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!

a day 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

a day 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.

August 30, 2019 at 12:33am

#965240

Life As We Know It

http://nautil.us/issue/17/big-bangs/the-unique-merger-that-made-you-and-ewe-and-...

The Unique Merger That Made You (and Ewe, and Yew)
All sophisticated life on the planet Earth may owe its existence to one freakish event.

Really, I debated with myself over linking this article here. It's not that it's very technical - it's not, at all; there's some biology lingo, but no math or physics. It's just, on the surface, not very relevant to much outside of biology. I, of course, find it fascinating.

But there's some speculation toward the end that's worth looking at. First, some highlights:

The transition from the classic prokaryotic model to the deluxe eukaryotic one is arguably the most important event in the history of life on Earth. And in more than 3 billion years of existence, it happened exactly once.

Well... [citation needed], I suppose. If it happened more than once, and the resulting organisms were out-competed or otherwise died off, we might never know. But I'll grant that the writer knows more about biology than I do; that's not a very high bar to clear.

If this story is true, and there are still those who doubt it, then all eukaryotes—every flower and fungus, spider and sparrow, man and woman—descended from a sudden and breathtakingly improbable merger between two microbes. They were our great-great-great-great-...-great-grandparents, and by becoming one, they laid the groundwork for the life forms that seem to make our planet so special. The world as we see it (and the fact that we see it at all; eyes are a eukaryotic invention) was irrevocably changed by that fateful union—a union so unlikely that it very well might not have happened at all, leaving our world forever dominated by microbes, never to welcome sophisticated and amazing life like trees, mushrooms, caterpillars, and us.

Now, those who like to deny evolution might ask something like "but if the eukaryotes were so successful, then why are there still prokaryotes?" I would suggest they read up on some basic biology; I'm certainly not going to waste time on that argument or its ilk.

“[Endosymbiosis] was taboo,” says Bill Martin. “You had to sneak into a closet to whisper to yourself about it before coming out again.”

That's legitimately funny.

The article goes on to lay out the case for eukaryotic origins - again, fascinating stuff, and it highlights how science gets shit done, but not really anything worth commenting on here - until we get to the end, where we find stuff that intersects with another of my favorite topics: extraterrestrial life.

This improbability has implications for the search for alien life. On other worlds with the right chemical conditions, Lane believes that life would be sure to emerge. But without a fateful merger, it would be forever microbial. Perhaps this is the answer to the Fermi paradox—the puzzling contradiction between the high apparent odds that intelligent life would exist elsewhere among the billions of planets in the Milky Way, and our inability to find any signs of such intelligence. As Lane wrote in 2010, “The unavoidable conclusion is that the universe should be full of bacteria, but more complex life will be rare.” And if intelligent aliens did exist, they would probably have something like mitochondria, too.

The "improbability" mentioned there is the unlikelihood of the formation of eukaryotes. Again, I'll point out that however improbable something is, once it happens, the probability of it having happened is unity - and since we eukaryotes wouldn't be around without it, we can't argue that something won't happen just because it's improbable. If that makes any sense.

I've also pointed out before that the "Fermi paradox" isn't a paradox. I don't accept the "high apparent odds that intelligent life would exist elsewhere." There's not a single rule, law, or requirement of evolution that those qualities that we call "intelligence" (please, can the "no intelligent life on Earth either" jokes; they're really tiresome) must occur, any more than there's a rule stating that eukaryotes must emerge.

So what this information does, really, is lower the odds of complex life existing in another biosphere. We don't know what those odds are, admittedly, because we simply don't have enough data points (we have precisely one data point - Earth).

Not that this idea is new to me; I've been sold on the Rare Earth Hypothesis for some years now. But this article goes into more depth than I've seen in the past, and does it without getting, as I said, too technical about it.

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