Argand diagrams and the polar form
Introduction





10.2





In this Block we introduce a geometrical interpretation of a complex number. Since a complex number z = x + iy 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 Block we introduce an alternative form, called the polar form, for representing complex numbers. We shall see that the polar form is particularly advantageous when multiplying and dividing complex numbers.

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6

x know what a complex number is

Prerequisites
Before starting this Block you should . . .
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y understand how to use trigonometric functions cos θ, sin θ and tan θ z understand what a polynomial function is { possess a knowledge of vectors
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Learning Outcomes

Learning Style

After completing this Block you should be able To achieve what is expected of you . . . to . . .  represent complex numbers on an Argand diagram  obtain the polar form of a complex number  multiply and divide complex numbers in polar form  allocate sufficient study time

 briefly revise the prerequisite material  attempt every guided exercise and most of the other exercises

1. The Argand diagram
In Block 10.1 we met a complex number z = x + iy in which x, y 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 block we examine a useful geometrical description of complex numbers. Since a complex number is specified by two real numbers x, y it is natural to represent a complex number by a vector in a plane. We take the usual xy plane in which the ‘horizontal’ axis is the x-axis and the ‘vertical’ axis is the y-axis (2, 3) y z (−1, 1) w −1 0 1 2 x 1

Thus the complex number z = 2 + 3i would be represented by a line starting from the origin and ending at the point with coordinates (2, 3) and w = −1 + i is represented by the line starting from the origin and ending at the point with coordinates (−1, 1). When the x-y plane is used in this way it is called an Argand diagram. With this interpretation the modulus of z, that is |z| is simply the length of the line which represents z. Now do this exercise Given that z = 1 + i, w = i represent the complex numbers z, w and 2z − 3w − 1 on an Argand diagram. Answer If we have two complex numbers z = a + ib, w = c + id then, as we already know z + w = (a + c) + i(b + d) that is, the real parts add together and the imaginary parts add together. But this is precisely 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 inthe x- and y-directions respectively. So, using vector addition: p + q = (a + c)i + (b + d)j We conclude from this that addition (and hence subtraction) of complex numbers is essentially equivalent to addition (subtraction) of two-dimensional vectors. Because of this, complex numbers (when represented on an Argand diagram) are slidable — as long as you keep their length and direction the same, you can position them anywhere on an Argand diagram. We see that the Cartesian form of a complex number: z = a + ib is a particularly 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.
Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

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y q p+q p 0 x vector addition Figure 1:

y w z+w z 0 x complex addition

2. The polar form of a complex number
We have seen, above, that the complex number z = a + ib can be represented by a line pointing out of the origin and ending at a point with coordinates (a, b). y P

r z θ 0 a x b

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 or equivalently, r= √ b = r sin θ

a2 + b2

b tan θ = . a

This leads to an alternative way of writing a complex number: z = a + ib = r cos θ + ir sin θ = r(cos θ + i sin θ) The angle θ is called the argument of z and written, for short, arg(z). The non-negative real number r is the modulus of z. 3
Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

Key Point If z = a + ib then z = r(cos θ + i sin θ) in which r = |z| = √ a2 + b2 and θ = arg(z) = tan−1 b a

Example Find the polar coordinate form of (i) z = 3 + 4i (ii) z = −3 − i Solution (i) Here r = |z| = √ 32 + 42 = √ 25 = 5 4 θ = arg(z) = tan−1 ( ) = 53.13◦ 3

so that z = 5(cos 53.13◦ + i sin 53.13◦) (ii) Here r = |z| = (−3)2 + (−1)2 = √ 10 ≈ 3.16 θ = arg(z) = tan−1 (−1) = tan−1 ( 1 ) 3 (−3)

Now, on your calculator (unless it is very sophisticated) you will obtain a value of about 18.43◦ for tan−1 ( 1 ). This is incorrect since if we use the Argand 3 diagram to plot z = −3 − i we get: y

θ −3 0 −1 The angle θ is clearly 180◦ + 18.43◦ = 198.43◦ . x

This example warns us to take care when determining arg(z) purely using algebra. You will always find it helpful to construct the Argand diagram to locate the particular quadrant into which your complex number is pointing. Your calculator cannot do this for you. Finally, in this example, z = 3.16(cos 198.43◦ + i sin 198.43◦). Try each part of this exercise Find the polar coordinate form of the following complex numbers.
Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

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Part (i) z = −i (i) Part (ii) z = 3 − 4i (ii) Answer Answer

Remember, to get the correct angle, draw your complex number on an Argand diagram.

