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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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

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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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March 18, 2021 at 12:01am

#1006580

Life Is

Wednesdays have become gaming days for me, and thinking too hard after a game gives me a headache. Still, as long as I'm not too wiped, I try to post something early Thursday. Today's random link is about defining life.

What Is Life? Its Vast Diversity Defies Easy Definition.
Scientists have struggled to formulate a universal definition of life. Is it possible they don’t need one?

The article, which as of this posting is recent, talks about someone's book. A lot of articles pushing books are annoying, so I'm dropping a warning, but as always, I'm not going to rag on someone for flogging their book, not on a site for writers.

“It is commonly said,” the scientists Frances Westall and André Brack wrote in 2018, “that there are as many definitions of life as there are people trying to define it.”

As an observer of science and of scientists, I find this behavior strange. It is as if astronomers kept coming up with new ways to define stars. I once asked Radu Popa, a microbiologist who started collecting definitions of life in the early 2000s, what he thought of this state of affairs.

“This is intolerable for any science,” he replied. “You can take a science in which there are two or three definitions for one thing. But a science in which the most important object has no definition? That’s absolutely unacceptable. How are we going to discuss it if you believe that the definition of life has something to do with DNA, and I think it has something to do with dynamic systems? We cannot make artificial life because we cannot agree on what life is. We cannot find life on Mars because we cannot agree what life represents.”

I imagine that this is made even more difficult when we only have one "kind" of life to study: Earth life. It's conceivable that we won't recognize it when we find it elsewhere (and, barring utter catastrophe, I'm sure that we will -- I mean simple life here, not little green women). So thinking about what makes something "alive" or "not alive" is worth pursuing.

With scientists adrift in an ocean of definitions, philosophers rowed out to offer lifelines.

I'm not sure that helps.

In 2011, Trifonov reviewed 123 definitions of life. Each was different, but the same words showed up again and again in many of them. Trifonov analyzed the linguistic structure of the definitions and sorted them into categories. Beneath their variations, Trifonov found an underlying core. He concluded that all the definitions agreed on one thing: life is self‐reproduction with variations. What NASA’s scientists had done in eleven words (“Life is a self‐sustained chemical system capable of undergoing Darwinian evolution”), Trifonov now did with three.

So the answer lies not in philosophy or science, but in linguistics? Color me skeptical. Especially since the "number of words" thing is spurious. Yes, fewer words may indicate elegance. But to be equivalent, the first definition has five words, not three, because "life is" precedes both.

I'd also argue that "chemical system" may or may not hold true; energy-based life has been hypothesized.

But perhaps I'm being pedantic.

In the 1940s, Wittgenstein argued that everyday conversations are rife with concepts that are very hard to define. How, for example, would you answer the question, “What are games?”

If you tried to answer with a list of necessary and sufficient requirements for a game, you’d fail. Some games have winners and losers, but others are open‐ended. Some games use tokens, others cards, others bowling balls. In some games, players get paid to play. In other games, they pay to play, even going into debt in some cases.

I only read this after doing my last Fantasy newsletter, which was about games.

A group of philosophers and scientists at Lund University in Sweden wondered if the question “What is life?” might better be answered the way Wittgenstein answered the question “What are games?” Rather than come up with a rigid list of required traits, they might be able to find family resemblances that could naturally join things together in a category we could call Life.

And this is where we find that the "lifeline" offered by philosophers, above, only muddies the waters. That's what philosophers do, though they seem to argue otherwise.

One philosopher has taken a far more radical stand. Carol Cleland argues that there’s no point in searching for a definition of life or even just a convenient stand‐in for one. It’s actually bad for science, she maintains, because it keeps us from reaching a deeper understanding about what it means to be alive. Cleland’s contempt for definitions is so profound that some of her fellow philosophers have taken issue with her. Kelly Smith has called Cleland’s ideas “dangerous.”

Nothing good has ever resulted from someone calling an idea "dangerous." Either the idea is false, in which case it eventually falls out of favor, or it's true, in which case it becomes the favored hypothesis. Since philosophers also tell us that there are things that are both true and false, or neither (such as the sentence "This sentence is a lie"), again - muddy water.

The article goes on to talk about the best evidence we have yet for past extraterrestrial life, found in a meteor. It's worth reading about the discussion; I won't paste it here because this is already longer than I wanted to type tonight. In summary, it's not incontrovertible evidence.

The trouble that scientists had with defining life had nothing to do with the particulars of life’s hallmarks such as homeostasis or evolution. It had to do with the nature of definitions themselves — something that scientists rarely stopped to consider. “Definitions,” Cleland wrote, “are not the proper tools for answering the scientific question ‘what is life?’”

Another example of philosophers fucking everything up. But I have to admit she has a point -- sometimes, definitions get in the way of understanding.

Anyway, I don't have anything profound to add. It's just worth reading, though I ended up with no desire to buy the book being promoted there. Just some things to think about. Personally, I think it's pretty clear that there's no binary "life" and "not life;" there's middle ground, like viruses. It's rare that something is truly a binary choice: heads or tails; no other options, and I think that any working definition of life needs to take these gray areas into account.

But it's not like I'm a scientist or a philosopher, and I don't have any skin in the game, so take my opinion with a big grain of non-living salt.

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