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.

2 Like

July 11, 2023 at 9:47am

#1052470

If Only Things Were Different

I would accompany this article from The Conversation with a significant amount of sodium chloride, but it's still interesting speculation.

8 billion people: how different the world would look if Neanderthals had prevailed

In evolutionary terms, the human population has rocketed in seconds. The news that it has now reached 8 billion seems inexplicable when you think about our history.

No, it seems inevitable when you think about our history.

There's a graph at the link showing population over time. Unsurprisingly, it's a classic exponential hockey-stick.

Neanderthals were more inclined to stay in their family groups and were warier of new people. If they had outcompeted our own species (Homo sapiens), the density of population would likely be far lower.

For homework, explain why this is a fallacious argument.

Hell, I'll save you the trouble: to outcompete sapiens, they would have had to be more social, and thus increase their population density.

But okay, let's play pretend.

The reasons for our dramatic population growth may lie in the early days of Homo sapiens more than 100,000 years ago. Genetic and anatomical differences between us and extinct species such as Neanderthals made us more similar to domesticated animal species. Large herds of cows, for example, can better tolerate the stress of living in a small space together than their wild ancestors who lived in small groups, spaced apart. These genetic differences changed our attitudes to people outside our own group. We became more tolerant.

I know this is going to raise objections from the down-on-humans brigade, but remember, "more tolerant" doesn't have to mean "very tolerant." We still have in-groups and out-groups.

The article then goes into what it says in the headline, speculating (based on data, not wild guesses) how things might be different. I don't find this particular speculation useful, but it still sheds light on the differences between the two human species... assuming, of course, that inferences of their behavior based on archaeology and other disciplines are correct.

As an aside, "species" can be a fuzzy concept. Usually it's meant to describe non-interbreeding populations. But, clearly, sometimes sapiens and neanderthalensis did interbreed. The whole designation thing is over my head, but for general purposes, calling them different species is good enough for communication; I'll let biologists hammer out the details.

The more technology humankind develops, the more our use of it harms the planet. Intensive farming is draining our soils of nutrients, overfishing is wrecking the seas, and the greenhouse gases we release when we produce the products we now rely on are driving extreme weather.

And yet, despite discussing both population and technology, the article stops short of what, to me, is a blindingly obvious connection there.

The vast majority of human population increase has occurred over the last 200 years, roughly coinciding with the industrial revolution. But that's not a coincidence. Advances in science have increased average life spans and reduced infant mortality, while at the same time fouling the environment. On the surface, reduced mortality seems like a good thing, but the increase in population requires extraction and processing of ever more resources, feeding back onto the "fouling the environment" part. "Intensive farming" is needed to feed the 8+ billion. So is "overfishing." It's not "the more technology humankind develops, the more our use of it harms the planet;" it's "the more people there are, the more we need that technology to survive and make more people."

So far, the benefits of technology have outweighed the downsides. But I doubt that will persist.

I feel like the article veers way too close to romanticizing the Neanderthal, kind of a "noble savage" conceit. Sure, maybe the ecosystem would be in better shape had they been more successful, from an evolution perspective, than we were, but that's like speculating about whether dinosaurs would have ended up building rockets if that asteroid hadn't fucked things up for them.

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