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The reality of “imaginary” numbers — discovery, not invention

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Over at YouTube, there is a bit of history of Math, on study of cubic functions — and yes there is as usual, some less than exemplary detail — that led to the “invention” of imaginary numbers:

Now of course, I contend that this was discovery not invention (I often don’t buy Veritas Sum’s narrative, but here is a way to see the story).

In News’ thread on i, I commented at 33:

your definition [– Eugene at 8: “there exists such a pair of real numbers (0, 1) that (0, 1) * (0, 1) = -1, where “*” is the specific multiplication rule defined for these types of pairs” –] is tantamount to describing the role of sqrt – 1 in a vector system of numbers, extensible to the ijk vectors and onward to quaternions etc. I often just say take the j* operator as rot 0x 90 degrees anticlockwise. do a double and we see j*j* x –> – x, i.e, the sqrt has a natural, rotation linked sense as j^2 = – 1. Where, oscillations and waves including transients [think here Laplace and Fourier transforms] are closely tied to rotations. Where of course waves are pervasive in quantum mechanics, the Schrodinger expression is about waves and of course energy with standing waves under constraints naturally being quantised as we know from school physics. Further to such, I showed that from {} –> 0 thence N,Z,Q,R,C,R* etc, once distinct identity of possible worlds obtains, core math is embedded in any possible world which gives it universal power as Wigner marvelled at.

So, we are back to Wigner’s wonder. END

Viola Lee:
The word and symbols we use are invented.
Wrong again. ALL information already existed. ET
The word and symbols we use are invented. What the math looks like devoid of a particular symbol system is impossible for us to comprehend, I think. Viola Lee
For what it's worth, I think of "i" in a simplistic way--one which holds up as one maneuvers from one branch of physics/mathematics to another. The following is in no way rigorous, but, it might perhaps be illuminating. Draw two axes on a piece of paper at right angles. Label one "x" and the other "y". Go out some distance on the horizontal (let this be "x") axis. Now, imagine drawing a circle around this point in the "y" and "z" direction (the axis out towards you and out and away from you). Redefine the axes of the circle. Call them "l" and "m." Now imagine reducing the circle around the x-axis until it collapses onto the corresponding point (distance) of the x-axis. Now, equate these two mathematical objects. The circle is described by L^2 + M^2, and the dot is y=0. We then have, L^2 + M^2 = 0. Or, L= {(+/-) sqrt (-1)}M; or, vice versa. I see the "i" as standing for "inner," or "interior." Real numbers are outside this interior circle. Inside the circle=point are the imaginary numbers. This is descriptive. If we assume that the universe has a boundary, then, being "interior" to this boundary, imaginary numbers would apply--relative to the boundary.... --> black holes .... --> anti de-Sitter space ... holographic principle. Again, this is simplistic; however, it can cut through a lot of mathematics at times. (BTW, transforming a point into a circle--with the point being equal to zero, is, in a very basic and simplistic view, how you arrive at string theory from QFT. It's a way of eliminating infinities.) (I hadn't thought of it before, but this could be why string theory has a landscape problem--any "background" you propose can itself be transposed via imaginary numbers into a different background. There would be no end to all of the backgrounds that would be possible.) PaV
Very good history lesson! Two thumbs up!! Of course there are lots of other applications of complex numbers but the particular one discussed is very impressive. JVL
The reality of “imaginary” numbers — discovery, not invention --> Happy Christmas kairosfocus

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