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🎶

about 22 hours 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.

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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.

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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.

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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.

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July 17, 2024 at 11:35am

#1074060

Turtles

I consciously saved this article from Big Think a while back. There were only a couple of things I had saved from that source; it's random coincidence (operating on a limited set of possibilities) that I got both of them to pop up in the same week. Considering my issues with the last one, which I discussed two days ago, it's just as well that I've cleared my queue of these; I'm not sure this one's much better.

Can the known particles and interactions explain consciousness?
At a fundamental level, only a few particles and forces govern all of reality. How do their combinations create human consciousness?

Again, I'm not sure that they're asking the right questions.

How odd is it that these particles and forces fit together so precisely as to enable conscious beings like us to exist? It’s a cutting-edge question, but one that we’re closer than ever to answering definitively.

We're also "closer than ever" to developing usable fusion power. Instead of 20 years away, as we were back in the 1960s, we're now only 19 years away.

Living creatures can be divided into cells; cells themselves are composed of organelles; organelles can be broken down into molecules; molecules are made up of atoms; atoms are comprised of electrons and atomic nuclei; electrons cannot be broken down further, but nuclei themselves consist of quarks and gluons.

As I've noted before, every hard science discipline is, at base, physics.

We should, therefore, be able to take these fundamental constituents of matter — quarks, gluons, and electrons — and assemble them in various ways to explain everything that we encounter in everyday life.

I'm not sure that follows. There are probably things we can never fully explain, but that's a good thing in my book.

The question that inspired that article is quoted within it, and it goes like:

“It has always puzzled me how come the particles & forces in the original quark gluon soup fit precisely to form:

nuclei, and when joined by electrons,
atoms (each with their distinct properties),
countless molecules (each with their distinct properties),
capable of forming life,
which can achieve consciousness,
and ultimately lead to astrophysicists?

This precise “fitting” can hardly be a pure coincidence.”

And that sounds to me like the question of a disingenuous religious person, trying to tie scientists up into a big knot of explanation to try to shore up his (the name is Ottho Heldring, so I'm assuming gender based on that) belief. Maybe I'm wrong about that, but believers have been known to do exactly that, with baseless questions like "if humans evolved from chimps, why are there still chimps?" Which is a question that rests on several false assumptions.

Coincidence or not, if such a history as he briefly described in the question had not taken place, we wouldn't be here to argue about it, now, would we? As I've said before, it doesn't much matter what the odds were after you've won the lottery.

The article itself isn't that long, but it manages to provide a brief history of the physical processes that, ultimately, led to the universe as we knew it, up to and including cats. I won't quote from the meat of it; it's there at the link, and while I have some quibbles, that's going to happen when you take something it requires a PhD to describe (which I definitely don't have) and boil it down to a couple thousand words.

Then, in summary:

But this is not necessarily miraculous. So long as there are a few simple rules and properties to nature:

And a restatement of what I said above:

If the laws of physics were so different that we couldn’t have come into existence, we never would have arisen to find these things out.

Pretty sure both of those are a version of the Anthropic Principle,

which has its own problems because it doesn't really explain anything.

Ultimately, we know that complexity can absolutely arise from simplicity. There's a random element to it, but it's not completely random; it's subject to physical (in the sense of "physics," not "physicality") rules.

And isn't that a more likely hypothesis than complexity arising from even greater complexity? The latter just raises the question of where the "greater complexity" comes from. "It was always there" is deeply unsatisfying, akin to "it's turtles all the way down."

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