Revolutionary Approach to Solving Indepently Problems Of Complex Numbers By Rotating Operator i.

Iota is a pure imaginary number and is equal to square root of -1 and is written as i . Squaring i we get -1 and raising to power 4 is +1. 

2. X and Y Axis are two perpendicular lines intersecting at a point O called origin . OX is a horizontal straight line extending towards right side or positive direction and is called positive X Axis. OX’ is a straight line extended towards left side or negative direction and is called negative X Axis . Similarly, straight line perpendicularly going up from origin O is positive Y Axis or OY and straight line going down from origin is called negative Y Axis or OY’ .  

 3.  Having said that obviously angle between + X Axis and + Y Axis is 90 degree or Pi by 2 and between X Axis and X’ Axis is 180 degree or Pi . That means positive real quantity is plotted on +X Axis and –Ve real quantity on –Ve X Axis . 5 being positive will be plotted in OX direction and
-4 being negative will be plotted in OX’ direction . 

4. Let us take the case of i . i square is -1 . Therefore , it represents a rotation of 180 deg of X Axis and will become –X Axis . Similarly, i raised to the power 4 will make rotation of 360 degree or full circle or OX will come to OX after completing full circle .Now since i square is -1 or 180 degree rotation. Therefore i will make 180/2=90 degree rotation. 
 And i raised to power ½ will make pi/4 angle rotation .

And i raised to power 1/3 will make pi/6 angle rotation.

And i raised to power 1/4 will make pi/8 angle rotation.

And i raised to power n will make n. pi/2 angle rotation.

 Therefore i is called 90 degree or Pi/2 operator .

90 degree rotation means OY Axis and –i means OY’ Axis.

That means i which represents Y Axis , imaginary quantities can be expressed on OY or OY’ as per their +Ve or –Ve signs.

5. Finding Roots And Power Of i By Rotation Of Operator. 

 Now let us calculate the valve of i under root .
As explained earlier , i is 90 deg rotation, therefore i under root will denote 90/2=45 deg rotation. That means , the quantity will be plotted on straight Line making 45 deg angle with X Axis . Let the quantity be unity . Then a unit length will be plotted on the line making 45 deg with X Axis. Therefore , its length on the X Axis will be 1xCos45 or 1 by under root two and on Y Axis, it will be 1xSin45 or 1 by under root two. 

Therefore i under root = Cos45 +i Sin45 = .707+.707 i 

Here .707 i is written to denote its length lying on imaginary or Y Axis .

6. But this is the basic or first value or principal value of square root of i .Since Y Axis is at 90 deg to X Axis and Y Axis will again reach if we rotate it further by 2pi or any integral multiple of 2pi .

Therefore Y Axis in general will be reached after rotation of of angle of 2k.Pi .
                                      i under root will in general correspond to angle ( 2k.pi + pi/2 )/2 where k is an integral real number 0,1,2,3—.  

Now if we put k=0 , we will get the value of i as calculated above .

If we put k=1 , then i under root = Cos(Pi +Pi/4)+i Sin(Pi+Pi/4)

     = -Cos Pi/4 -i Sin Pi/4.

Therefore i square root = -.707-.707i 

7. By mathematical induction ,        i raised to the power n = Cos(2k.pi+pi/2).n+ iSin(2k.pi+pi/2).n

8. Applying this formula , let us find the value of iota raised to the power iota .

 This is equal to Cos(2k.pi+pi/2).i+i Sin(2k.pi+pi/2).i
    = Cosh(2k.pi+pi/2) –Sinh(2k.pi+pi/2) 

Taking the principal value , i raised to power i will be Cosh pi/2 -Sinh Pi/2.

Therefore i raised to power i  is a real quantity .

9. Reducing a complex number to i raised to power n And  Finding  Its Roots And Power By Rotation Of Operator. 

Let us say there is a complex number a+ib where a, b are real quantities .

This can be converted into the form as M(Cosx+iSinx )

Where   M= square root of (a square + b square)

 And tanx =b/a. or Angle x=tan inverse b/a.

 We have seen above that principal value of i raised to the power n

= Cos n.pi/2+iSin n.pi/2

Therefore x=n.pi/2 or n=2x/pi.

Or a+ib= M(I) , where I is i raised to power 2x/pi.

Thus any complex number can be written as i raised to power n .
After reducing it to the form i raised to power n , its roots , powers can be worked out as explained in para 5 above. i raised to power n can be written as M(CosX+iSinX) or M(I) , where I is i raised to power 2x/pi and M is magnitude . 

Principal value of Imaginary number raised to the power n= Cos n.pi/2+ iSin npi/2 .

In this way , every complex number can be reduced to i raised to power n , problems  of complex numbers like raising to the power , roots etc  can be solved independently by operator i without resorting to Euler Formula or De Moivre’s Theorem.   

Author: Narinder

I am graduate in Electronics and Electrical Communication Engineering from Punjab Engineering College, Chandigarh. I worked as Scientist in Solid State Physics Lab DELHI then National Fertilizers Ltd , Bhatinda as Instrumentation Engineer, then Ministry of Labour, Employment, Training as Deputy Director. Thereafter, I joined Civil Services in the year 1986, worked in different capacities on administrative posts and retired on September 30, 2013 as Secretary to Government Punjab from Indian Administratve Services. My interests are Astronomy, Physics, History, Music, Law, Spirituality, Administration and writings. I believe in hard work, determination and consistency in efforts. I love to write on topics related to Astronomy Daily life experience and human sufferings. My favourite writers are Leo Tolstoy, Rabindranath Tagore, Mulk Raj Anand and Munshi Prem Chand.

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