Reciprocal System Research Society

ALGEBRA OF THE IMAGINARY OPERATOR 'i' -- KVK Nehru

Simply speaking the operator "i" signifies the

Quote:

"turning of a vector by 90 degrees counter-clockwise in the X-Y plane."

If "i" is applied twice it produces a 180 degrees rotation in the X-Y plane. That is why

Quote:

i² = i.i = -1

and "i" is defined as the square-root of -1, the so-called ‘imaginary’ square-root of -1. The word ‘imaginary’ in this context has no other significance, like shifting from the Time Region to the Time-Space Region etc. Remember that the square root of a negative number is ALGEBRAICALLY not defined. But "i" is given a GEOMETRICAL meaning noted in statement above.

If we apply "i" to the X-axis it turns it into the Y-axis. Then again if you apply "i" to the Y-axis it turns it into the -X-axis. Summarizing

Quote:

i.X = Y; i.Y = -X; i.(-X) = -Y and i.(-Y) = X

Or alternatively,

Quote:

i² = -1; i³ = -i and i⁴ = 1

Next imagine a vector, v1 = 2 + i³, with value 2 of the X-component and value 3 of the Y-component. Applying "i" to v1 we get, v2 = i.v1 = i.(2 + i³) = -3 + i². Obviously v2 is v1 turned by 90 degrees counter-clockwise in the X-Y plane.

Continuing with the discussion of the properties of "i", let us see what is implied in the ‘division’ by this operator,

Quote:

1/i = i/i² = -i

which means "turning a vector by 90 degrees CLOCKWISE in the X-Y plane," the inverse operation of course.

Now if the vector is not limited to the X-Y plane but extends in three dimensions, involving X-, Y- and Z-axes, two more imaginary operators, "j" and "k", will become necessary. The operator "j" signifies

Quote:

"turning the vector by 90 degrees in the Y-Z plane, from Y-axis toward Z-axis"

and the operator "k" signifies

Quote:

"turning the vector by 90 degrees in the Z-X plane, from Z-axis toward X-axis"

Applying any two of the operators "i", "j" and "k" in a sequence produces the effect of the third operator. For example

Quote:

i.j = k; j.k = i and k.i = j

Further see that i² = -1, j² = -1 and k² = -1. But this DOES NOT MEAN that "i", "j" and "k" are the same. This is because the individual operators signify rotations in DIFFERENT planes.