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!

about 11 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.

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Adherennium Plotting something

about 13 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.

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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.

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.

June 27, 2020 at 12:07am

#986616

Something about Nothing

Sometimes in here I talk about science. Sometimes I talk about philosophy. Other times I talk about booze. Of the three, booze is the most important to me.

But here's an article about science and philosophy anyway. Well, "article" is being generous. It's more like a book promotion.

https://blogs.scientificamerican.com/cross-check/making-sense-of-quantum-mechani...

Making Sense of Quantum Mechanics
Philosopher David Albert thinks there might be a “clear and straightforward” way of thinking about quantum phenomena

Clickbait for nerds! And I bit hard on that hook.

I’ll always be grateful to David Albert. In the early 1990s, when I was struggling to write an article called “Quantum Philosophy,” Albert, a philosopher of physics who specializes in quantum mechanics, took pity on me. He served as my guide through the netherworld of quantum interpretations, elucidating the relative merits of the Copenhagen, Bohmian pilot-wave and many-worlds interpretations.

The problem with popularizations of quantum mechanics is that they too often stray into the territory of New Age nonsense -- something that this article addresses later. So here's somebody who, maybe, has a pretty good grasp on both QT and philosophy.

Horgan: ... What is philosophy’s chief contribution to humanity?

Albert: Clarity. Wonder. A certain kind of anxiety. The obvious and enormous and innumerable consequences for the history of the world that I mentioned above. I don't know.

See? Only an actual scientist is comfortable saying "I don't know." He could have led with that, of course, but "I don't know" doesn't sell books or generate ad revenue.

But I take issue with the "clarity" answer. Philosophy always muddies the water, and discussions thereof always descend into epistemology, and you start to question whether anything -- including yourself -- is real.

I say any philosophy that doesn't start with "I am real" is merely mental gymnastics. Nothing wrong with that, per se; a lot of great discoveries have come out of mental gymnastics. But no amount of wishing away the world around us will make the chair under your ass disappear. Or your ass. That takes diet and exercise.

Here’s my question: Will science ever tell us why there is something rather than nothing?

It already has, in the most basic sense. That is, if there were nothing, then we wouldn't be around to ask the question. Of course, "There is something, rather than nothing, so that we could ask the question," is admittedly unsatisfying and an affront to both philosophy and science.

Horgan: I’ll take that as a no. You’ve spent a lot of time pondering quantum mechanics. Have you figured it out yet?

Albert: Yup. Sort of. I think I understand it much better than I used to, and better than I used to imagine I ever would.

I've always been of the opinion that anyone who claims to have figured out quantum mechanics is lying, at the very least to themself.

And the interpretations of quantum mechanics that I like (although “interpretation” is really the wrong word here - since the various so-called “interpretations” on offer are really different physical theories, which often make different empirical predictions) are the ones that show, by explicit example, that an account like that can still be had - interpretations (that is) like Bohmian Mechanics, and theories of spontaneous localization. And among those I have no favorite. I think all of them are interesting, and promising, and since they make different predictions about the outcomes of certain performable experiments, it will be up to those experiments, and not to philosophers like myself, to decide which of them, if any, are actually true.

So... no. You haven't figured it out. If you had, there'd be only one theory, and you'd have a Nobel Prize. That, or superpowers from having hacked the source code of the universe. Well... I suppose if you have the latter you can always get the former.

It turns out that if you imagine that the fundamental physical space of the world is something other, and larger, and different than the three-dimensional space of our everyday experience, then everything that has always seemed uncanny about quantum mechanics suddenly becomes clear and straightforward and understandable and in some sense to have been expected.

Even I, an utter amateur at this sort of thing, can see how postulating more dimensions can explain a lot of things (the spin of an electron, e.g., or certain features of gravity) But like I said, I know next to nothing, so I leave it to the scientists to figure out exactly how many dimensions and how they relate to each other. "But if there are other dimensions, why can't we perceive them?" Same reason we can't perceive the earth as round without experiments or a change in perspective -- neither of which change the fact that the earth is, indeed, round.

Horgan: Why does quantum mechanics inspire so much New Age nonsense?

Albert: Precisely (I guess) because quantum mechanics had for so long been understood - incorrectly - to have overthrown every hope of understanding the world in an objective and literal and realistic and mechanical way.

Uncertainty is not ineffability.

Horgan: Fair point. What’s your utopia?

Albert: I see that some of your other respondents talked about someplace where they could sit around in bathing suits and have nice conversations with interesting people. I like that too.

While drinking booze. There is no utopia without booze. HA! I got a reference to booze in here.

Anyway, the link is there if you want to read it. I found it interesting but certainly not clear, straightforward, or possessing the ability to help one "make sense" out of anything. Just remember it's basically one person's opinion -- albeit the opinion of someone more knowledgeable about this sort of thing than I am.

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