Complex NumbersPolar Form

MTH 362: Advanced Engineering MathematicsLecture 1

Jonathan A. Chavez Casillas1

1University of Rhode IslandDepartment of Mathematics

September 7, 2017

Jonathan Chavez

Complex NumbersPolar Form

Course Name and number:

MTH 362: Advanced Engineering Mathematics

Instructor: Jonathan︸ ︷︷ ︸First Name

Allan︸ ︷︷ ︸Middle Name

Chavez Casillas︸ ︷︷ ︸Last Name

Office Hours: Wednesday 11 am - 2pm (or by appointment)

Office Room: Lippitt Hall 200A

E-mail: jchavezc@uri.edu

Jonathan Chavez

Complex NumbersPolar Form

Jonathan Chavez

Complex NumbersPolar Form

Jonathan Chavez

Complex NumbersPolar Form

Jonathan Chavez

Complex NumbersPolar Form

Jonathan Chavez

Complex NumbersPolar Form

1 Complex Numbers and the Complex Plane

2 Polar Form of Complex Numbers. Powers and Roots

Jonathan Chavez

Complex NumbersPolar Form

The Complex NumbersSince high school we have been taught to solve equations such as:

Exercise: Solve:x2 + 2x = −1

x3 + 6x2 − 40x = 0

The solution to such equations belong to the real numbers.

Jonathan Chavez

Complex NumbersPolar Form

The Complex Numbers

However, if we modify slightly the equations, the solutions are not real anymore: Exercise: Try to solve:x2 = −1

x3 − 10x2 − 40x = 0

The solutions of the previous equations belong to a set of numbers called the Complex Numbers

Jonathan Chavez

Complex NumbersPolar Form

The Complex Numbers

By definition, a complex number has two parts: A real part and an imaginary part. Because of this duo,there are two basic ways of writing complex numbers:

Complex numbers notation:A complex number z consists of an ORDERED pair of real numbers x and y . That is, z = (x , y) isthought as a complex number and x is called the real part and y is called the imaginary part. In notation,x = Re z and y = Im z. The imaginary unit, (0, 1), is denoted by i . Thus, a more common notation is:

z=(x,y)=x+iy

It is important to notice that two complex numbers z1 = (x1, y1) = x1 + iy1 and z2 = (x2, y2) = x2 + iy2 areequal ONLY if x1 = x2 AND y1 = y2.

Jonathan Chavez

Complex NumbersPolar Form

Arithmetic of Complex Numbers

Addition: The addition of two complex numbers z1 = (x1, y1) = x1 + iy1 and z2 = (x2, y2) = x2 + iy2 isdefined as:

z1 + z2 := (x1 + x2, y1 + y2) = x1 + x2 + i(y1 + y2)

Multiplication: The product of two complex numbers z1 = (x1, y1) = x1 + iy1 and z2 = (x2, y2) = x2 + iy2is defined as:

z1z2 := (x1x2 − y1y2, x1y2 + x2y1) = x1x2 − y1y2 + i(x1y2 + x2y1)

Particular Example: A very important example and a shortcut to multiply complex numbers is that ifz1 = (0, 1) = i and z2 = (0, 1) = i , then z1z2 = i · i = i2 = (−1, 0) = −1. That is

i2 = −1.

Jonathan Chavez

Complex NumbersPolar Form

Arithmetic of Complex Numbers

Subtraction: The subtraction of two complex numbers z1 = (x1, y1) = x1 + iy1 and z2 = (x2, y2) = x2 + iy2is defined as:

z1 − z2 := (x1 − x2, y1 − y2) = x1 − x2 + i(y1 − y2)

Quotient: The quotient of two complex numbers z1 = (x1, y1) = x1 + iy1 and z2 = (x2, y2) = x2 + iy2 isdefined as:

z1z2

:=(

x1x2 + y1y2

x22 + y2

2,

x2y1 − x1y2

x22 + y2

2

)= x1x2 + y1y2

x22 + y2

2+ i x2y1 − x1y2

x22 + y2

2

An easy way to remember the quotient is to multiply and divide by the conjugate (see below) of the divisor.

Jonathan Chavez

Complex NumbersPolar Form

The Complex PlaneAs the notation z = (x , y) suggests, we can identify the complex number z with the pair (x , y) and thus,we can plot it in the plane as we would for a point in the two dimensional Cartesian plane. This planecomposed of two coordinates axis (the horizontal, or x−axis, represents the real part and the vertical ory−axis represents the imaginary part) is called the complex plane.

Exercise: For the complex numbers z1 = (3, 1) = 3 + i and z2 = (2, 4) = 2 + 4i , plot z1, z2, z1 + z2, z1 − z2.How does the addition and substraction relates to the one of vectors in R2?

Jonathan Chavez

Complex NumbersPolar Form

The Complex ConjugateThe complex conjugate of a complex number z = (x , y) = x + iy is defined as z = (x ,−y) = x − iy . It isobtained geometrically by just reflecting about the real (x-)axis.

