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

March 8, 2023 at 8:57am

#1046080

Behave

Time now for another prompt from "JAFBG"

[XGC]

There are many influences on a person's behaviour and personality - their upbringing, stress, mental health issues, pain, etc. At what point do you feel a person has to take responsibility for their own behaviour?

One might be tempted to conclude that someone, like myself, who doesn't accept the idea of free will, would think that the logical conclusion is that none of us are responsible for what we do.

This is arrant nonsense. Not that we don't have free will—that debate won't be settled anytime soon—but that we're not responsible.

The way I look at it, what we do have is knowledge of potential consequences. As a simplistic example, I like to drink, and I like to drive, but I never combine the two because I know that a) it's hazardous to other, uninvolved (some might say "innocent," but that's misleading) people, and b) there are severe penalties for it.

Lack of free will is no excuse for behaving badly. It's not a ticket to licentiousness or a justification for choosing that pizza over a big steaming plate of broccoli. (It also doesn't mean that our actions are predictable, any more than you can predict where an individual raindrop will fall during a storm. But that's irrelevant to the prompt.)

It doesn't mean we can be irresponsible or avoid facing consequences. The awareness of potential consequences is baked into the background of every choice we seem to make. In other words, just because we could not, in hindsight, have taken a different course, doesn't mean that we can't strive to do better in the future, with updated experience and knowledge.

That said, there are people who are genuinely unaware of consequences: the very young, for example, or the mentally ill. Where those lines are drawn can be (and often is) debated. So you ask me: at what point does a person have to take responsibility for their own behavior? Well, at all points... with exceptions. Even the US legal system recognizes that; you're not fully responsible until you're 18, and you can be declared incompetent even as an adult. The default, though, is that every person is responsible once they pass that magical milestone. Except, of course, when it comes to drinking; then, it's 21.

We've also decided, as a society, that self-inflicted disinhibition doesn't absolve one of responsibility; see, for example, my drinking and driving note above. That extends, in my view, to acting like an asshole because "that's just who I am."

So, to sum up, I don't fucking care what your excuse is; if you know it's wrong, don't do it. And if you do, accept the consequences.

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