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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Write_Mikey_Write!

about 15 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 17 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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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