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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 3 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 5 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.

1 Like

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 21, 2021 at 12:07am

#1008886

Mind Matters

Well, I haven't mentioned this subject in a while. The article is from 2018, but I just found it recently.

Could consciousness all come down to the way things vibrate?

I think the last time I ragged on panpsychism was about a year ago. This article has the added benefit of the word "vibrate" in the headline, which makes me want to make vibrator jokes. But those would be too easy, so I'll try to restrain myself.

Why is my awareness here, while yours is over there? Why is the universe split in two for each of us, into a subject and an infinity of objects? How is each of us our own center of experience, receiving information about the rest of the world out there? Why are some things conscious and others apparently not? Is a rat conscious? A gnat? A bacterium?

"Infinity?" [citation needed]

Over the last decade, my colleague, University of California, Santa Barbara psychology professor Jonathan Schooler and I have developed what we call a “resonance theory of consciousness.” We suggest that resonance – another word for synchronized vibrations – is at the heart of not only human consciousness but also animal consciousness and of physical reality more generally. It sounds like something the hippies might have dreamed up – it’s all vibrations, man! – but stick with me.

Because if there's any place where such a hypothesis could emerge, it would be Santa Barbara.

Based on the observed behavior of the entities that surround us, from electrons to atoms to molecules, to bacteria to mice, bats, rats, and on, we suggest that all things may be viewed as at least a little conscious. This sounds strange at first blush, but “panpsychism” – the view that all matter has some associated consciousness – is an increasingly accepted position with respect to the nature of consciousness.

I've laid out my problem with panpsychism before. In brief: where's the evidence, for or against? Is it falsifiable? Can we do experiments on it? Because otherwise it's just an idea. Maybe it's a good idea - I mean, can it really hurt to consider all things as related? But to me it has one great flaw, which is that if you claim that every atom has some kind of consciousness, you dilute any definition of consciousness. Like when you call time an illusion, you change the definition of "illusion."

Still, the article is worth reading, because who knows? Maybe they're onto something (as opposed to on something, which is the usual state necessary to claim "It's all ONE, dude.") From a purely physical standpoint, it's clear that there's a deep connection between all life, and for that matter, all matter. What's not clear to me is that consciousness, which from the standpoint of parsimony might be better described as an emergent property of an individual's neural connections, also comes from some sort of lower-level consciousness.

It's a lot to think about, and it's not like I can come up with a more compelling idea. So I keep reading on the subject, and thus, sharing what I find.

After all, that's what group consciousness is all about, yes?

2 Like • Comment

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