Exercise: Show that for any two complex numbers z1 = (x1, y1) and z2 = (x2, y2):

• z1z1 = x21 + y2

1 . • Re z1 = x1 = 12 (z1 + z1).

• Im z1 = y1 = 12 (z − z1). • (z1 + z2) = z1 + z2 and z1z2 = z1z2.

Jonathan Chavez

Complex NumbersPolar Form

Exercise: Let z1 = 4 + 3i and z2 = 1− i . Find:

• (2z1 + 3z2)2. • Re(1/z1).

• Im (1 + i)z2. • z2/(z1 − 3).

Jonathan Chavez

Complex NumbersPolar Form

Table of Contents

1 Complex Numbers and the Complex Plane

2 Polar Form of Complex Numbers. Powers and Roots

Jonathan Chavez

Complex NumbersPolar Form

Polar Form of Complex Numbers

As in two dimensions, when plotting a complex number, we may look at its polar form. That is, the angleit makes with the real axis and the distance from the origin the point has. Remember that for a point(x , y) in the cartesian plane, its polar coordinates are:

x = r cos θ y = r sin θ

Thus, it is natural to assume that the polar representation of the complex number z = x + iy is given by

z = r(cos θ + i sin θ).

r = |z| =√

x2 + y2 is called the modulus of the complex number and θ = arg(z) = arctan yx is the

argument of the complex number z. As in calculus, all angles are measured in radians and a positive angletraverse in counterclockwise sense.

Jonathan Chavez

Complex NumbersPolar Form

Polar Form of Complex NumbersRemember that you may write the same angle in a lot of different ways. For example, π/3 is the same angleas π/3 + 6π, but all angles are unique up to a multiple of 2π, which is the periodicity of the trigonometricfunctions sine and cosine. Thus, a Principal Argument, denoted as Argz, is defined for complex numbers.

The principal argument satisfy the inequality

−π < Argz ≤ π

Please notice that the equal sign is on the right side of the inequality!

Then, the way to think of the Principal argument is as follows:

Jonathan Chavez

Complex NumbersPolar Form

Polar Form of Complex Numbers

Exercise: Obtain the polar form of z1 = −2− 2i and z2 = 1−√

3i . Find their modulus and all possiblearguments. Find also their principal argument.

Jonathan Chavez

Complex NumbersPolar Form

Multiplication and Division in Polar FormSo far, we understand the geometric meaning of adding and subtracting two complex numbers. But, whatdoes it mean geometrically to multiply and divide two complex numbers?

Home Exercise: Show that if z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), then

z1z2 = r1r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)] z1z2

= r1r2

[cos(θ1 − θ2) + i sin(θ1 − θ2)]

Exercise: Using the formulas above show that

|z1z2| = |z1||z2| and∣∣ z1

z2

∣∣ = |z1||z2|

arg(z1z2) = arg(z1) + arg(z2) and arg( z1

z2

)= arg(z1)− arg(z2) (up to multiples of 2π)

Jonathan Chavez

Complex NumbersPolar Form

Integer Powers of z and De Moivre’s Formula

Using the product formula with z1 = z2 = z = r(cos θ + i sin θ) we obtain that z2 = r2[cos(2θ) + i sin(2θ)].We can continue by induction to show that for n a positive integer,

zn = rn(cos(nθ) + i sin(nθ)) (1)Home Exercise: Show that

1 Equation (1) holds also for negative integers n.

2 (De Moivre’s Formula) (cos θ + i sin θ)n = cos(nθ) + i sin(nθ).

3 Using De Moivre’s formula with n=2, expand the binomial in the left and equate the real animaginary parts from both sides of the formula. Do you recognize those formulas?

Hint: For (a) use Equation (1) and the quotient formula with one complex number being 1. For (b) useagain Equation (1) for a complex number with modulus 1.

Jonathan Chavez

Complex NumbersPolar Form

Roots of Complex numbers

If P(z) is a polynomial of degree n (i.e., P(z) = anzn + an−1zn−1 + . . .+ a1z + a0), then the equationP(z) = 0 has ALWAYS n complex solutions (Is this true if we restrict z to be just real?)

Then, it should follow that the equation zn = w , where z and w are complex numbers, should have also nroots. Each of those roots is represented as

z = n√w

Thus, the expression above has not just value but n.

The natural question is to find out which are those n−values.

Jonathan Chavez

Complex NumbersPolar Form

Roots of Complex numbersExercise: Use the polar form of a complex number and De Moivre formula to find the n roots of theequation zn = w . In particular, find the n roots of the unity, i.e., what are the roots of the equation zn = 1.

Jonathan Chavez

Complex NumbersPolar Form

Roots of Complex numbersExercise: As seen above,

n√1 = cos(2kπ

n

)+ i sin

(2kπn

), k = 0, 1, 2, . . . , n − 1.

These n values are called the nth roots of unity. They lie on the circle of radius 1 and center 0, called theunit circle, and constitute the vertices of a regular polygon of n sides (inscribed in the circle of radius 1).The next figure depicts the roots of unity when n = 3, 4, 5.

Jonathan Chavez