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Thursday, Nov. 14, 2024

1:32am EST

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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

A signature for exclusive use of winners at the 2019 Quill Awards

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

Show Recent Comments

10 Most Recent Blog Comments:

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.

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

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

1 Like

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.

2 Like

September 18, 2024 at 9:31am

#1076947

It Came From Outer Space

Today's article, from Popular Mechanics, is a few years old, but I'm not aware of any related developments since then.

Some Scientists Believe All Life Started on Mars
Maybe it's not as crazy as it sounds.

Let me state right off the bat that I have multiple issues with the way the headline is phrased. I'll give it a pass for now, though, because what's really important is the actual article.

Could life as we know it have begun on Mars instead of Earth?

Could it? Sure. Now, how about some evidence?

A handful of scientists believe so, and even more think we should at least consider the possibility.

One of my issues with the headline, and this sentence, is the use of the word "believe." It's as out of place in science as a clown at a black-tie ball.

This special case of the overall theory of panspermia, where life on Earth began somewhere else and traveled or was planted here, has some prominent supporters.

"Traveled or was planted" could imply that some sentience or conscious guidance was involved, so I don't like that in science either. As I've said before, if aliens planted life on Earth, where did said aliens come from? There are some good hypotheses as to how life emerged from non-life (though the distinction isn't a bright bold line) right here on Earth; where are the competing hypotheses for extraterrestrial origin?

I've described it before as "kicking the can down the road," and I suspect the underlying impetus for wanting to "believe" this is either a) an attempt to bring religious beliefs into science, or b) a conviction that life on Earth is not "special." Or maybe both.

In a new Salon article, these proponents say the theory makes intuitive sense based on what the two planets are like.

If there's one thing you need to know about science, it's this: the whole point is to question intuition. Lots of solid scientific theories and discoveries very much contradict intuition.

On the other talon, there's some evidence (as the article later points out) that Mars cooled off to the point where it had liquid water—essential for life as we know it—way before Earth did, so, okay, perhaps the chemical reactions that turned non-life into life took place there and not in our own oceans, only getting kicked off that planet by asteroid impacts. There's also some evidence that simple life can survive such a journey. So I can accept the possibility. But again: evidence? "Can" isn't the same thing as "did."

There is, of course, at least one other possibility: life started on Mars, and also started on Earth, perhaps later, but separately.

BUT. If you're going to assert that sentient aliens did it, then you also have to ask when and how said aliens got their start. Or, of course, you can hand-wave it and say "God did it," but then it's not science, but religion.

And I have one more quibble with the article:

Either way, Mars’s once-molten core slowed and solidified, reducing the planet’s gravity and atmosphere to nearly nothing and removing essential protections for any life form of which we know.

Solidifying a molten core would do exactly jack and squat to the planet's gravity, which is correlated to its mass. I guess maybe a tiny fraction of a bit, due to energy/mass equivalence, but come on. I think they meant "magnetic field," not gravity. It's the magnetic field (which on Earth is probably generated by processes in the molten interior) that keeps the worst of the dangerous emanations of the accursed daystar from knocking complex molecules (like proteins and DNA) apart.

Now, don't get me wrong, here; I'd be as excited as anyone to hear that real evidence for life, now or in the past, on Mars has been confirmed (by which I mean microbes or their equivalent, not pop-fiction sentient Martians). And maybe then we can compare it to Earth life and have a better idea if the one came from the other. But right now, it's all speculation, and, again, we have some very compelling, evidence-based ideas for how inert molecules might have come together to initiate the self-replicating process we call life, right here on this planet.

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