# Complex numbers/Real part, conjugation, modulus/Introduction/Section

## Definition

The set with and , with componentwise addition and the multiplication defined by

is called the *field of complex numbers*. We denote it by

So the addition is just the vector addition in , but the multiplication is a new operation. We will see in fact a more geometric interpretation for complex multiplication.

From now on we write

and in particular we put
,
this number is called the *imaginary unit*. It has the important property

From this property and the rules of a field one can deduce all algebraic properties of the complex numbers. This also helps in memorizing the multiplication rule as we have

We consider a real number as the complex number . Hence . It does not make a difference whether we add or multiply real numbers as real numbers or as complex numbers.

## Definition

For a complex number

we call

the *real part* of and

*imaginary part*of .

However, one should not think that complex numbers are less real than real numbers. The construction of the complex numbers starting from the reals is by far easier that the construction of the real numbers starting from the rational numbers. On the other hand, it was a long historic process until complex numbers were accepted as numbers; they form a field, but not an ordered field, and so at first glance they are numbers which do not measure anything.

One should think of complex numbers as points of the plane; for the additive structure we have directly
.
The horizontal axis is called the *real axis* and the vertical axis is called the *imaginary axis*.

## Lemma

The real part and the imaginary part of complex numbers fulfill the following properties (for

and in ).- .
- .
- .
- For
we have
- holds if and only if holds, and this holds if and only if holds.

### Proof

## Definition

The mapping

*complex conjugation*.

For the number is called the *complex-conjugated number* of . Geometrically the complex conjugation to
is simple the reflection at the real axis.

## Lemma

For the complex conjugation the following rules hold (for arbitrary

).- .
- .
- .
- For we have .
- .
- We have if and only if holds.

### Proof

The square of a real number is always nonnegative, and the sum of two nonnegative resl numbers is again nonnegative. For a nonnegative real number there exists a unique nonnegative *square root* , see
exercise
Hence the following definition yields a welldefined real number.

## Definition

For a complex number

the *modulus* is defined by

The modulus of a complex number is due to the *Pythagorean theorem* the distance of to the zero point
.
The modulus is a mapping

The set of all complex numbers with a certain modulus form a circle with the zero point as center and with the modulus as radius. In particular, all complex numbers with modulus form the *complex unit circle*. The numbers of the complex unit circle are by the *formula of Euler* in relation to the complex exponential funxtion and to the trigonometric functions, see
fact
and
fact.
We further mention that the product of two complex numbers of the unit circle may be computed by adding the angles starting from the positive real axis counter clockwise.

## Lemma

The modulus of complex numbers

fulfils the following properties.- .
- For real the real and the complex modulus are the same.
- We have if and only if .
- .
- .
- For we have .
- .
- .

### Proof

We only show the triangle inequality, for the other statements see exercise. Because of (6) we have for every complex number the estimate . Therefore

and hence

By taking the square root we get the stated estimate.