Presentation on theme: "Complex Numbers If we wish to work with , we need to extend the set of real numbers Definitions i is a number such that i2 = -1 C is the set of."— Presentation transcript:
1 Complex NumbersIf we wish to work with , we need to extend the set of real numbersDefinitionsi is a number such that i2 = -1C is the set of numbers Z, of the form where a and b are real numbers.a is called the real part of Z and we write a = R(z) of a = Re(z)b is called the imaginary part of Z and we write b = i(z) or b = Im(z)
2 GivenandAddition is defined by:Multiplication is defined by:We may write a + bi or a + ib, whichever we find more convenient
11 Argand DiagramsThe complex number is represented on the plane by the point P(x,y). The plane is referred to as “The Complex Plane”, and diagrams of this sort are called Argand Diagrams.ryxpAny point on the x-axis represents a purely Real NumberAny point on the y-axis represents a purely imaginary number
12 The size of the rotation is called the amplitude or argument of z. It is often denoted Arg z.This angle could beWe refer to the value of Arg z which lies in the range -< asthe principal argument. It is denoted arg z, lower case ‘a’.ryxpBy simple trigonometry:This is referred to as the Polar form of z.
13 a) Find the modulus and argument of the complex number Since (3,4) lies in the first quadrant, n = 0b) Find the modulus and argument of the complex numberSince (-3,-4) lies in the third quadrant, n = -1
20 Polar Form and Multiplication Note arg(z1z2) lies in the range (-, ) and adjustments have to be made by adding or subtracting 2 as appropriate if Arg(z1z2) goes outside that range during the calculation.
30 By De Moivre’s theorem, when finding the nth root of a complex number we are effectively dividing the argument by n. We should therefore study arguments in the range (-n, n) so that we have all the solutions in the range (-, ) after division.The position vectors of the solution will divide the circle of radius r, centre the origin, into n equal sectors.
34 Page 106 Exercise 7: Question 2 plus a selection from 1
35 PolynomialsIn 1799 Gauss proved that every polynomial equation with complex coefficients, f(z) = 0, where z C, has at least one root in the set of complex numbers. He later called this theorem the fundamental theorem of algebra. In this course we restrict ourselves to real coefficients but the fundamental theorem still applies since real numbers are also complex.
36 We need to find z2, z3 and z4And substitute them into theOriginal equation.