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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 9 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 11 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

February 3, 2022 at 12:03am

#1025928

Emotion Notion

Some of my entries this month, including this one, will be based on prompts from "Journalistic Intentions"

[18+]. As usual, I'm picking them at random.

The prompts for that activity are all quotes this round. My goal is to do these entries without looking up the source of the quote until after I'm done, because these are meant to be "blind quotes." We'll see if I can avoid the temptation, because sometimes I want context.

And thus, the first entry for JI:

"And it hit me, that to me, those are two of my deepest-felt emotions. Justice, equality, fairness, mercy, longsuffering, Work, Passion, knowledge, and above all else, Truth. Those are my primary emotions."

Fortunately, math isn't an emotion. I count nine there, not two.

I realize most people don't consider those things to be emotions. I have it on good authority, though, that some people indeed feel them as emotions, which seems a bit odd to me, but then, most things involving emotion do. So I simply accept that yes, for some people those are emotions.

Which is something I think more people might try to understand: that other people feel emotions differently than they do. It's not exactly empathy, because I'm not asking anyone to feel what they feel, but just to accept that they do feel it.

Maybe it's a bit like being blind. As I was losing my eyesight last year, I naturally started to think about what it might be like to be blind in a society that's so visually-oriented. So you're blind, and someone's describing something to you visually... you can't see it directly, but you can accept that they see what they see, right? And in a sense (pun intended, as always), we're all blind when it comes to others' emotions. We can sometimes read it in their faces or their body language, a certain stress in their voice, but to get a better idea of what they're feeling, they have to tell us.

Many's the time I've been stressed, and people ask me "what are you angry about?" Well, I wasn't angry until you misread my emotional state; I guess I'm just one of those people who project anger when stressed. It doesn't help that I don't really smile; I physically can't show my teeth when I'm grinning, the way other people do. (At least this helps when I play poker, and when I face apex predators in the forest.) (Just kidding; I don't do forests.)

Worse, though, is when I'm depressed. I mean, I guess depression is an emotion, as it's a mental state associated with certain internal feelings Although again, different people process it differently -- some cry, some can't get out of bed, some experience anhedonia (which is something I've never had; even when I'm depressed, I enjoy certain things, such as beer). What I'm getting at, though, is if I bother to tell someone I'm depressed, I get, "What are you depressed about?" Grr. Now I'm annoyed AND depressed, because it's not about anything; it's a brain chemical neuroreceptor thing or whatever. Worst still is when I get, "Are you thinking about suicide?" No, but now I'm thinking about homicide. (Not seriously, though.)

Just to be clear, the only times I've ever "thought about" suicide, it's been in the context of fiction writing, much as the writer of a murder mystery "thinks about" murder.

The point, insofar as there is one, is that the only thing we can really rely on when it comes to assessing someone else's emotional state is a) what they tell us and b) if they're being honest about it. And I tend to default to "yes" for part (b) there unless I have some reason to think they're lying.

Consequently, if someone tells you that the emotion they're feeling right now is one of those nine, fucking believe them.

Now, I don't feel any of those things as emotions. Well, I guess "passion" is the exception there (or used to be, back when I had shits to give), but the other eight are what are, for me, abstract principles or, as in the case of "work," activities.

Most of them are things I strive for. There's no external source of justice, so I try to be just and I feel satisfaction when someone acts justly. Similarly, there's no fairness, so it's up to me to act as fairly as I can, which also makes me feel good. And so on. The exception for me is work, to which I'm strongly allergic. But regardless, most of these ideas can create emotion in me -- anger when I see injustice or inequality, for example, or happiness when I find some new knowledge or achieve some epiphany of truth -- but I don't feel them directly as emotions. Again, though, I accept that others do.

Maybe it's sort of like synesthesia, you know, the people who can hear colors or smell numbers. Just different neuronal firing in their brains.

As an aside, I have to wonder why some of those qualities are capitalized, and others aren't. This might be a matter of context, and like I said, I'm just going by the provided quote, rather than trying to find the source.

Anyway, I feel that to be... you know... fair, I should present a contrary view, so here it is.

New research based on brain scans shows that people who care about justice are swayed more by reason than by emotion.

All I can say to rebut that, not being an expert in the field, is that it may be true for the majority, but for something like this, we also have to consider individual differences and outliers. What I mean is, I don't think we should be reading an article like that and assuming that it applies to everyone. It's like... if someone did a study about coffee, and the study showed that people like coffee (not an unreasonable result). So you think, "everyone likes coffee!" Well, perhaps most people do, but it doesn't change the fact that I despise the stuff.

With that, I've banged on long enough. And I managed not to look up the quote, so... yay me?

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