Complex numbers are written in the shape: a+ bi. Here a
and b – real numbers, and i
– an imaginary unit, i.e. i
2
= –1. A real number
a is called an abscissa of complex number
a+ bi, and b – an ordinate of complex number a+ bi. Two
complex numbers
a+ bi and a – bi are called the conjugate complex
numbers.
Main agreements:
1. A real number
a can also be written in the shape of a complex number:
a+ 0
i or a – 0
i. For example, the
records 5 + 0 i and 5 – 0 i
mean the same real number 5 .
2. A complex number 0+
bi is called a pure imaginary number.
The record
bi means
the same as 0+ bi.
3. Two complex numbers
a+ bi and c+ di
are considered as equal ones, if
a=c and
b=d.
Otherwise, the complex numbers aren’t equal.
Addition.
A sum of complex numbers
a+ bi and c+ di is called a complex number ( a+ c ) + (
b+ d ) i. So, at addition of complex numbers their
abscissas and ordinates are added
separately. This definition corresponds to the
rules of operations at usual polynomials.
Subtraction. A difference of two complex numbers
a+ bi
( a minuend ) and c+ di ( a subtrahend ) is called a complex
number ( a
– c ) + (
b – d )
i. So, at subtraction of two complex
numbers their abscissas and ordinates are
subtracted separately.
Multiplication. A product of complex numbers a+ bi and c+ di is called a complex number: (
ac – bd ) + (
ad + bc ) i.
This definition follows from two requirements:
1) the numbers
a+ bi and c+ di must be multiplied as algebraic binomials,
2) a number
i has a main property: i ²
= –1.
E x a m p l e . (
a+ bi )(
a – bi )= a ² + b ².
Hence it follows, that
a
product
of two conjugate complex numbers
is a real positive number!
Division. To divide a complex number
a+ bi ( a dividend ) by another c+ di ( a divisor ) means to find the third number e+ f i ( a quotient ), which being
multiplied by the divisor
c+ di, results the dividend a+ bi. If divisor isn’t equal to zero, then division is always
valid.
E x
a m p l e . Find ( 8 + i ) : ( 2 – 3i ) .
S o l u t
i o n . Rewrite this quotient as a fraction: 
Multiplying its numerator and
denominator by 2 + 3i and
executing
all operations, we’ll receive:
Geometric representation
of complex numbers. Real numbers are represented by
points in a numerical line:
Here a point A means a number
–3, a point B – a number 2, and O – zero. In contrast this complex numbers
are represented by points in a “numerical” ( coordinate ) plane. For this
we select a system of rectangular ( Cartesian ) coordinates with the same scale
in both axes. Then, a complex number a+ bi
will be represented by point P
with abscissa a
and ordinate b ( see figure ). This
coordinate system is called a complex plane.
___
Modulus
of a complex number is a length of
vector OP, representing this complex number in a coordinate
(complex) plane. Modulus of complex number a+ bi is signed as | a+ bi
| or by letter r and equal to :
Conjugate complex numbers have
the same modulus.
___
Argument of a complex number is the angle 
between x-axis and
vector OP,
representing this complex number. 
Trigonometric form of a complex
number. Abscissa
a and ordinate b
of the complex number
a + bi can
be expressed by its modulus r and argument
:
Operations
with complex numbers, represented in the trigonometric form.
This is
the famous Moivre’s
formula.
Here k is any integer. To receive
n different values of the n-th degree
root of z it’s
necessary to give n consecutive
values for k ( e.g., k = 0, 1, 2,…, n – 1
) .
