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 Radians


A Radian to the power of N is N Radians. 


Leonhard Euler (1707 - 1783)
e ^ (i x Pi) = (e ^ i) ^ Pi = Radian ^ Pi = Pi x Radian = Pi Radians = - 1


A Radian to the power of Pi is Pi Radians.


Rotation x Rotation = Rotation + Rotation


Angle x Angle = Angle + Angle


(Amplitude x Amplitude = Amplitude x Amplitude)


(Line x Line = Line x Line)


90 Deg. x 90 Deg. = 90 Deg. + 90 Deg. = 180 Deg.


270 Deg. x 270 Deg. = 270 Deg. + 270 Deg. = 540 Deg. = 180 Deg.


(Read multiplication by 'x' as 'of' in the above to help make things clearer).


Therefore: SQRT 180 Deg. = 90 Deg. & 270 Deg.


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.