The role of multiplicity of waves in the theory:


Under multiplication, lines and amplitudes are one and the same, angles and rotations are also one and the same.


There is no magic or mystery regarding the square root of minus one, it is simply logical common sense if we multiply rotations (angles) by amplitudes (lines) and rotations (angles) by rotations (angles).


rotation (angle)^power =
power x rotation (angle)


example: radian^N = N x radian =
N radians


A radian to the power of N is
N radians. 


Leonhard Euler (1707 - 1783)
e^(i*pi) = (e^i)^pi = radian^pi =
pi x radian = pi radians = -1


A radian to the power of pi is pi radians.


rotation*rotation =
rotation+rotation


angle*angle = angle+angle


amplitude x amplitude =
amplitude x amplitude
(normal multiplication)


line x line = line x line
(normal multiplication)


90 degrees*90 degrees = 90 degrees+90 degrees = 180 degrees


270 degrees*270 degrees =
270 degrees+270 degrees =
540 degrees = 180 degrees


therefore: SQRT 180 degrees =
90 degrees & 270 degrees


Read multiplication by 'x' and '*' as 'of' in the above equations to make things clearer.


The above is the most logical explanation of the square root of minus one (SQRT -1).


To multiply two or more numbers simply add the rotations (angles) and multiply the amplitudes (lines), it's Abraham De Moivrer's (1667 - 1754) theorem without the algebra.


That concludes the role of multiplicity of waves in the theory.


End of page 9.