10.2. Argand Diagrams and the Polar Form. Introduction. Prerequisites. Learning Outcomes
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1 Argand Diagrams and the Polar Form 10.2 Introduction In this Section we introduce a geometrical interpretation of a complex number. Since a complex number = x +i is comprised of two real numbers it is natural to consider a plane in which to place a complex number. We shall see that there is a close connection between complex numbers and two-dimensional vectors. In the second part of this Section we introduce an alternative form, called the polar form, for representing complex numbers. We shall see that the polar form is particularl advantageous when multipling and dividing complex numbers. Prerequisites Before starting this Section ou should... Learning Outcomes After completing this Section ou should be able to... 1 know what a complex number is 2 understand how to use trigonometric functions cos θ, sin θ and tan θ 3 understand what a polnomial function is 4 possess a knowledge of vectors represent complex numbers on an Argand diagram obtain the polar form of a complex number multipl and divide complex numbers in polar form
2 1. The Argand Diagram In Section 10.1 we met a complex number = x +i in which x, are real numbers and i= 1. We learned how to combine complex numbers together using the usual operations of addition, subtraction, multiplication and division. In this Section we examine a useful geometrical description of complex numbers. Since a complex number is specified b two real numbers x, it is natural to represent a complex number b a vector in a plane. We take the usual Ox plane in which the horiontal axis is the x-axis and the vertical axis is the -axis (2, 3) ( 1, 1) w x Thus the complex number = 2+3iwould be represented b a line starting from the origin and ending at the point with coordinates (2, 3) and w = 1+iisrepresented b the line starting from the origin and ending at the point with coordinates ( 1, 1). When the Ox plane is used in this wa it is called an Argand diagram. With this interpretation the modulus of, that is is simpl the length of the line which represents. Given that = 1+i, w =irepresent the complex numbers,w and 2 3w 1 on an Argand diagram. Your solution HELM (VERSION 1: March 18, 2004): Workbook Level 1 2
3 Noting that 2 3w 1=2+2i 3i 1=1 iou should obtain the following diagram. 1 Argand diagram w 0 1 x 1 2 3w 1 If we have two complex numbers = a +ib, w = c +id then, as we alread know + w =(a + c)+i(b + d) that is, the real parts add together and the imaginar parts add together. But this is precisel what occurs with the addition of two vectors. If p and q are 2-dimensional vectors then: p = ai + bj q = ci + dj where i and j are unit vectors in the x- and -directions respectivel. So, using vector addition: p + q =(a + c)i +(b + d)j q p + q w + w p 0 x vector addition complex addition We conclude from this that addition (and hence subtraction) of complex numbers is essentiall equivalent to addition (subtraction) of two-dimensional vectors. Because of this, complex numbers (when represented on an Argand diagram) are slidable as long as ou keep their length and direction the same, ou can position them anwhere on an Argand diagram. We see that the Cartesian form of a complex number: = a +ib is a particularl suitable form for addition (or subtraction) of complex numbers. However, when we come to consider multiplication and division of complex numbers, the Cartesian description is not the most convenient form that is available to us. A much more convenient form is the polar form which we now introduce. 3 HELM (VERSION 1: March 18, 2004): Workbook Level 1
4 2. The Polar Form of a Complex Number We have seen, above, that the complex number = a +ib can be represented b a line pointing out of the origin and ending at a point with cartesian coordinates (a, b). P r b θ a To locate the point P we introduce polar coordinates (r, θ) where r is the positive distance from 0 and θ is the angle measured from the positive x-axis, as shown in the diagram. From the properties of the right-angled triangle there is an obvious relation between (a, b) and (r, θ): a = r cos θ b = r sin θ or equivalentl, r = a 2 + b 2 tan θ = b a. This leads to an alternative wa of writing a complex number: = a +ib = r cos θ +irsin θ = r(cos θ + isin θ) The angle θ is called the argument of and written, for short, arg(). The non-negative real number r is the modulus of. Wenormall consider θ to lie in the interval π <θ π although an value θ +2kπ for integer k will be equivalent to θ. Itisnormal to express θ in radians not degrees. If = a +ib then in which Ke Point = r(cos θ +isin θ) r = = a 2 + b 2 and θ = arg() =tan 1 b a HELM (VERSION 1: March 18, 2004): Workbook Level 1 4
