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!

a day 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.

1 Like

Adherennium Plotting something

a day 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.

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

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

May 17, 2019 at 12:29am

#959121

Life Sucks And Then You Die

A lot of medical research today focuses on developing cures to ageing. Presumably, with the right breakthrough, humans could live forever. How do you feel about this? Write an entry describing the advantages and disadvantages of extremely long life using facts and opinions to support your answer.

If I were forced to sum up my current view on good writing in a single, pithy sentence, that sentence would be: "Every solution has a problem."

I've been reading science fiction for almost as long as I've been reading. Sure, some of it is just fun space opera, and that has its place. Most of it - the good stuff - delves into the interface of humans with technological advancement. And one recurring trope is the idea of life extension.

Thing is, "forever" is a long time. Longer than five billion years, which is the approximate amount of time for the sun to run out of hydrogen to fuse, with catastrophic consequences for Earth. Now, are we talking immortality, or indestructibility? Because being "alive" for eternity on a bare ball of molten rock circling a helium-fusing red giant - and then to experience even more solar cataclysms, eventually resulting in that rock cooling to near absolute zero - well... that would suck.

Not to mention the eventual heat death of the universe.

As an aside for the less science-minded out there: you might have heard the phrase "heat death of the universe" before. The phrase conjures up a vision of something very hot where life can't survive. It's actually a thermodynamics thing: "heat" is the transfer of energy from a high-energy place (e.g. a star) to a low-energy place (e.g. a planet). When there are no stars left, some trillion years from now (or something of that order; I can't be arsed to look it up), there's no more energy transfer, and everything in the universe settles down to its average temperature: really fucking cold. Life can't exist in such a state, either. So we upload our consciousness into computers? No, computers can't work in such a realm. Perhaps in a trillion years we'd be clever enough to work around this, but that's purely speculation.

"Forever" is longer than a trillion years. Infinitely longer.

So, it would suck.

Okay, so what if we take away the "indestructibility" part?

Over the last couple hundred years or so, science - specifically, medical science - has advanced to heights undreamed-of by our ancestors of two hundred years ago. We're living, on average, twice as long (though much of that has to do with a reduction in infant mortality, which brings the average up, but that's not important right now). Notably, the discovery/invention of antibiotics wiped out whole categories of "stuff that kills you." I, personally, would have died at least twice over were it not for antibiotics (and a few more times without other advancements such as cardiac stents). This led people to start thinking, "What if we could wipe out, say, cancer, too?"

Well, that's a harder problem, but we're working on it. Thing is... well, what's the number one cause of death right now? Depends on age group, but it's either cancer or heart disease. Call it cancer. So we wipe out cancer - then there's still a number one cause of death: heart disease. Now say we get a global fix for that. Something else would become the number one cause of death. Alzheimer's, maybe; there's some evidence that everyone will get Alzheimer's if they live long enough. Okay, so that becomes the priority, and we find a way to cure that, and then something else becomes #1... and so on... you see where I'm going with this. And that's not even counting the completely predictable, but somehow not predicted, evolution of bacteria to become antibiotic-resistant.

Eliminate disease in general, and the #1 cause of death becomes accident. I think I read somewhere that if this were possible, if we could wipe out all disease (not likely given the surge of anti-vaccination sentiment), we'd still have an average age at death of something like 200 years - because accidents happen.

Don't quote that number as gospel, because I pulled it out of my ass, but let's use it for the sake of argument. The "average" means that some people will live beyond that, while others will not reach 200. Okay, so maybe we also advance technology to the point where what are now fatal accidents can be nonfatal. The number then increases.

Still, it doesn't increase to infinity, which is what "forever" would mean.

Okay, and I still haven't gotten into the social or environmental chaos that could result from age-increasing technologies.

Social change happens because generations swap out. Imagine we already lived 200 years on average. Then, there would be people alive today who were alive during the American Civil War. Lots of them. Some people from the Revolutionary War, less than 250 years ago, would still be alive. Given the prevalent mindsets through that "four score and seven year" time frame, do you think we'd have made the advancements we did on, say, civil rights? What future advancements, in society or even in technology, would we have to forego if all the old-school farts, set in their ways, including those with real power and the means to keep it, stuck around beyond their natural spans?

As to the environment, we'd be completely screwed. 7.5 billion people fart and belch on the planet today. I've heard arguments that that's too many; I've heard other arguments that we can handle more. But could we handle three times as many? Because that's what would happen if the average death age shifted from about 70 to about 200. Or maybe it wouldn't, as per Malthus, and then we're back to lowering the average death age again.

Okay, so maybe we limit births. Yeah... like that kind of totalitarianism would fly in most of the world. China tried it, as you know, and they're still feeling the unintended consequences.

The more you think about this sort of thing, the more rabbit-holes you fall into. I know, because I have thought about it; I write science fiction as well.

Safety valves like migrating to other planets? Might work, for a time, but then you get resentment.

Uploading consciousness into computers? Hell, who knows what that would really be like, if it's even possible as the transhumanists like to claim - do you honestly think they'd program in simple pleasures like the relief you feel after you take a shit?

Much has been made of the distinction between natural, unnatural, and supernatural. I, for one, reject the last one (though it's a mildly amusing TV show), and the other two have a blurry line between them - it wasn't that long ago that a large number of people thought interracial marriage was "unnatural," and don't even get me started on homosexuality being labeled the same. "This food is all-natural!" Yeah... tell that to Socrates. But death? Death is natural. I'm not saying we shouldn't do everything we can to extend active years for people, but immortality? Let's leave that in the realm of science fiction... and the supernatural.

Of course, that's not going to stop people from trying, and maybe one of them will succeed... with my luck, it'll happen about 12 seconds after I kick it. And that would really suck: being the last human to die.

The only consolation is I probably wouldn't know it.

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