Trigonometric form of complex numbers
Définition : Conjugate of a complex number
We call complex conjugated of the complex number
the complex number
Fondamental : Property of the conjugation :
Définition : Modulus of a complex number
be a complex number,
is a real positive number.
We call modulus of complex number
the positive real number
Expression of the modulus:
If
Fondamental : Properties of the modulus
which is also true for
with the convention
If
more generally if
:
and
Fondamental : Argument of a complex nomber
• Case of a complex number of modulus 1
Theorem
For any complex number of modulus
there exists a unique element
such that
Definition
The real number
is called the main argument of the complex number
For any relative integer
we have
The real numbers
belonging to
are called argments of the complex number
• Case of a complex number of arbitrary modulus
Let
be a non zero complex number.
The complex number
has a modulus of
The argument of the complex number
is, by definition, the one of the complex number
Denoting by
the positive real number
and by
an argument of
we have:
We denote by
an arbitrary argument of the complex number
Fondamental : Properties of the argument
By also assuming that
More generally, by assuming
non zero:
Définition : Trigonometric notation
We denote by
the complex number of modulus
and of argument
With this notation, the previous formula becomes:
Fondamental : Moivres's formula
