.

The second chapter will start with the definition of modulus and argument of a complex number. Then students will learn how to convert from cartesian form to trigonometric form. We will also introduce the Euler formula and the exponential form of a complex number. Then we will see how properties of modulus and arguments can be used in multiplication and division of complex number. Students will also learn the geometrical interpretation of addition and subtraction of complex numbers.

In the third chapter, we will formally introduce the De Moivre's Theorem and see how to use it to find powers of complex numbers and also to solve equations of the form z^n=a+ib

The fourth chapter will teach students some useful applications of De Moivre's Theorem. In particular, we will use it to express sin(nx) and cos(nx) as polynomial of sin x and cos x. Powers of sin x and cos x can also be expressed in terms of multiple angles We will also see how it can be used to evaluate trigonometric series using geometric series or other summation methods.

The last chapter will be focusing on common loci in Argand diagram. We will look at the circle, perpendicular bisector and the half line.  Students will learn how to use properties of circles and geometry/trigonometry to find maximum and minimum values of modulus and argument from a given point.