Algebra of the Imaginary Operator 'i'

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bperet
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Algebra of the Imaginary Operator 'i'

Post by bperet » Thu Aug 05, 2004 12:51 am

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.
Last edited by bperet on Sun Sep 23, 2018 3:00 pm, edited 1 time in total.
Reason: Fix superscripts
Every dogma has its day...

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