1.4 Some Soothing Exercises

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You will probably be feeling a bit gobsmacked still. This is quite normal,

and is cured by the following procedure: Do the next lot of exercises slowly

and carefully. Afterwards, you will see that everything I have said so far is

dead obvious and you will wonder why it took so long to say it. If, on the

other hand you decide to skip them in the hope that light will dawn at a

later stage, you risk getting more and more muddled about the subject. This

would be a pity, because it is really rather neat.

There is a good chance you will try to convince yourself that it will be enough

to put o_ doing these exercises until about a week before the exam. This

will mean that you will not know what is going on for the rest of the course,

but will spend the lectures copying down the notes with your brain out of

1.4. SOME SOOTHING EXERCISES 19

gear. You won't enjoy this, you really won't.

So sober up, get yourself a pile of scrap paper and a pen, put a chair somewhere

quiet and make sure the distractions are somewhere else. Some people

are too dumb to see where their best interests lie, but you are smarter than

that. Right?

Exercise 1.4.1 Translate the complex numbers (1+i0), (0 +i1), (3-i2) into

the other two forms. The _rst is often written 1, the second as i.

Exercise 1.4.2 Translate the complex numbers _ 2 0

0 2 _; _ 0 1

􀀀1 0 _; _ 0 􀀀1

1 0 _;

and _ 2 􀀀1

1 2 _ into the other two forms.

Exercise 1.4.3 Multiply the complex number

_ 0

1 _

by itself. Express in all three forms.

Exercise 1.4.4 Multiply the complex numbers

_ 2

3 _ and _ 2

􀀀3 _

Now do it for

_ a

b _ and _ a

􀀀b _

Translate this into the (a+ib) notation.

Exercise 1.4.5 It is usual to de_ne the norm of a point as its distance from

the origin. The convention is to write

k _ a

b _k = pa2 + b2

20 CHAPTER 1. FUNDAMENTALS

In the classical notation, we call it the modulus and write

ja + ibj = pa2 + b2

There is not the slightest reason to have two di_erent names except that this

is what we have always done.

Find a description of the complex numbers of modulus 1 in the point and

matrix forms. Draw a picture in the _rst case.

Exercise 1.4.6 You can also represent points in the plane by using polar

coordinates. Work out the rules for multiplying (r; _) by (s; _). This is a

fourth representation, and in some ways the best. How many more, you may

ask.

Exercise 1.4.7 Show that if you have two complex numbers of modulus 1,

their product is of modulus 1. (Hint: This is very obvious in one representation

and an amazing coincidence in another. Choose a representation for

which it is obvious.)

Exercise 1.4.8 What can you say about the polar representation of a complex

number of modulus 1?

Exercise 1.4.9 What can you say about the e_ect of multiplying by a complex

number of modulus 1?

Exercise 1.4.10 Take a piece of graph paper, put axes in the centre and

mark on some units along the axes so you go from about _ 􀀀5

􀀀5 _ in the

bottom left corner to about _ 5

5 _ in the top right corner. We are going to

see what happens to the complex plane when we multiply everything in it by

a _xed complex number.

I shall choose the complex number 1

p2

+ i 1

p2

for reasons you will see later.

Choose a point in the plane, _ a

b _ (make the numbers easy) and mark it

with a red blob. Now calculate (a + ib) _ (1=p2 + i=p2) and plot the result

1.4. SOME SOOTHING EXERCISES 21

in green. Draw an arrow from the red point to the green one so you can see

what goes where,

Now repeat for half a dozen points (a+ib). Can you explain what the map

from C to C does?

Repeat using the complex number 2+0i (2 for short) as the multiplier.

Exercise 1.4.11 By analogy with the real numbers, we can write the above

map as

w = (1=p2 + i=p2)z

which is similar to

y = (1=p2) x

but is now a function from C to C instead of from R to R.

Note that in functions from R to R we can draw the graph of the function

and get a picture of it. For functions from C to C we cannot draw a graph!

We have to have other ways of visualising complex functions, which is where

the subject gets interesting. Most of this course is about such functions.

Work out what the simple (!) function w = z2 does to a few points. This is

about the simplest non-linear function you could have, and visualising what it

does in the complex plane is very important. The fact that the real function

y = x2 has graph a parabola will turn out to be absolutely no help at all.

Sort this one out, and you will be in good shape for the more complicated

cases to follow.

Warning: This will take you a while to _nish. It's harder than it looks.

Exercise 1.4.12 The rotation matrices

_ cos _ 􀀀sin _

sin _ cos _ _

are the complex numbers of modulus one. If we think about the point representation

of them, we get the points _ cos _

sin _ _ or cos _ + i sin _ in classical

notation.

22 CHAPTER 1. FUNDAMENTALS

The fact that such a matrix rotates the plane by the angle _ means that

multiplying by a complex number of the form cos _ + i sin _ just rotates the

plane by an angle _. This has a strong bearing on an earlier question.

If you multiply the complex number cos _+i sin _ by itself, you just get cos 2_+

i sin 2_. Check this carefully.

What does this tell you about taking square roots of these complex numbers?

Exercise 1.4.13 Write out the complex number p3=2+i in polar form, and

check to see what happens when you multiply a few complex numbers by it. It

will be easier if you put everything in polar form, and do the multiplications

also in polars.

Remember, I am giving you these di_erent forms in order to make your life

easier, not to complicate it. Get used to hopping between di_erent representations

and all will be well.

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