Complex Numbers

This article discusses some introductory ideas associated with complex numbers, their algebra and geometry. This includes a look at their importance in solving polynomial equations, how complex numbers add and multiply, and how they can be represented. Finally we look at the nth roots of unity, that is, the solutions of the equations z = 1. 1 The Need For Complex Numbers The shortest path between two truths in the real domain passes through the complex domain Jacques Hadamard (1865-1963) All of you will know that the two roots of the equation ax + bx+ c = 0 are x = −b± √ b2 − 4ac 2a (1) and solving quadratic equations is something that mathematicians have been able to do since the time of the Babylonians. When b − 4ac > 0 then these two roots are real and distinct; graphically they are where the curve y = ax+ bx+ c cuts the x-axis. When b− 4ac = 0 then we have one real root and the curve just touches the x-axis here. But what happens when b − 4ac < 0? In this case there are no real solutions to the equation, as no real number squares to give the negative b − 4ac. From the graphical point of view, the curve y = ax + bx+ c lies entirely above or below the x-axis.