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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QPdoll

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With all that walking, I hope you stayed hydrated.

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WakeUpAndLive️~Happiness

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Elisa: Middle Aged Stik

a day ago
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WakeUpAndLive️~Happiness

a day ago
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WakeUpAndLive️~Happiness

a day ago
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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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October 11, 2023 at 11:24am

#1057187

The Old Tongue

Seems to me the world's oldest language wouldn't have needed a name, because there was only one of them.

What’s the World’s Oldest Language?
Debate rages over which languages can claim to have the earliest origin

I suppose it's possible that, as with genetics, all languages might trace their origins back to one first language, but I don't know enough about the field to even speculate that.

The globe hums with thousands of languages. But when did humans first lay out a structured system to communicate, one that was distinct to a particular area?

Well, so much for the "only one of them" comment above. My next doubt is that the "structured system" was in any way planned. How do you plan without language to describe things?

Scientists are aware of more than 7,100 languages in use today.

And no, they didn't originate at the Tower of Babel. That's an origin myth.

Alternatively, if we assume that most languages can be traced back to an original, universal human language, all languages are equally old. “You know that your parents spoke a language, and their parents spoke a language, and so forth. So intuitively, you’d imagine that all languages were born from a single origin,” Hieber says.

Okay, so even linguists don't know that for sure.

But it’s impossible to prove the existence of a proto-human language—the hypothetical direct ancestor of every language in the world. Accordingly, some linguists argue that the designation of the “oldest language” should belong to one with a well-established written record.

I'd be careful calling something impossible. That tends to bite people on the ass. Still, some things are impossible.

More speculation from me: language was probably a thing for a very long time before any system of writing was developed.

Among these languages are Sumerian and Akkadian, both dating back at least 4,600 years.

And humans have been around for, depending on who you talk to, maybe 300,000 years ago. That's a lot of time for languages to evolve into Sumerian and thus be written down.

Part of the problem of pinning down exact origin points is that with evolution, whether biological or linguistic (there are parallels, though they're not the same process), there usually isn't just one thing you can hold up and say "this is the moment when x ended and y started." It's a gradual process.

As for the oldest language that is still spoken, several contenders emerge. Hebrew and Arabic stand out among such languages for having timelines that linguists can reasonably trace, according to Hieber.

According to Kabbalistic mystics, it was definitely Hebrew. This, too, is an origin myth.

Bowern adds Chinese to the list of candidates. The language likely emerged from Proto-Sino-Tibetan, which is also an ancestor to Burmese and the Tibetan languages, around 4,500 years ago, although the exact date is disputed.

I knew it was going to have to be in there somewhere.

Deven Patel, a professor of South Asia studies at the University of Pennsylvania, says the earliest written records of Sanskrit are ancient Hindu texts that were composed between 1500 and 1200 B.C.E. and are part of the Vedas, a collection of religious works from ancient India. “In my view, Sanskrit is the oldest continuous language tradition, meaning it’s still producing literature and people speak it, although it’s not a first language in the modern era,” Patel says.

I'm going to go out on a limb here and guess, based on the name, that Deven Patel's ancestry is South Asian. As with the Kabbalists I mentioned above, of course one would be inclined to make the case for one's own language.

That, of course, doesn't mean he's wrong.

Disagreements about the age of Sanskrit and Tamil illustrate the broader issues in pinpointing the world’s oldest language. “To answer this question, we’ve seen people create new histories, which are as much political as they are scientific,” Patel says. “There are bragging rights associated with being the oldest and still evolving language.”

In theory, science, including linguistics, should be non-tribal; you work from evidence, not cultural assumptions. In practice, naturally, it's done by humans, with all of their attendant biases (one of which is always going to be linguistic).

This, incidentally, is why it's essential to seek diversity in the pursuit of science. Biases should, in theory, cancel each other out.

Anyway. So they're not after the "oldest language" at all, and that's probably beyond our capabilities unless we invent a time machine. "Oldest still evolving language" makes more sense to pursue, though the nature of linguistic evolution makes it difficult—consider how unintelligible the English of Beowulf is to modern English speakers, and yet it's still called English.

In any case, I really, really hope there's more to the pursuit than mere "bragging rights."

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