1.5 Some Classical Jargon

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We write 1 + i0 as 1, a+i0 as a, 0+ib as ib. In particular, the origin 0 + i0

is written 0.

You will often _nd 4 + 3i written when strictly speaking it should be 4 + i3.

This is one of the di_erences that don't make a di_erence.

We use the following notation: <(x + iy) = x which is read: 'The real part

of the complex number x+iy is x.'

And

=(x + iy) = y which is read : 'The imaginary part of the complex number

x+iy is y.' The = sign is a letter I in a font derived from German Blackletter.

Some books use 'Re(x+iy)' in place of <(x + iy) and 'Im(x+iy)' in place of

=(x + iy).

We also write

x + iy = x􀀀iy, and call _z the complex conjugate of z for any complex number

z.

1.5. SOME CLASSICAL JARGON 23

Notice that the complex conjugate of a complex number in matrix form is

just the transpose of the matrix; reect about the principal diagonal.

The following 'fact' will make some computations shorter:

jzj2 = z_z

Verify it by writing out z as x + iy and doing the multiplication.

Exercise 1.5.1 Draw the triangle obtained by taking a line from the origin

to the complex number x+iy, drawing a line from the origin along the X axis

of length x, and a vertical line from (x,0) up to x+iy. Mark on this triangle

the values jx + iyj, <(x + iy) and =(x + iy).

Exercise 1.5.2 Mark on the plane a point z = x + iy. Also mark on 􀀀z

and _z.

Exercise 1.5.3 Verify that __z = z for any z.

The exercises will have shown you that it is easy to write out a complex

number in Polar form. We can write

z = x + iy = r(cos _ + i sin _)

where _ = arccos x = arcsin y, and r = jzj.

We write:

arg(z) = _ in this case. There is the usual problem about adding multiples

of 2_, we take the principal value of _ as you would expect. arg(0 + 0i) is

not de_ned.

Exercise 1.5.4 Calculate arg(1 + i)

I apologise for this jargon; it does help to make the calculations shorter after

a bit of practice, and given that there have been four centuries of history to

accumulate the stu_, it could be a lot worse.

24 CHAPTER 1. FUNDAMENTALS

In general, I am more concerned with getting the ideas across than the jargon,

which often obscures the ideas for beginners. Jargon is usually used to keep

people from understanding what you are doing, which is childish, but the

method only works on those who haven't seen it before. Once you _gure out

what it actually means, it is pretty simple stu_.

Exercise 1.5.5 Show that

1

z

=

_z

z_z

Do it the long way by expanding z as x + iy and the short way by cross

multiplying. Is cross multiplying a respectable thing to do? Explain your

position.

Note that z_z is always real (the i component is zero). Use this for calculating

1

4 + 3i

and

1

5 + 12i

Express your answers in the classical form a+ib.

Exercise 1.5.6 Find 1

z when z = r(cos _ + i sin _) and express the answers

in polar form.

Exercise 1.5.7 Find 1

z when

z = _ a 􀀀b

b a _

Express your answer in classical, point, polar and matrix forms.

Exercise 1.5.8 Calculate

2 􀀀 i3

5 + i12

Express your answer in classical, point, polar and matrix forms.

It should be clear from doing the exercises, that you can _nd a multiplicative

inverse for any complex number except 0. Hence you can divide z by w for

any complex numbers z and w except when w = 0.

This is most easily seen in the matrix form:

1.5. SOME CLASSICAL JARGON 25

Exercise 1.5.9 Calculate the inverse matrix to

z = _ a 􀀀b

b a _

and show it exists except when both a and b are zero

The classical jargon leads to some short and neat arguments which can all

be worked out by longer calculations. Here is an example:

Proposition 1.5.1 (The Triangle Inequality) For any two complex numbers

z, w:

jz + wj _ jzj+jwj

Proof:

jz + wj

2 = (z + w)(z + w)

= (z + w)(_z + _ w)

= z_z + z _ w + w_z + w _ w

= jzj

2 + z _ w + w_z + jwj

2

= jzj

2 + z _ w + z _ w + jwj

2

= jzj

2 + 2<(z _w) + jwj

2

_ jzj

2 + 2j<(z _ w)j + jwj

2

_ jzj

2 + 2jz _wj+jwj

2

_ (jzj+j _wj)2

Hence

jz + wj _ jzj+jwj

since jwj = j _ wj. 2

Check through the argument carefully to justify each stage.

Exercise 1.5.10 Prove that for any two complex numbers z;w; jzwj = jzjjwj.

26 CHAPTER 1. FUNDAMENTALS

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