a) Complex numbers in rectangular form
b) Argand plane
c) Algebraic operations on complex numbers

**Complex Numbers**

We have seen that the equation *x*^{2} +1 = 0 does not have a
solution in real number system.

In general there are polynomial equations with real coefficient
which have no real solution.

We enlarge the real number system so as to accommodate solutions
of such polynomial equations.

This has triggered the mathematicians to define complex number
system.

In this section, we define

a)
Complex numbers in rectangular form

b)
Argand plane

c)
Algebraic operations on complex numbers

The complex number system is an extension of real number system
with imaginary unit *i *.

The imaginary unit *i *with the property *i*^{2} = âˆ’1 , is combined with two
real numbers *x *and *y* by the process of addition and multiplication,
we obtain a complex number *x *+ *iy*. The symbol '+ ' is treated as vector addition. It was introduced by Carl
Friedrich Gauss (1777-1855).

**Definition 2.1
(Rectangular form of a complex number)**

A complex number is of
the form ** x **+

If *x *=
0 , the complex number is said to be purely imaginary. If *y *= 0 , the complex number
is said to be real. Zero is the only number which is at once real and purely
imaginary. It is customary to denote the standard rectangular form of a complex
number *x *+
*iy *as *z *and we write *x *=
Re(*z*) and *y *= Im(*z*) . For instance, Re(5 âˆ’ *i*7) = 5 and Im (5 âˆ’ *i*7) = âˆ’7 .

The numbers of the form *Î±** *+
*i**Î²** *, *Î²** *â‰ 0 are called imaginary
(non real complex) numbers.

The equality of complex numbers is defined as follows.

**Definition 2.2**

Two complex numbers *z*_{1}** **= *x*_{1}** **+ *iy*_{1}** **and *z*_{2}** **= *x*_{2}** **+ *iy*_{2}** **are said to be equal
if and only if

**Re( z**

For instance, if *Î±** *+
*i**Î²** *= âˆ’7 + 3*i *, then *Î±** *= âˆ’7 and *Î²** *= 3 .

A complex number *z *=
*x *+ *iy *is uniquely
determined by an ordered pair of real numbers ( *x*, *y *) .

The numbers 3 âˆ’ 8*i*, 6 and âˆ’4*i *are equivalent to (3, âˆ’8), (6, 0), and (0, âˆ’4) respectively. In this way we are able to associate a complex
number *z *= *x *+ *iy *with a point ( *x*, *y *) in a coordinate plane.

If we consider *x *axis as real axis and *y *axis as
imaginary axis to represent a complex number, then the *xy *-plane is
called complex plane or Argand plane. It is named after the Swiss mathematician
Jean Argand (1768 â€“ 1822).

A complex number is represented not only by a point, but also by
a position vector pointing from the origin to the point. The number, the point,
and the vector will all be denoted by the same letter *z . *As usual we
identify all vectors which can be obtained from each other by parallel
displacements. In this chapter, **C** denotes the set of all complex
numbers. Geometrically, a complex number can be viewed as either a point in **R**^{2} or a vector in the
Argand plane.

Here are some complex numbers: 2 + *i*, âˆ’ 1 + 2*i*, 3 -
2 *i*, 0 - 2 *i*, 3+ âˆš-2, -2-3*i*, cos (Ï€/6) + isin(Ï€/6) and
3 + 0*i*. Some of them are plotted in Argand plane.

In this section, we study the algebraic and geometric structure of
the complex number system.

We assume various corresponding properties of real numbers to be
known.

If *z *= *x *+ *iy *and *k *âˆˆ
**R**, then we define

k z =
(kx) + (ky )*i* .

In particular 0 *z *=
0 , 1 *z *=
*z *and (âˆ’1)*z *= âˆ’*z *.

If *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + *iy*_{2} , where *x*_{1} , *x*_{2} , *y*_{1} , and *y*_{2} âˆˆ
**R**, then we define

*z*_{1} + *z*_{2} = (*x*_{1} + *iy*_{1}) + (*x*_{2} + *iy*_{2})

= (*x*_{1} + *x*_{2}) + *i*(*y*_{1} + *y*_{2})

*z*_{1} + *z*_{2}= (*x*_{1} + *x*_{2}) + *i*(*y*_{1} + *y*_{2})

We have already seen that vectors are characterized by length and
direction, and that a given vector remains unchanged under translation.
When *z*_{1} = *x*_{1} + *iy*_{1} and *z*_{2} = *x*_{2} + *iy*_{2} then by the
parallelogram law of
addition,thesum *z*_{1} + *z*_{2} = ( *x*_{1} + *x*_{2} ) + *i *( *y*_{1} + *y*_{2}* ) *corresponds to the point ( *x*_{1}* *+ *x*_{2}* *, *y*_{1}* *+ *y*_{2}* *) . It also corresponds to a vector with those coordinates
as its components. Hence the points *z*_{1} , *z*_{2} , and *z*_{1} + *z*_{2} in complex plane may
be obtained vectorially as shown in the adjacent Fig. 2.11.

