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.
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.