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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Turkey DrumStik

about 7 hours 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.

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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

November 6, 2018 at 1:45am

#945003

All A-Clone

One important thing about science fiction - and here, I'm talking about real science fiction, not space opera, though that can be fun - is that it gives us a chance to explore the ramifications of certain technologies before those technologies get a chance to be realized in our lives.

A common theme in SF when I was young was the topic of cloning. The idea had already been around for a long time, and had in fact been used to great success in plants. Every banana you eat, for example, is genetically identical. But of course, that led philosophers and science fiction writers (the distinction is sometimes murky) to speculate on cloning animals, including humans.

And boy, did a lot of them get it wrong. I don't mean wrong in the sense that the writers who predicted flying cars were wrong, but wrong on a basic, fundamental level. Stories abounded of exact duplicates of a human being, down to the last hair and quirky mannerism. Now, I'm not saying that sort of thing will never happen - I wrote a story set in the far future that features this sort of thing - but it's not happening anytime soon.

With exact duplicates of that sort, all sorts of ethical and existential questions get raised, not the least of which is the attempt to define the word "I." It's a simple word, just one letter, and in our normal frames of reference the pronoun has a simple, singular referent. Analogy: You have two thumb drives. You copy your budgeting spreadsheet from one drive to the other, and make changes to one of them. Then your SO goes looking for it to check if you have hookers and blow as line items somewhere. Which one was the original, and which the copy? They both exist, in the sense that binary information can be said to exist.

Well, animals are a great deal more complex than a budgeting spreadsheet, and we simply aren't anywhere near the level of technology required to treat them as such. No, in the real world, cloning is essentially asexual reproduction - you don't end up with a fully-formed Minerva, but with a zygote, then an embryo, then a fetus, then a baby, with all of the life-support requirements that these stages have. It's no different from having a standard baby, except that it got all of its genes from you, and not from you and your hookers-and-blow-budgeting SO.

(Okay, there are differences involving telomeres and possibly environmental changes to genes, but I'm not a biologist so we'll ignore them for this discussion.)

The result is, basically, your identical twin - only several years younger.

But wait, there's more. Check this

out.

Nearly a decade ago, on December 22, 2001, scientists at Texas A&M University announced the birth of CC, the world’s first cloned cat. CC, as in Copy Cat, has the same DNA blueprint as her genetic mother, Rainbow. So, like identical twins they share the same genetic code. That means everything’s the same—their looks, their mannerisms, their…. But wait.

Okay, they named the clone Copy Cat and I have some faith in humanity restored because of that.

A closer view in color reveals they don’t actually look identical. In fact, technically they’re not even the same color. Rainbow, with her spackling of orange mixed in with patches of black accentuated by a white belly and legs is calico while CC, who has no orange coloring at all, is a tiger-tabby.

The article goes on to explain that no, they don't have the same mannerisms, either. An animal clone, then, is clearly a completely separate entity. There is no fuzzy boundary on the pronouns; the two cats are distinctly different.

That article is obviously old now, which means the cats involved are almost certainly ex-cats and that the technology has advanced since then - but I've heard absolutely nothing to convince me that "cloning," as we know it, is anything other than a type of asexual reproduction.

The important takeaway here is that, even if the technology is there to do that with humans, if you were to go out and get yourself cloned tomorrow, there'd be a new entity born about nine months later (probably to a surrogate mother), and the only question would be whether that entity is your sibling or your offspring, or both - and even that is hardly a new conundrum if you live in, say, Alabama. Point is, it's not "you," and you have no say over it beyond, possibly, parental rights and obligations. We already know what to do with a new life; we've been doing it since long before you could call us "human."

(I'm not a lawyer, either, but this is the ethical argument, not the legal one.)

Now, there's a somewhat related question, one that's also been done to death in science fiction - that of artificially grown organs or limbs that could potentially be used as replacement parts. I think I've already established that we can't use clones for that because they are individuals, at least not without their consent (and if you think such consent would be invalid, consider how many people have donated a kidney or part of a liver to a compatible recipient).

Short answer: This is not an ethical question.

Long answer: We're not talking about a separate, conscious, living entity here, but a collection of cells grown in a vat (or whatever medium they use these days). No central nervous system, no thoughts, nothing to have thoughts with, and not even the potential to become an independent organism.

I will add one more thing, as food for thought (pun absolutely intended). But it's a little squicky, so if you're squeamish, stop reading here.

...

Still here?

Fine. You were warned.

There's a persistent cultural taboo in most societies against cannibalism. There are good reasons for that, it's not just a matter of "eating people is icky," but I won't go into them now. Still, in extreme situations, people have been known to violate this taboo: The Donner party, for example, or that soccer team or whatever that crashed in the Andes. You do what you have to in order to survive. So here's the ethical conundrum, if you can look at it without the weight of cultural baggage: what's the actual difference between consuming human flesh to survive, and accepting a donor organ to prolong your life? I mean besides the method by which you get the meat into your body, which to me is a distinction without a difference.

As I see it, the cloning thing is settled ethics. The organ donor thing also seems to be settled by everyone else, but I don't really understand why people get weird about cloning and not about organ donation.

Feel free to discuss.

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