Similarly the difference *z*_{1} âˆ’ *z*_{2} can also be drawn as
a position vector whose initial point is the origin and terminal point is ( *x*_{1} âˆ’ *x*_{2} , *y*_{1} âˆ’ *y*_{2} ) . We define

*z*_{1} - *z*_{2} = *z*_{1} + (- *z*_{2} )

= (*x*_{1} + *iy*_{1}) + (-*x*_{2} - *iy*_{2})

= (*x*_{1} - *x*_{2}) + *i*(*y*_{1} - *y*_{2})

*z*_{1} + *z*_{2}= (*x*_{1} - *x*_{2}) + *i*(*y*_{1} - *y*_{2})

It is important to note here that the vector representing the
difference of the vector *z*_{1} âˆ’
*z*_{2} may also be drawn
joining the end point of *z*_{2} to the tip of *z*_{1} instead of the origin.
This kind of representation does not alter the meaning or interpretation of the
difference operator. The difference vector joining the tips of *z*_{1} and *z*_{2} is shown in (green)
dotted line.

The multiplication of complex numbers *z*1 and *z*2 is
defined as

*z*_{1} *z*_{2} = (*x*_{1} + *iy*_{1} )(*x*_{2} + *iy*_{2} )

= (*x*_{1} *x*_{2} âˆ’ *y*_{1} *y*_{2} ) + *i*(*x*_{1} *y*_{2} + *x*_{2} *y*_{1} )

*z*_{1} *z*_{2} = (*x*_{1} *x*_{2} âˆ’ *y*_{1} *y*_{2} ) + *i*(*x*_{1} *y*_{2} + *x*_{2} *y*_{1} ) .

Although the product of two complex numbers *z*_{1} and *z*_{2} is itself a complex
number represented by a vector, that vector lies in the same plane as the
vectors *z*_{1} and *z*_{2} . Evidently, then, this product is neither
the scalar product nor the vector product used in vector algebra.

If *z *= *x *+ *iy *, then

*iz *= *i*(*x *+ *iy*)

= âˆ’ *y *+ *ix*

The complex number *iz*
is a rotation of *z* by 90Ëš or Ï€/2 radians
in the counter clockwise direction about
the origin. In general, multiplication of a complex number *z* by *i* successively gives
a 90Â° counter clockwise rotation successively about the origin.

Let *z*_{1} = 6 + 7*i *and *z*_{2} = 3 âˆ’ 5*i *. Then *z*_{1} + *z*_{2} and *z*_{1} âˆ’ *z*_{2} are

(i) (3 - 5*i*) +
(6 + 7*i*) = (3 + 6) + (-5+7)*i *= 9+2*i*

(6 + 7*i*) - (3 -
5*i*) = (6 - 3) + (7 - (-5))*i *= 3+12*i*

Let *z*_{1} = 2 + 3*i *and *z*_{2} = 4 + 7*i *. Then *z*_{1} *z*_{2} is

(ii) (2 + 3*i*)( 4 + 7*i*) = (2x4 â€“
3x7) + *i*(2x7 + 3x4)

= (8-21) + (14+12)*i*
= -13 +26*i*

**Example 2.2**

Find the value of the real numbers *x *and *y*, if the
complex number (2 +
*i*)*x *+
(1âˆ’ *i*) *y *+ 2*i *âˆ’ 3 and *x *+ (âˆ’1+ 2*i*) *y *+1+ *i *are equal

**Solution**

Let *z*_{1} = (2 + *i*)*x *+
(1- *i*) *y *+ 2*i *- 3 = (2*x *+ *y *- 3) + *i *(
*x *- *y *+ 2) and

*z*2 = *x *+ (-1+ 2*i*) *y *+1+ *i *= ( *x *-
*y *+1) + *i *(2 *y *+1).

Given that *z*_{1} = *z*_{2} .

Therefore (2*x *+ *y *- 3) + *i *( *x *- *y
*+ 2) = ( *x *- *y *+1) + *i *(2 *y *+1) .

Equating real and imaginary parts separately, gives

2*x *+ *y *- 3= *x *- *y *+1 â‡’ *x *+ 2 *y *= 4 .

*x *- *y *+ 2 = 2 *y *+1 â‡’ *x *- 3*y *=
-1 .

Solving the above equations, gives

*x *= 2 and *y *=
1.

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