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

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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

October 22, 2020 at 12:06am

#996443

Big Brother

So, I lied.

Big gods came after the rise of civilisations, not before, finds study using huge historical database

I said yesterday that I was done with religion for now, but then this came up at random. (Note: this article uses British spellings. I will try to stick to American in my own commentary.)

When you think of religion, you probably think of a god who rewards the good and punishes the wicked.

If only.

But the idea of morally concerned gods is by no means universal. Social scientists have long known that small-scale traditional societies – the kind missionaries used to dismiss as “pagan” – envisaged a spirit world that cared little about the morality of human behaviour.

That's probably because, as the article theorizes later, in a smaller society, everyone knows everyone else, so anyone who transgresses against social norms is known to do so; they lack the protection of anonymity. Unlike today, when anyone can get on the internet and become a troll.

I should also point out that I read this article as someone who understands that humans created their gods, rather than vice-versa.

Nevertheless, the world religions we know today, and their myriad variants, either demand belief in all-seeing punitive deities or at least postulate some kind of broader mechanism – such as karma – for rewarding the virtuous and punishing the wicked. In recent years, researchers have debated how and why these moralising religions came into being.

The whole "good are rewarded while evil people are punished" thing falls apart in those religions that stress faith alone as the key to salvation. You can be a mass murderer or a child rapist, and if you repent on your deathbed, according to some theologies, you still get the wings and a harp thing. But, in general, at least in the way things are interpreted for the average person, I'll concede the point.

Now, thanks to our massive new database of world history, known as Seshat (named after the Egyptian goddess of record keeping), we’re starting to get some answers.

Which is great and all, except that one wonders about the inputs. Your conclusions are only as good as your observations.

The database uses a sample of the world’s historical societies, going back in a continuous time series up to 10,000 years before the present, to analyse hundreds of variables relating to social complexity, religion, warfare, agriculture and other features of human culture and society that vary over time and space. Now that the database is finally ready for analysis, we are poised to test a long list of theories about global history.

Again, this says nothing about the quality of the data itself.

One of the earliest questions we’re testing is whether morally concerned deities drove the rise of complex societies.

I would put this differently, inserting "a belief in" between "whether" and "morally." But I got the idea.

In other words, gods who care about whether we are good or bad did not drive the initial rise of civilisations – but came later.

This makes more sense than the other way around, really. Sometimes you build something and only later realize that it needs a security system.

We are now looking to other factors that may have driven the rise of the first large civilisation.

Pretty sure it was beer. That, or a desire for mutual protection.

If the original function of moralising gods in world history was to hold together fragile, ethnically diverse coalitions, what might declining belief in such deities mean for the future of societies today? Could modern secularisation, for example, contribute to the unravelling of efforts to cooperate regionally – such as the European Union? If beliefs in big gods decline, what will that mean for cooperation across ethnic groups in the face of migration, warfare, or the spread of xenophobia? Can the functions of moralising gods simply be replaced by other forms of surveillance?

We already have such systems in place, but it's an open question whether they're as useful as religion in controlling peoples' behavior. Think star ratings on Uber, or credit scores, or the Chinese social score.

People game those systems just like they game religion, but I'm not sure to what extent. Like, every once in a while, in my travels, I'll see a sign for a hotel or restaurant or other service that displays an icthys, you know, the fish symbol. Always, without fail, those that do so are more greedy, underhanded, or hateful, to the point where I've quit patronizing those establishments. One hotel I went to turned away a minority family, blatantly and obviously racist. The next people in line, a white couple, noted that "those people" "always bring in their friends and trash the place." The desk clerk nodded knowingly. Yes, I turned around and found another place to stay.

There are none so evil as those who believe themselves to be righteous.

Perhaps I'm also being prejudiced in avoiding such places based on a few bad experiences, but so be it. I don't claim to be righteous. Point is, even with their "eye in the sky," some of them still act like greasy tools.

I've seen it asked before: "If you don't believe in God, what keeps you from murdering or stealing?" Well, heck, if belief in God is the only thing keeping you from doing these things, or worse, then please, for gods' sake, keep believing. I'll just be over here quietly drinking a beer and not murdering anyone.

There's a landmine of an argument I'm not prepared or willing to engage in right now, concerning exactly what is good and what is evil. I've been thinking about it a bit lately, but haven't come to any firm conclusions. Obviously, I don't believe in those concepts as external forces; I'm thinking in terms of human behavior. "I know it when I see it" just doesn't cut it, but I'm not sure what does. But it's clear to me that the kind of religion this article speaks of, with the rewarding / punishing deity, doesn't make much difference.

Perhaps actual surveillance, and the threat of purely human retribution, does.

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