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 3 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 5 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.

September 13, 2020 at 12:50am

#993223

Infinite Diversity in Infinite Combinations

Today's 30DBC prompt is delayed, and I would like to get back to what I do best (drinking), so I'm just going to rant a bit until I see the prompt.

So my thing these days has been going back through episodes of Star Trek that, in some cases, I haven't seen in decades. Right now, that means the Animated Series. I did this sort of thing a few years ago with Doctor Who, which has been around slightly longer, so some of my observations still apply.

This is not just for funsies, either. A writer needs to... well, write, first and foremost, but also read and watch what other writers have done. Trek has lasted as long as it has for reasons, and I'm trying to tease out what these reasons are.

It certainly wasn't consistent quality. I mean, yes, absolutely, some of the episodes of the original series, and even some of the animated episodes, were excellent.

Most were not.

And yet, it's still around.

What I can figure from this so far is: character matters. That is, maybe you have crap stories, but as long as you keep your characters true to themselves, those can be sort of forgotten.

Also, worldbuilding matters. In the case of Trek, it's not just one world but an entire galaxy's worth. DW, on the other hand, spans "all of time and space," and hoo boy I can't imagine trying to plan all of that, or even keep it consistent over nearly 60 years.

The writers solve this by not worrying too much about consistency. After all, when you indulge in time fuckery, continuity can go right out the window.

Anyway. Just some random thoughts until the prompt shows up. Which just happened.

PROMPT September 13th

Write about friendships that are more like family. What makes a great friend?

Okay, this is one of my hot-button topics. I'd tell you to buckle your seatbelts, but neither the Enterprise nor the TARDIS has those.

You've heard the expression "blood is thicker than water." Most people, I think, take that to mean something equivalent to "family first." Specifically, "genetic relatives first."

This is bullshit.

First of all, the full expression, as noted by this site

and others, is: “The blood of the covenant is thicker than the water of the womb.” That particular link goes on to explain: "This actually means that blood shed in battle bonds soldiers more strongly than simple genetics. Although we commonly use it to suggest the strength of family ties, it doesn’t refer to family at all."

I'll leave it to more religious readers to specify chapter and verse. I can't be arsed. Somewhere in the NT. But whatever the source, the point is, we get it wrong.

The site I linked gets it wrong, too. "It doesn't refer to family at all" is patently untrue. It does refer to family - and rejects family in favor of those with whom we share other bonds.

So, when it comes to "friendships that are more like family," I take exception -- friendships, those relationships that we choose, are far more important, in my opinion, than accidents of birth. This is especially true when one's family is poisonous. I'm not saying mine was, but I see so many people with not just issues, but entire collections of National Geographic, who refuse to break ties with their relatives because "but they're family."

So what?

Which is not to say that you should only hang out with people who agree with you. No, you should only hang out with people who genuinely care about you -- and for that, DNA is irrelevant. A good friend won't always agree with you, nor should they -- and vice versa. A good friend will stick with you when times are bad, do stuff with you when times are good, and, most importantly, accept you for who you are while at the same time supporting you should you want to change.

Family can do all that too, sure, but they don't always.

I've mentioned before that I was adopted as an infant. My parents -- the people who raised me -- did the best they could, supported me almost to a fault. As a result, I've never had more than passing curiosity about my birth parents. No need; one set of family drama is more than enough, thank you.

Ironically, the latest incarnation of Doctor Who, which I mentioned above, carries this idea. I say ironically, because I mentioned it before I got the prompt, but it ended up fitting so well. And Trek? Well, Trek has always been about the bonds between people who found themselves on a starship together, family notwithstanding.

To put family on a pedestal, to buy into that myth that "blood is thicker than water," is to hold the wide range of diversity of culture, expression, ability, and opinion as a lower priority than the accidental bonds of close kinship, and I think that might be one of the things wrong with the world today.

As the Vulcans say, "infinite diversity in infinite combinations." Sure, it's a fictional alien race in a fictional SF series that was sometimes really, really cringeworthy, but what is fiction if not aspirational? An actual, living human being came up with that idea, and I assert that it's a good one.

Your opinion, of course, may be different. But that's part of diversity.

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