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.

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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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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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February 27, 2021 at 12:02am

#1005335

Schrödinger’s Human

As you probably know by now, sometimes I like to toss in a quantum bomb just for the fun of it.

This Twist on Schrödinger’s Cat Paradox Has Major Implications for Quantum Theory
A laboratory demonstration of the classic “Wigner’s friend” thought experiment could overturn cherished assumptions about reality

And it's a whole lot of fun to talk about overturning "cherished assumptions about reality."

Before I get into the article, though -- and trust me, I'm not going to get too deep into it; it's there if you want to read it, but I just have a few observations to make -- a couple of quick disclaimers:

1) This stuff is way above my pay grade, but it doesn't stop me from being fascinated by it.
2) It is a Bad Idea to make general conclusions about the nature of reality from reading about quantum physics.

To give you an example of (2), consider the phenomenon of "entanglement." Entanglement is famous for appearing to violate the lightspeed limit of the universe. So SF writers, philosophers and others like to hand-wave FTL communication by using entangled particles. But the thing is, entanglement doesn't actually violate FTL limitations because there is no way, even theoretically, for it to allow the transfer of information faster than light.

You know what else violates the speed limit? Your vision. Go outside at night. Look at a star. Then look at another star. Congratulations, your vision just traveled faster than light. But so what? It's a limit on information, not observation.

Anyway, the point is, maybe I draw some of those general conclusions myself; mostly, I'm just trying to understand.

Take his fellow physicist Erwin Schrödinger’s famous thought experiment in which a cat is trapped in a box with poison that will be released if a radioactive atom decays. Radioactivity is a quantum process, so before the box is opened, the story goes, the atom has both decayed and not decayed, leaving the unfortunate cat in limbo—a so-called superposition between life and death. But does the cat experience being in superposition?

Here's the thing that always bugged me about the Schrödinger’s Cat "experiment." Well... apart from the cavalier disregard for a poor kitty's life; fortunately, it was only a thought experiment and, as far as I know, no one has attempted it in a laboratory. The other thing that bugs me is this: If, as the theory suggests, it takes an observer to collapse the wave function and determine "alive" or "dead," how do we define "observer?" I mean, other people have asked this quesiton, too, but I'm talking about it here anyway.

What I mean is, are we really going to assert that it requires a human consciousness? Because I know cats, and I can tell you that cats are conscious animals. They probably don't understand quantum mechanics, and they probably don't have our obsession with questions of the meaning of existence or of life and death, but the cat in Schrödinger’s box would "know" whether it is alive or dead (well, if dead, it wouldn't know anything because it's dead, but you get my point). To further clarify this, replace, in the thought experiment, a cat with a human. Does it really matter if an outside observer doesn't see the result? Because the person in the box does.

So, the article gets into this and even marries the cat-in-the-box thing with quantum entanglement, and like I said, I don't pretend to understand all of it. And it further describes several competing interpretations, or ways to understand what's "actually" going on.

It's easy to draw mystical conclusions from this, and quacks have done so. "Look, science says that nothing really exists until we see it, so we can manipulate reality by choosing what to see" or some such dreck. No. That's not how it works. I don't know how it works, but that's not it.

I'm also still not entirely convinced by many-worlds interpretations -- the article describes this too, but in brief, the idea is that any quantum event that can have more than one outcome (which as I understand it is all of them) produces every possible outcome, each splitting off its own universe. This may fit the data, according to physicists, but there are other possible explanations and, well, that just seems like exactly what Occam's Razor is for: "entities should not be multiplied unnecessarily."

It is, though, great fodder for science fiction. However, some writers have interpreted this to mean "every decision we make splits off new universes," but I'm also not convinced that we actually make decisions; rather, we do what we must, and make up stories about it afterward.

So yeah, I'm not quoting any more of the article. It's there if you want to read it. I just wanted to add my own thoughts.

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