Week 4 – Complex Numbers

Cartesian and polar form of a complex number. The Argand diagram. Roots of unity. The relationship between exponential and trigonometric functions. The geometry of the Argand diagram. 1 The Need For Complex Numbers All of you will know that the two roots of the quadratic 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? Then there are no real solutions to the equation as no real 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.