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.

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').

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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.

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

October 30, 2020 at 12:14am

#997135

All The Long Years

Do you want to live forever?

What if We Could Live for a Million Years?
Vastly extended life spans would bring dazzling opportunities—and daunting risks

Well, a million years isn't forever. Not even close. Not even in the same ballpark. Not even the same sport.

Recently, scientists discovered bacteria that had been buried beneath the ocean floor for more than a hundred million years and was still alive.

This sounds like openers on a really bad horror novel.

What would change if we could live for even just a million years? Two thoughts immediately come to mind. First, tenure in academia would have to be capped.

Oh, sure, the first thing I think of is academic tenure. Sure. Don't tell me, let me guess. "Abraham (Avi) Loeb is the Frank B. Baird, Jr., Professor of Science at Harvard University."

Quelle surprise.

Second, a birthday cake cannot hold a million candles.

And sure, that's the second thing I'd think of, too. Uh huh.

With advances in bioscience and technology, one can imagine a post-COVID-19 future when most diseases are cured and our life span will increase substantially.

Science fiction has imagined such things for at least as long as I've been alive, which is not even an eyeblink compared to a million years.

Given the luxury of pursuing longer-term plans, we could accomplish more ambitious tasks. We could decide to care more about our planetary environment and interpersonal cooperation, since pollution or hostilities carry long-term dangers.

Nah. We'd just find ways to make sex kinkier.

But even with shrewd strategies, survival is by no means guaranteed. For example, the known correlation between brain size and body weight did not make dinosaurs smart enough to deflect the asteroid that killed them.

That's... what? Loeb is an astronomer, not a biologist, and I'm neither, but while brain size may or may not be statistically correlated with body weight, it doesn't seem to be correlated with those qualities that we deem "intelligence." Octopuses are intelligent, though not technological, and don't even have the same kind of brain that we do. Or that dolphins do.

Increasing our fertility period in proportion to our life span will bring the risk of overpopulating Earth.

Do you want to tell him, or should I?

Alternatively, travel ports could launch people into space to balance the birth rate and maintain a terrestrial population suitable for the available supply of food and energy.

More science fiction. First, we have to find places for them to live, and we still haven't found such places. It's entirely possible that we could create them, but that's science fiction right now, too. (I'm not ragging on SF. I love SF. But I have some idea about what's plausible and what isn't.)

The good news is that over a lifetime as long as a million years, space travel can take us to the nearest stars using existing chemical rockets. It would take merely 100,000 years to arrive at the habitable planet around Proxima Centauri with a space vehicle that travels at the speed of NASA’s New Horizons spacecraft.

First off, habitable? Perhaps he knows something I don't; after all, like I said, his specialty is astronomy. Last I heard, there is indeed a planet (or possibly a number of planets) in the presumed habitable zone around PC, but merely being in a habitable zone of a star doesn't make a planet habitable by humans, or give it compatible biology if there is life there, which is still an open question. Venus and Mars are both technically in the Sun's habitable zone, and they'd require massive terraforming efforts.

Second, the logisitical hurdles involved in crafting a livable spaceship that will last 100K years, or even 50K or 10K (assuming we could increase the speed, which isn't out of the question), are formidable. Could we overcome them? Probably. Would you like to live inside a spinning spaceship for thousands of years? I wouldn't. All this changes if we manage to invent warp drive, of course.

And the passengers will have to maintain a stable mindset for their journey’s goal and not lose faith, like a fisherman who, after a long hiatus without finding any fish, asks whether “the real purpose of fishing is catching fish.”

Third, if I spend 100,000 years on a massive, self-contained, self-sustaining spaceship, I'd be sorely tempted to stay there rather than risking the unknowns of a planet that may or may not be compatible with human life.

What does a mature technological civilization look like after such a long time? Can it survive the destructive forces that its technologies unleash? One way to find out is to search for technosignatures of alien civilizations, dead or alive. Inevitably, all forms of life eventually disappear. The universe cools as it expands, and all stars will die 10 trillion years from now. In the distant future, everything will freeze; there will be no energy left to support life.

Bypassing the discussion of the possibility of finding technologically sophisticated alien life -- I've harped on that nonsense before -- even if we find some, there's absolutely no guarantee that it will illuminate our own path. Alien life is, by definition, alien, not large men in forehead prosthetics growling at each other in Klingon.

In principle, one could imagine a life that lasts a billion years, during which stars turn on and off in the sky just like light bulbs. Against the backdrop of that long-term perspective, our current concerns about the world would seem as naive as the first thought in the head of a newborn baby.

Would our sense of time continue to alter? A year in your 50s is noticeably shorter than a year in your teens. This says nothing profound about time itself; it's either a result of a year being a proportionally shorter fraction of one's lived life, or related to the idea that new experiences tend to stretch our perception of time and we have fewer new experiences as we get older. Or a bit of both. Either way, it's not like we'd probably experience a day during one's 100,000th year of life as we experience a minute now.

In any case, this is all pure speculation -- which, don't mistake me, is a good thing. Consider the most common causes of death

here in the US:

Heart disease: 655,381
Cancer: 599,274
Accidents (unintentional injuries): 167,127
Chronic lower respiratory diseases: 159,486
Stroke (cerebrovascular diseases): 147,810
Alzheimer’s disease: 122,019
Diabetes: 84,946
Influenza and pneumonia: 59,120
Nephritis, nephrotic syndrome, and nephrosis: 51,386
Intentional self-harm (suicide): 48,344

In the past, this list would have been different. Childbirth might have been on it, for example. Or measles or something. We've reduced the impact of those things, letting us live long enough so that heart disease and cancer could climb to the top spots.

So, say we somehow eliminate heart disease, preferably without forcing everyone to adopt a plant-based diet and never smoke. Then you have cancer in the top spot. Okay, we've made great strides against cancer, and I can believe that if we last long enough as a society, we might be able to do something about it.

That might put "accidents" as the leading cause of death. And I don't see how to fully eliminate those, not without taking away some of the things that make life worth living. Driving, to name one common cause of accidental death. We each have a 1/100 chance, or thereabouts

, of dying as a result of an automobile accident over the course of our current lifespan. Sure, that's lower than it used to be, and could probably go lower, but the only way to eliminate it entirely is to ban cars (self-driving cars might increase those odds, but no transportation system is perfectly safe). Point being, and I can't be arsed to do the probabilistic calculation here, but all else being equal, people living in a hypothetical world free of disease have a really fucking slim chance of making it to 1000, let alone 1,000,000.

Of course, all else would not be equal, but the point is there's no way to avoid death from accident entirely. Hell, people have died stepping out of the shower at home, or falling down stairs, so even staying home has its risks -- risks that are cumulative year over year, making it extraordinarily unlikely that, even having eliminated disease, anyone would make it to a million years old.

Oh, sure, one can imagine technologies that mitigate those risks. But attempting to eliminate them entirely reaches a point of diminishing returns.

So while these things are fun to think about, let's not take the ideas too seriously. We're a little more complex than bacteria.

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