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Sunday, Nov. 17, 2024

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

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Signature for those who have won a Quill Award at the 2020 Quill Awards

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10 Most Recent Blog Comments:

Turkey DrumStik

a day ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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.

1 Like

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').

1 Like

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

May 14, 2024 at 9:58am

#1071051

Not One Thing

While we spent yesterday discussing the origin of Pop-Tarts, today's article, from Scientific American, touches on something almost as profound.

Is There a Thing, or a Relationship between Things, at the Bottom of Things?
Quantum mechanics inspires us to speculate that interactions between entities, not entities in themselves, are fundamental to reality

Let me start with this: I was, at first, a bit thrown off by the headline. Maybe it's just me, being used to seeing headlines phrased as yes/no questions, which I thought this was at first. Then I realized that they're asking which is more fundamental: things, or relations? Just in case some reader experiences the same confusion, I thought I'd attempt to interpret.

Which makes it still a binary question, but I'll run with it for now.

What’s at the bottom of things?

It's turtles all the way down.

If we keep asking “Why?” where do we end up?

If my childhood was representative, the answer to that is "In our rooms, alone."

The monotheistic faiths assert that our questions must culminate in God, a solitary, supernatural creator.

No, I'm not going to get into that argument today.

Dissatisfied with that hypothesis, physicists postulate that everything stems from a single primordial force or particle, perhaps a supersymmetric string, from which flow the myriad forces and particles of our fallen world.

One could still, presumably, reconcile those two worldviews by assuming, for example, that God created the "primordial force or particle."

I don't buy that either, by the way. But like I said, I'm not here to argue about it.

(From what I understand, supersymmetric string theory is all but ruled out, though I still maintain that a perfectly logical "string theory" is: "The Universe is a big ball of string, and God is a cat.")

Notice that, for all their differences, religion and physics share the ultrareductionist conviction that reality comes down to one thing.

Yeah, okay, but as one of those "things" is supposedly infinitely complex, and the other is infinitesimally simple, that's a pretty damn big difference.

Call this the oneness doctrine.

This may be confusing, too, as, at least to me, "oneness" conjures up images of hippies sitting around going, "Whoa, it's, like, all ONE, man."

So, I’m intrigued by the conjecture that at the heart of reality there are at least two things doing something to each other.

This may also be a bias due to our species having evolved via two things doing something to each other.

In other words, there is an interaction, a relationship. Call this the relationship doctrine.

Another point of clarification: These days, you see the word "relationship," and the implication is sexual and/or romantic. But the word "relationship" is much broader than that, meant to describe how any thing relates to any other thing. Like how the Earth and the Moon relate via gravity.

This word definition creep is the same sort of thing that leads us to make jokes about names like the town of Intercourse. When the town, or village, or whatever, was first named, intercourse described interaction between people. Later, people started talking about "sexual intercourse," and because all we do is think about sex, when this was later shortened to "intercourse," the original meaning was all but forgotten, except among pedants like me.

Point being, erase that connotation of "relationship" from your mind when you think about how the author uses the word. Yes, despite my joke above about "two things doing something to each other."

Anyway, the article cites a bunch of thinkers who saw the "relationship" bit as being fundamental. Then:

Part of me finds the relationship doctrine, and especially Gefter’s you-centered metaphysics, beautiful and consoling, a welcome alternative to mindless materialism. The relationship doctrine also seems intuitively sensible. Just as words must be defined by other words, so we humans are defined, and to a certain extent brought into existence, by other human beings.

If there's only one thing to know about science, it's this: it doesn't care whether you find it beautiful and consoling or not. No matter how appealing an idea is, it needs to be falsifiable, and it needs to be tested.

Moreover, as I mentioned above, I have a long-standing aversion to the oneness doctrine. This antipathy dates back to a drug trip in 1981, during which I felt myself becoming a solitary consciousness, the only one in the universe.

I knew that was coming. I bet you saw it coming a light-year away, too.

I thought: What is the difference between one thing and nothing? One thing only exists in relation to something else.

I mean, I don't disagree with the philosophy, but my main takeaway here is, "Wow, someone actually remembered details of their acid trip."

And yet I have doubts about the relationship doctrine, as I do about all metaphysical systems that privilege mind, consciousness, observation, information. They smack of narcissism, anthropomorphism and wishful thinking.

This is good. Having doubts and expressing them is what saves this article.

In conclusion, for me anyway, the headline question is less meaningful than it seems. Leaving aside the imprecision of the word "thing" (which I've harped on in a recent entry), it's meaningless to consider objects without some sort of relationship between them; and you can't speak of the relationships between objects without acknowledging the objects. Also, as the drug-tripping author notes, if there's only one object, it's not much of a universe at all.

So my philosophy is "both." We have to consider both things, and the interactions between things, to get anywhere.

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