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 an hour 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 3 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.

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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.

May 12, 2023 at 8:06am

#1049518

Mind, the Gap

A little more depth than usual, today, but I'll try to make it worth your time.

Descartes and the Discovery of the Mind-Body Problem
The French philosopher René Descartes is often credited with discovering the mind-body problem, a mystery that haunts philosophers to this day. The reality is more complicated than that.

Descartes was, of course, more than a philosopher. Probably most famous for "I think, therefore, I am," he was also a scientist and mathematician (he's the guy who decided it would be cool to represent points on a two-dimensional grid with x and y axes, ever after known as the Cartesian coordinate system from his name). And he had the best hair of any philosopher. Newton's was arguably better, but his focus was mostly science, math, and alchemy; plus, I suspect his was a wig.

Consider the human body, with everything in it, including internal and external organs and parts — the stomach, nerves and brain, arms, legs, eyes, and all the rest.

Yeah, I know what you were thinking about with "all the rest."

Even with all this equipment, especially the sensory organs, it is surprising that we can consciously perceive things in the world that are far away from us.

Eh, not really. Most animals can do that. Kind of necessary for avoiding predators and finding prey.

For example, I can open my eyes in the morning and see a cup of coffee waiting for me on the bedside table. There it is, a foot away, and I am not touching it, yet somehow it is making itself manifest to me. How does it happen that I see it? How does the visual system convey to my awareness or mind the image of the cup of coffee?

"Even more importantly, I live alone!"

I am conscious of the cup, we might even say, though it is not clear what this means and how it differs from saying that I see the cup.

Everyone treats consciousness like it's gotta be this massive, complex thing, but if it turns out to be really simple, I'll laugh my ass off (from beyond the grave, if the dualists turn out to be right).

How did my neurons contact me or my mind or consciousness, and stamp there the image of the cup of coffee for me?

It’s a mystery. That mystery is the mind-body problem.

By this point in the article, the author could have, you know, drank the coffee instead. Such are the perils of philosophy: your coffee gets cold while you wax philosophical about it.

Our mind-body problem is not just a difficulty about how the mind and body are related and how they affect one another. It is also a difficulty about how they can be related and how they can affect one another.

Plot twist: they're the same thing.

According to Descartes, matter is essentially spatial, and it has the characteristic properties of linear dimensionality. Things in space have a position, at least, and a height, a depth, and a length, or one or more of these.

Hence, Cartesian coordinates (extended into a third dimension).

Mental entities, on the other hand, do not have these characteristics. We cannot say that a mind is a two-by-two-by-two-inch cube or a sphere with a two-inch radius, for example, located in a position in space inside the skull. This is not because it has some other shape in space, but because it is not characterized by space at all.

*bong hit* duuuuuude.

Okay, but in seriousness, I believe this next bit is the actual crux of the matter under discussion:

What is characteristic of a mind, Descartes claims, is that it is conscious, not that it has shape or consists of physical matter. Unlike the brain, which has physical characteristics and occupies space, it does not seem to make sense to attach spatial descriptions to it. In short, our bodies are certainly in space, and our minds are not, in the very straightforward sense that the assignation of linear dimensions and locations to them or to their contents and activities is unintelligible. That this straightforward test of physicality has survived all the philosophical changes of opinion since Descartes, almost unscathed, is remarkable.

I don't know about that last sentence. There are philosophers that will tell you, with straight faces (if you can see said faces behind their beards) that what we know as matter is an illusion, and mind is the only thing that's real.

Oh, wait, that's from Descartes, too.

Descartes is untroubled by the fact that, as he has described them, mind and matter are very different: One is spatial and the other not, and therefore one cannot act upon the other.

And yet, one does act upon the other, as we prove every moment of every day, so if it didn't trouble him, it's not going to trouble me, either.

This is analogous to Zeno's paradoxes.

The way they're formulated, nothing can move, and two people can never get close enough to kiss, and no one would ever be able to enter a room. All of Zeno's paradoxes were later resolved by calculus and limit theory (which came along a generation after Descartes and built on his work), but I mean that in a philosophical sense, when your philosophy doesn't mesh with reality, it's not reality that's wrong; it's your philosophy.

Descartes is surely right about this. The “nature” of a baked Alaska pudding, for instance, is very different from that of a human being, since one is a pudding and the other is a human being — but the two can “act on each other” without difficulty, for example when the human being consumes the baked Alaska pudding and the baked Alaska in return gives the human being a stomachache.

Okay, that was legitimately funny.

The difficulty, however, is not merely that mind and body are different. It is that they are different in such a way that their interaction is impossible because it involves a contradiction.

I would say: an apparent contradiction. Again, it just means that our understanding of one or the other is faulty.

My money's on "mind."

Mind is consciousness, which has no extension or spatial dimension, and matter is not conscious, since it is completely defined by its spatial dimensions and location.

Unless you subscribe to the panpsychism philosophy, which is that all matter has some rudimentary consciousness. (I do not, and have banged on about it at length in previous entries.)

Was there really no mind-body problem before Descartes and his debate with his critics in 1641? Of course, long before Descartes, philosophers and religious thinkers had spoken about the body and the mind or soul, and their relationship. Plato, for example, wrote a fascinating dialogue, the Phaedo, which contains arguments for the survival of the soul after death, and for its immortality.

I begin to see the problem. It starts with a false premise: that the mind, or soul, exists independently of the body and can survive the body's demise.

This is like believing that a candle's light can survive the snuffing of the candle. No. Except in the metaphorical sense, as those who saw the candle's flame can remember it.

What happens, if anything, for example, when we decide to do even such a simple thing as to lift up a cup and take a sip of coffee?

Well, after spending all your time philosophizing about it, that's when you discover it's cold. And I'm still not clear on how it appeared on your bedside table to begin with.

Anyway, I get around all this by understanding that the mind is a construct of the physical body. But what do I know? I'm not a philosopher.

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