Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Elycia Lee ☮

about 3 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

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Wordsmitty ✍️

about 16 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

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Annette in France

about 19 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

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Elisa: Middle Aged Stik

about 21 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

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Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

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Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

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Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

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Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

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Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

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Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

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April 28, 2020 at 12:05am

#982240

I Drink Therefore I Am

A while back, I talked about the concept of panpsychism. It's in this entry: "Universal Consciousness"

. In it, I asserted that there was no real evidence for the idea. And if you want to know what the idea is, read that entry that I just linked, and/or take a look at today's article:

https://aeon.co/ideas/why-panpsychism-fails-to-solve-the-mystery-of-consciousnes...

Why panpsychism fails to solve the mystery of consciousness

So, this article is suggesting that not only is there no evidence for it, there might be evidence against it.

Is consciousness everywhere? Is it a basic feature of the Universe, at the very heart of the tiniest subatomic particles? Such an idea – panpsychism as it is known – might sound like New Age mysticism, but some hard-nosed analytic philosophers have suggested it might be how things are, and it’s now a hot topic in philosophy of mind.

Wait, hang on, just hit pause a minute. I need to wrap my head around the phrase "hard-nosed analytic philosophers."

While I'm processing that, this particular article gives a pretty good overview of what exactly panpsychism is, and the "problems" that it would address. Then:

I remain unpersuaded, and I’m not alone in this. Even if we accept that basic physical entities must have some categorical nature (and it might be that we don’t; perhaps at bottom reality is just dispositions), consciousness is an unlikely candidate for this fundamental property.

Yesterday, I talked about the elegance of complexity arising from simplicity. If one argues for a consciousness that pervades all things, that goes out the window. Might as well drop the fancy language and call it God, which turns it into a much older idea.

[Consciousness] appears to be a specific state of certain highly complex information-processing systems, not a basic feature of the Universe.

"Appears to be" is hardly evidence one way or the other.

Panpsychism offers no distinctive predictions or explanations. It finds a place for consciousness in the physical world, but that place is a sort of limbo. Consciousness is indeed a hard nut to crack, but I think we should exhaust the other options before we take a metaphysical sledgehammer to it.

Without predictions or explanations, it's philosophy, not science. This doesn't mean it's necessarily wrong, but, much like ideas such as "honor" or "liberty," it's nothing you can pin down.

So I’m not a panpsychist. I agree with panpsychists that it seems as if our experiences have a private, intrinsic nature that cannot be explained by science. But I draw a different conclusion from this. Rather than thinking that this is a fundamental property of all matter, I think that it is an illusion.

I dislike applying the term "illusion" to something we can call an "emergent property." As Descartes noted, "I think, therefore I am;" as far as I'm concerned -- I know some philosophers will disagree with me -- my consciousness is one of the few things that I can be almost absolutely certain is real. To call it an illusion is to break the entire definition of the word "illusion."

Consciousness, in that sense, is not everywhere but nowhere. Perhaps this seems as strange a view as panpsychism. But thinking about consciousness can lead one to embrace strange views.

It can also lead one to drink. Speaking of which, it's about that time...

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