Multiplication and division using polar coordinates
The reader will perhaps be wondering why we have bothered to introduce the polar form of a complex number. After all, the calculation of arg(z) is not particularly 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 why, let us consider two complex numbers in polar form: z = r(cos θ + i sin θ) w = t(cos φ + i sin φ)

Then the product zw is calculated in the usual way zw = {r(cos θ + i sin θ)}{t(cos φ + i sin φ)} = rt[cos θ cos φ − sin θ sin φ + i(sin θ cos φ + cos θ sin φ)] = rt(cos(θ + φ) + i sin(θ + φ) in which we have used the standard trigonometric relations 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 multiply together whilst the arguments arg(z) = θ and arg(w) = φ add together. Now do this exercise If z = r(cos θ + i sin θ) and w = t(cos φ + i sin φ) then find the polar expression for z w Answer

We see that in calculating the quotient that the moduli r and t divide whilst the arguments arg(z) = θ and arg(w) = φ subtract. Key Point If z = r(cos θ + i sin θ) and w = t(cos φ + i sin φ) then zw = rt(cos(θ + φ) + i sin(θ + φ)) z r = (cos(θ − φ) + i sin(θ − φ)) w t

We conclude that addition and subtraction are most easily carried out in Cartesian form whereas multiplication and division are most easily carried out in polar form. 5
Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

Complex numbers and rotations
We have seen, when multiplying one complex number by another, that moduli multiply and arguments add. If, in particular, z is a complex number with a unit modulus z = cos θ + i sin θ and if w is any other complex number w = t(cos φ + i sin φ) then wz = t(cos(θ + φ) + i sin(θ + φ)) so that the effect of multiplying w by z is to rotate the line representing the complex number w through an angle θ. This is illustrated in the following diagram. (i.e. r = 1)

y

y wz w multiply by z θ+φ x 0 x

φ 0

This result would certainly 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◦ + i sin 180◦ ) we see that multiplying a number by −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 More exercises for you to try 1. Display, on an Argand diagram, the complex numbers 1 − i, 1 + 3i and 2i − 1. (1 + 3i) 2. Find the polar form of (i) 1 − i, (ii) 1 + 3i (iii) 2i − 1. Hence calculate (2i − 1) 3. On an Argand diagram draw the complex number 1 + 2i. By changing to polar form examine the effect of multiplying 1 + 2i by, in turn, i, i2 , i3 , i4 . Represent these new complex numbers on an Argand diagram. 4. By utilising the Argand diagram convince yourself that |z + w| ≤ |z| + |w| for any two complex numbers z, w. This is known as the triangle inequality. Answer
Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

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3. Computer Exercise or Activity
For this exercise it will be necessary for you to access the computer package DERIVE. In DERIVE the basic complex object i is denoted by ˆ You i. can use this in any expression by keying ctrl +i or by clicking on the ˆ icon in the Expression dialog box. The conjugate of i a complex number z is written conj(z) in DERIVE and the modulus of z is written abs(z). DERIVE will help you verify your complex number solutions to the Block exercises. DERIVE does not have a facility for plotting complex numbers on an Argand diagram. Neither is it capable of expressing complex numbers in polar form. However, it will sometimes present a complex number in exponential form which is a method of writing complex numbers which we will not meet until the next block. MAPLE can handle complex numbers also. Its standard notation for i is I. Its notation for the conjugate z ∗ is conjugate(z). Conveniently, it can also convert complex numbers in Cartesian form to polar form. To do this you will first need to load the appropriate library readlib(polar). For example to convert 1 + i into polars we would include the following program segment: > readlib(polar) : > z := 1 + I : > convert(z, polar); MAPLE responds with polar √ π 2, 4

Here the first term in the list represents the modulus of z and the second term represents the argument of z expressed in radians. As a useful exercise in the use of MAPLE you should verify all of the Cartesian to polar form conversions in this block and check your solutions to the exercises. End of Block 10.2

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Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

You should obtain the following diagram. y 1 w 0 z 1 2z − 3w − 1 x

Argand diagram

−1

Back to the theory

Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

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z = 1(cos 270◦ + i sin 270◦) Back to the theory

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Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

z = 5(cos 306.86◦ + i sin 306.86◦) Back to the theory

Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

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r z = (cos(θ − φ) + i sin(θ − φ)) w t

Back to the theory

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Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

1. y 2i − 1 1 + 3i

0 1−i

x

√ 2(cos 315◦ + i sin 315◦) (ii) 10(cos 71.57◦ + i sin 71.57◦ ) (iii) √ 5(cos 116.57◦ + i sin 116.57◦). Also √ (1 + 3i) √ = 2(cos(−45◦ ) + i sin(−45◦ )) = 2(cos(45◦ ) − i sin(45◦ )) = (1 − i). 3. Each time (2i − 1) you multiply through by i you effect a rotation through 90◦ of the line representing the complex number 1 + 2i. After four such products you are back to where you started, at 1 + 2i. 4. This inequality 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 1 for the geometrical interpretation. 2. (i) Back to the theory

√

Engineering Mathematics: Open Learning Unit Level 1 10.2: Complex numbers

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