5MM001 - Calculus and Linear Algebra Lecture 3

Complex Numbers 3: Euler’s Form

The aim of this lecture is to continue our investigation of complex numbers, in particular using the modulus-argument form.

Revision

Recall the compound angle identities

We recall how we use sine and cosine when angles are greater than . To find the sine and cosine of angles greater then we use the corresponding acute angle made with the x axis.

Example 1

Find and

The corresponding acute angle is . The sine ratio for in the second quadrant is positive so that

The cosine ratio for in the second quadrant is negative so that

The diagram below shows where the sine and cosine are negative and positive.

Euler’s Form of a Complex Number

We can now define z to obtain Euler’s form of a complex number

This formula enables you to express complex numbers in different forms.

Example 2

We find the Euler form of the complex number . On the Argand diagram we have:

Thus,

Example 3

If and express z in Cartesian form.

Using Euler’s form

Products of Complex Numbers using Euler’s Form

Euler’s form of a complex number is useful when multiplying complex numbers.

Let and so that

This gives the results

and

To summarise, to multiply two complex numbers given in polar or Euler’s form, multiply the moduli and add the arguments.

Example 4

If and find in modulus and argument, Euler and Cartesian form.

So that modulus-argument form is

, .

Euler’s form

and Cartesian form

Division of Complex Numbers using Euler’s Form

A similar result to that of multiplication can be found for division.

Let and then

To summarise, to divide two complex numbers given in polar or Euler’s form, divide the moduli and subtract the arguments.

Note: care must be taken so that the resulting argument is given within the principal value set.

Example 5

If and find in modulus and argument form.

so that modulus and argument form is

, .

The argument is outside the principal value set and therefore we need to find equivalent principal value form. This can be found by drawing a suitable diagram.

The equivalent argument is therefore -5 π/6.

Example 6

Find the square root of .

Let . Finding the modulus and argument

Using Euler’s form we know that .

From

Therefore

Equating the modulus and argument

Solving

It follows that

Tutorial Questions

The tutorial questions consist of three sections: Warm-up questions, section A and section B.

· The warm-up questions are designed to allow you attempt some simple questions on the lecture material and test your own learning. They also help prepare you for the section A and B questions.

· Section A questions show the level of learning required to answer the section A questions on the test and exam.

· Section B questions show the level of learning required to answer the section B questions on the test and exam.

Note that not every set of tutorial questions will contain section A and B questions.

Warm-up Questions

1) By first expressing in Euler's form find the Cartesian form of the complex numbers with the following polar coordinates

a b c

d e f .

2) Give the form and the modulus and argument of the following

a b

c d .

Section A Questions

3) Given that and find the following in Euler's form

a b c d

e f g h .

4) Write down the moduli and arguments of and and hence express in Euler's form. Find and .

5) If show that and hence find the square root of .

6) Simplify .

7) Show that the modulus of is .

Section B Questions

8) Find the real and imaginary parts of .

Express and in Euler's form.

Hence show that and find an exact expression for .

9) The complex numbers α and β are given by and .

Show that α=2+2i.

Show that |α|=|β|. Find arg α and arg β.

Find the modulus and argument of αβ. Illustrate the complex numbers α, β and αβ on a single Argand diagram.

Describe the set of points in the Argand diagram representing the complex numbers z for which |z- α|=|z- β|. Draw this set of points on your previous Argand diagram.

Show that z= α+β satisfies |z- α|=|z- β|. Mark the point representing α+β on your diagram, and find the exact value of arg(α+β).

Solutions

1) a b c

d e f .

2) a

d .

3) a b

c d

e f 432

g h .

4) , , ,

5) , Note: 2 roots

6) .

Sample worked solution

(The last step is essential to ensure we have a principal argument in the correct range.)

The equation is satisfied when the complex number z is equidistant from α and β. So it is the perpendicular bisector of the line from α to β

We know that , so z= α+β satisfies |z- α|=|z- β|.