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🎶

about 22 hours 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').

1 Like

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

November 8, 2024 at 9:10am

#1079661

Bungalow Bash

Today, we have an exercise in metaphor-stretching:

Caveat Bungalow
The ostracizing of normal

This starts out looking like a piece about housing:

When I was a kid, I lived in a bungalow for a while. You know, one of those houses that’s compact and square and has a half-upper floor that’s basically just a loft with sloped ceilings.

Yeah, well, the house I lived in as a kid was originally a shotgun shack.

Now if someone told me to live in a loft with sloped ceilings I’d ask if there was any regular house available, please.

Nice to be able to be picky, isn't it?

Something else I’ve noticed that’s shaped like a bungalow is the way we talk to each other. Human speech.

And that's the metaphor part, which, as it turns out, is what the post is really about.

We waste an unbelievable amount of time — in our daily lives, on podcasts, in interviews, in blogs and articles, even in Tweets and Notes for God’s sakes — qualifying everything we say with caveats. We say “now I know not all x are y, and I know that historically abc, and I’m not trying to say that lmnop so please don’t take this wrong…”

Granted, I avoid X/Twatter like the plague it is, and I don't listen to podcasts. But while I've seen what this dude's talking about, and even engaged in it on occasion (for instance, when I note that I might be talking about something with a US bias), I don't think disclaimers like that rise to the level of wasting "an unbelievable amount of time."

Saying anything even remotely controversial on the internet is terrifying.

Is it though? Is it more terrifying than saying it in a crowd of Americans with guns?

No matter what argument you make on the internet, you will get people who reject it wholesale because you forgot an asterisk. Because you forgot to mention their particular edge case.

Yes, and?

We all know that not all heterosexual dating advice or sex advice applies to transgender people. It doesn’t even apply to all straight people. We are all intelligent enough to know that.

[citation needed] on the "intelligent" bit.

People have managed to make it insensitive to speak about things that the majority of people deal with. People have managed to make it insensitive to be normal.

And what, exactly, is "normal?"

And then the tyrannical minority ostracizes you for it, and in turn makes it okay for everyone else to ostracize you.

"Tyrannical minority?" Loud, sometimes, maybe, but that seems like an oxymoron. Unless the "minority" is an actual, political tyrant, a minority of one.

If you drove down a street consisting entirely of overly-decorated bungalows, with nice upper windows and big, furnished porches, you’d call bullshit on the entire street.

No, I don't think I would.

Everywhere I look, I see people decorating their speech with nuance when all they really want to say is some simple, normal thing. It feels like bullshit.

Well, it's not everywhere I look. Perhaps examine your own biases, first?

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.

Some people are just actively searchlighting for reasons to get outraged. They aren’t worth listening to.

On that, we can agree.

People who need that much nuance weren’t going to learn from your argument anyway.

Aaaannnd you've lost me again.

As I've said before — we as a culture have become profoundly unserious.

To me, the opposite of "serious" is "funny." So the opposite of "unserious" would be "unfunny." Me? I'd rather be funny.

Maybe if we just start saying what we mean and placing the impetus on the reader to read nuance into the topic, we’ll all grow up a little.

Well, sure, calling me childish certainly helps your argument.

People are tired of having to decorate their speech to make it marketable.

I think by "people," he means "I." As in him, not me.

I think we’re past peak Woke, and I think part of what that means is that we’re past peak not-being-able-to-speak-like-adults.

Ah, there it is.

I'm not dismissing his argument, mind you. I read the whole thing, top to bottom (it's really not that long). I simply don't agree with most of it, though I accept that my opinion could change.

If you're going to do a metaphor, make sure it's one that we can actually relate to. Bungalows may not be exotic or ritzy, but they're generally better than no home at all.

So that's me, saying what I mean.

4 Like • Comment • View 2 comments

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