Complex Number

Complex numbers

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as i2 = -1.

The real part of a complex number a + bi is the real number a, and the imaginary part is the real number b multiplied by i.

 For example, 2+5i is a complex number, where 2 is a real number (Re) and 5i is an imaginary number (Im).

The set of all complex numbers is denoted by the symbol C

Note:-1)Euler was the first mathematician to introduce the symbol i (iota) for the square root of -1 with property. i2=-1  He also called this symbol as the imaginary unit.

2)  Iota (i) is neither 0, nor greater than 0, nor less than 0.


3) The square root of a negative real number is called an imaginary unit.


4) For any positive real number a, we have √-a=i√a


5)The property √a√b = √ab  is valid only if at least one of a and b is non-negative. If
a and b are both negative then √a√b = -√ab

Integral powers of iota (i) :

 Since i= $\sqrt{-1}$1   hence we have i2=-1,i3=I and i4=1 .

 In general we have the following results

i4k+1 = i.i4k+2 = -1 i4k+3 =  -i.i4k = 1

Where k can have an integral value (positive or negative).

Similarly, we can find for the negative power of i, which are as follows;

i-1 =-i, i-2=-1,i-3=i and i-4=1

Important Tips
→The sum of four consecutive powers of i is always zero i.e ik+ik+1+ik+2+ik=3=0

(i+1)2=2i , (1-i)2=-2i

 $\frac{1+i}{1-i}=i,\frac{1-i}{i+i}=-i$1+i1i =i,1ii+i =i 

Real Part of a Complex Number

If x and y are two real numbers, then a number of the form z=x+iy is called a complex
number. Here ‘x’ is called the real part of z. The real
part of z is denoted by Re(z)

Imaginary Part of a Complex Number

If x and y are two real numbers, then a number of the form z=x+iy. Here ‘y’ is called the Imaginary part of z. The Imaginary
part of z is denoted by Im(z)

If z = 5 – 4i, then Re(z) = 5 and Im(z) = – 4

Note :

  • A complex number z is purely real if its imaginary part is zero i.e., Im(z) = 0 and
    purely imaginary if its real part is zero i.e., Re(z) = 0.
  •  I can be denoted by the ordered pair (0,1).
  •  The complex number (a, b) can also be split as (a, 0) + (0, 1) (b, 0)

 

Equality of Two Complex Numbers.

Two complex numbers  z1 = x1 + iy1 and z2 = x2 + iy2 are said to be equal if and only if their

real parts and imaginary parts are separately equal.

i.e., z1= z2 Þ x1 + iy1 = x2 + iy2 Û x1 = x2 and  y1= y2

Thus , one complex equation is equivalent to two real equations.

 Note :

  •  A complex number z=x+iy.=0 iff . x=0, y=0
  •  The complex number do not possess the property of order i.e., (a + ib) < (or) > (c + id) is

not defined. For example, the statement 9 + 6i > 3 + 2i makes no sense.

Representation of Complex Number.

A complex number can be represented in the following from:

(1) Geometrical representation (Cartesian representation): The complex number z = a + ib =(a, b)

is represented by a point P whose coordinates are referred to rectangular axes XOX ¢ and YOY ¢ which are called real and imaginary axis respectively. Thus a complex number z is represented by a point in a plane, and corresponding to every point in this plane there exists a complex number such a plane is called argand plane or argand diagram or complex plane or gaussian plane.

Note :

  •  Distance of any complex number from the origin is called the modules of complex  number and is denoted by |z|, i.e |z|=  $\sqrt{a^2+b^2}$a2+b2  
  •  Angle of any complex number with positive direction of x– axis is called amplitude or argument of z. i.e., amp (z)= arg(z)= tan -1(b/a)

 

(2)Trigonometrical (Polar) representation : In D OPM, let  OP = r  , then a= rcosq   and b=. r sinq

Hence z can be expressed as z = r(cosq + i sinq )

where r = |z| and q = principal value of argument of z.

For general values of the argument z = r[cos(2np +q )+ i sin(2np +q )]

 

(3) Vector representation :

If P is the point (a, b) on the argand plane corresponding to the complex number z = a + ib .

Then OP̅=aî+bĵ, \| OP̅|=√a²+b² and arg z = direction of the vector OP̅= tan-1(b/a)

Therefore, complex number z can also be represented by OP̅

 

(4) Eulerian representation (Exponential form) :

Since we have eiq =cosq + i sinq and thus z can be expressed as z = r eiq  , where r = |z| and q =  arg(z).

Logarithm of complex number

Geometry of complex number

Square Root of a Complex Number