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 14 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 15 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.

April 21, 2022 at 12:01am

#1031096

The Forest for the Trees

Here's another one for "Journalistic Intentions"

[18+]

Large Tree Planting Projects

"They're going to make us dinner."

So my cleric said to the party upon encountering a group of trolls tonight.

Sure, I suppose it was within the realm of possibility for the DM to have planned for the trolls to cook a nice venison roast and serve it to us. More likely, though, they were going to turn us into the main course.

The way English works, though, the sentence is indeed ambiguous. Is "us" the direct or indirect object of "make?" Other languages have no such ambiguity. "They're going to make dinner for us" is distinct from "They're going to make us for dinner."

So, coming off a D&D high and pulling this prompt at random from the list, initially I read it as (Large Tree) (Planting Projects). So I imagined much bigger versions of the trolls (they were big to begin with) running around and planting large trees everywhere. But it could also mean that people are planting little trees that will eventually grow up to be large trees. As opposed to, I dunno, dogwoods, which never get too huge.

One could, I suppose, also read it as (Large Tree Planting) Projects, which, I don't know, could be read to imply that the trees are the things doing the planting? Though I guess you could do that with the first example, too. These two permutations aren't really all that different.

Then there's the interpretation that, I assume, was actually intended: Large (Tree Planting Projects).

This one implies that the tree planting projects are large in scope. In other words, "large" is actually modifying "projects" as opposed to "tree."

That would make more sense, wouldn't it? A small project to plant trees wouldn't be very effective. It would be like if one person in the world recycled, and no one else did. Maybe you could initiate numerous such projects, but that would involve getting people who don't give a shit to give a shit, which ain't gonna happen.

According to sources whittled down into Wikipedia

, "In the 12,000 years since the start of human agriculture, the number of trees worldwide has decreased by 46%." Also, "...about 15 billion trees are cut down annually and about 5 billion are planted." But that last quote doesn't tell me anything about tree life cycles. With or without us, trees get seeded, grow, reproduce, and die.

I almost typed "natural tree life cycles" there, and then I remembered a) beavers and b) anything we, as part of nature, do is also part of nature.

But taking those numbers at face value, that's a net loss of 10 billion trees a year. Which certainly seems like a lot -- it exceeds the human population of the planet -- but according to the same source, there's probably about 3 trillion trees. That's... oh, hell, I'm entirely too wiped tonight to do the math. Let's just say that we're in no danger of running out of trees anytime soon.

Which doesn't mean, of course, that any net loss is a good thing. And I get the impression that it's not just the number, but the distribution. Deforesting the Amazon rainforest probably has a much bigger negative impact than, say, losing them more or less evenly all over the globe.

That would, of course, imply that it matters where any of these large tree planting projects are taking place. Which may be obvious, because, for example, you can plant all the trees you want in the Sahara; they won't grow.

So whatever these projects are, assuming I parsed the phrase correctly, I hope they're successful.

My D&D party was, by the way. Successful, I mean. It took a while, but we eventually dispatched the trolls. Now if only we could do that with the ones on the internet...

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