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1/j != -j: Negatives are not inverses

Posted: Mon May 23, 2016 1:46 pm
by bperet
gvoe wrote: 1/i = - i

got it, so it is correct
Want to know something funny? This is actually wrong! I've wondered about it since High School, because if 1/i = -i, then they would be at the SAME tick mark on the i axis, making the center both zero and one at the same time.

I discovered the error while I was analyzing Eric Dollard's stuff from the 80s and applying it to RS2 concepts. An omission was made--that of the "unit circle" on the complex plane. In the RS, you have to have it because your datum is unity, not zero. Took some digging, but what I found was this--a bit of a scam from electrical engineering, calculating impedance:

\frac{1}{j} * \frac{j}{j} = \frac{j}{-1} = -j

The incorrect assumption is j/j = 1, which it does not, because "j" is an operator, not a variable. That's like saying +/+ = 1.

When I corrected this error by NOT making 1/j = -j, all of Dollard's stuff fell into place. I made a detailed diagram, which I'll post in the relevant topic.

Re: 1/j != -j: Negatives are not inverses

Posted: Sat Mar 04, 2017 1:47 pm
by bperet
I just stumbled upon a mathematical relation that may clear up the 1/j = -j error... it comes from the effect that 1-dimensional motion in the time region has on the time-space region (the region beyond the unit space boundary, commonly called space-time), documented in the Inter-atomic distance chapter of Basic Properties of Matter. In that chapter, Larson shows that the outside effect is the integral of 1/t dt, or ln(t), and it turns out that:

ln(t) = -ln(1/t)
ln(1/t) = -ln(t)

Since you are actually dealing with angular velocity in the time region (being all rotational), the natural log got substituted for the rotational operator, j:

ln(1/t) = -ln(t)
1/j = -j

(Make "t" imaginary by changing to "j" and drop the log.) And if you have dealt with LC circuits, the curves are basically logarithmic in their frequency (speed) response. It appears to be a conversion between a zero-based coordinate datum and a unity-based scalar datum.