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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Brandiwyn🎶

a day ago
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.

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Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

April 9, 2023 at 12:02pm

#1047885

Revisited: "Once More With Feeling"

It's Sunday (some say Easter Sunday), so it's time once again to forget the worries of the present and focus on the disaster of the past.

Apparently, back in 2020 (a pretty bad year from most perspectives), I'd had a few blog entries, most of which I vaguely remember, talking about panpsychism, the persistent belief (for it is a belief, not scientific knowledge) that consciousness arises in complex life because every thing in the universe, including subatomic particles, has a rudimentary form of consciousness. I may not be getting it exactly right, but that's the gist of it.

Naturally (pun intended), I push back—not from any deep-seated need to be special, but because it's not, as far as I've been able to tell, a testable, falsifiable hypothesis. Apparently, I pushed back a few times, based on the title of the entry ("Once More With Feeling"

), and the first line:

This is probably the last bit about panpsychism I'll be linking. For now. Maybe.

And I can't be arsed to scour every day since August 11, 2020 to see if I talked about it again until now.

But let's turn this into a Merit Badge Mini-Contest.

Yeah, I know, I haven't done one of these in a while. Interest seemed to peter out. But maybe I'll start them up again, perhaps once a month. Not today, though.

Anyway, the link (available at the original entry above) is still active, if you're interested in what this publication, and one philosopher, has to say about it.

This entry is about what I have to say about it.

In the link above, I copied the "elevator pitch" for panpsychism:

In our standard view of things, consciousness exists only in the brains of highly evolved organisms, and hence consciousness exists only in a tiny part of the universe and only in very recent history. According to panpsychism, in contrast, consciousness pervades the universe and is a fundamental feature of it.

And today, rereading this, I realized that I didn't address the misconception about evolution: that there are "highly evolved organisms" as opposed, apparently, to less-evolved organisms.

That point of view is dangerously self-centered.

Every living being on Earth shares a common ancestor. Every living thing on Earth has therefore been subject to evolutionary pressure for the same amount of time (3-4 billion years; the exact time is irrelevant to this discussion so I won't bother splitting hairs on it). We have certain adaptations that have made us very good at becoming an invasive species, even to the extent of being able to live in places like Antartica and, for at least a little while, on the moon.

But that doesn't make us "highly evolved," any more than a bacterium is "highly evolved" because it acquired antibiotic resistance from its progenitors.

None of this means that panpsychism is right. I'm just pointing out that the "standard view" noted in the quote is kind of a straw man. I can accept the idea that any living thing has some form of consciousness, but every nonliving thing? I'mma need evidence. "Consciousness pervades the universe" is more in the realm of theology, which requires no evidence.

Anyway, the original entry goes deeper into other arguments, which I won't rehash here. Bottom line is, sure, it's a legitimate philosophy; it's also older than recorded language in humans (in the form of animism). It makes for good creative writing, and excellent (if later forgotten) stoned dorm room conversations.

What it's not, is science.

It may be, someday, when we know more. Which is why we do science.

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