5 Example Find the polar coordinate form of (i) =3+4i (ii) = 3 i Solution (i) Here r = = = 25=5 θ = arg() =tan 1 ( 4 3 )=53.13 so that = 5(cos isin ) (ii) Here r = = ( 3) 2 +( 1) 2 = ( 1) θ = arg() =tan ( 3) = tan 1 ( 1) 3 Now, on our calculator (unless it is ver sophisticated) ou will obtain a value of about for tan 1 ( 1 ). This is incorrect since if we use the Argand diagram to plot = 3 iweget: 3 θ 3 1 The angle θ is clearl = This example warns us to take care when determining arg() purel using algebra. You will alwas find it helpful to construct the Argand diagram to locate the particular quadrant into which our complex number is pointing. Your calculator cannot do this for ou. Finall, in this example, =3.16(cos isin ). Find the polar coordinate form of the complex numbers (i) = i (ii) =3 4i Your solution (i) = 1(cos 270 +isin 270 ) 5 HELM (VERSION 1: March 18, 2004): Workbook Level 1
6 Your solution (ii) = 5(cos isin ) Remember, to get the correct angle, draw our complex number on an Argand diagram. Multiplication and division using polar coordinates The reader will perhaps be wondering wh we have bothered to introduce the polar form of a complex number. After all, the calculation of arg() isnot particularl straightforward. However, as we have said, the polar form of a complex number is a much more convenient vehicle to use for multiplication and division of complex numbers. To see wh, let us consider two complex numbers in polar form: = r(cos θ +isin θ) w = t(cos φ +isin φ) Then the product w is calculated in the usual wa w = [r (cos θ +isin θ)][t (cos φ +isin φ)] rt [cos θ cos φ sin θ sin φ +i(sin θ cos φ + cos θ sin φ)] rt [cos(θ + φ)+isin(θ + φ)] in which we have used the standard trigonometric identities cos(θ + φ) cos θ cos φ sin θ sin φ sin(θ + φ) sin θ cos φ + cos θ sin φ. We see that in calculating the product that the moduli r and t multipl together whilst the arguments arg() = θ and arg(w) = φ add together. If = r(cos θ +isin θ) and w = t(cos φ +isin φ) then find the polar expression for w Your solution (cos(θ φ)+isin(θ φ)) w = r t We see that in calculating the quotient that the moduli r and t divide whilst the arguments arg() = θ and arg(w) = φ subtract. HELM (VERSION 1: March 18, 2004): Workbook Level 1 6
7 Ke Point If = r(cos θ +isin θ) and w = t(cos φ +isin φ) then w = rt(cos(θ + φ)+isin(θ + φ)) w = r (cos(θ φ)+isin(θ φ)) t We conclude that addition and subtraction are most easil carried out in Cartesian form whereas multiplication and division are most easil carried out in polar form. Complex numbers and rotations We have seen, when multipling one complex number b another, that moduli multipl and arguments add. If, in particular, is a complex number with a unit modulus = cos θ +isin θ (i.e. r=1) and if w is an other complex number then w = t(cos φ +isin φ) w = t(cos(θ + φ)+isin(θ + φ)) so that the effect of multipling w b is to rotate the line representing the complex number w through an angle θ. This is illustrated in the following diagram. w φ multipl b w θ + φ This result would certainl be difficult to obtain had we continued to use the Cartesian form. Since, in terms of the polar form of a complex number 1 =1(cos 180 +isin 180 ) 7 HELM (VERSION 1: March 18, 2004): Workbook Level 1
8 we see that multipling a number b 1 produces a rotation through 180. In particular the product of ( 1) with itself i.e. ( 1)( 1) rotates the number 180 again, totalling 360 which is equivalent to leaving the number unchanged. Hence the introduction of complex numbers has explained the accepted (though odd) result ( 1)( 1) =+1 Exercises 1. Displa, on an Argand diagram, the complex numbers 1 i, 1 + 3i and 2i Find the polar form of (i) 1 i, (ii) 1 + 3i (iii) 2i 1. Hence calculate (1 + 3i) (2i 1) 3. On an Argand diagram draw the complex number 1+2i. B changing to polar form examine the effect of multipling 1 + 2i b, in turn, i, i 2,i 3,i 4. Represent these new complex numbers on an Argand diagram. 4. B utilising the Argand diagram convince ourself that + w + w for an two complex numbers,w. This is known as the triangle inequalit. Answers i 2i 1 1 i 2. (i) 2(cos 315 +isin 315 ) (ii) 10(cos isin ) (iii) 5(cos i sin (1 + 3i) ). Also (2i 1) = 2(cos( 45 )+isin( 45 )) = 2(cos(45 ) i sin(45 ))=(1 i). 3. Each time ou multipl through b i ou effect a rotation through 90 of the line representing the complex number 1 + 2i. After four such products ou are back to where ou started, at 1 + 2i. 4. This inequalit states that no one side of a triangle is greater in length than the sum of the lengths of the other two sides. See the second part of Figure?? for the geometrical interpretation. HELM (VERSION 1: March 18, 2004): Workbook Level 1 8
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