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🎶

a day 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.

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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.

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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.

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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.

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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').

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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.

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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.

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October 5, 2023 at 10:15am

#1056798

Recursion (n.) 1. see Recursion

Another one from LitHub today. Another book ad, that is, but what the hell; it's free, relevant, and fairly short.

What Makes Language Human?
Caleb Everett on Syntax and Recursion

There seems to be a need, for some people, to "prove" that we humans aren't really so special. We use tools! Oh, but so do crows. We have language! So do whales. We can solve complex problems! So can octopodes. We're the only technological species! But the Universe is enormous, so there's probably others.

Still, it's good to investigate these things, if only to attempt to push back against tribal superiority complexes.

And we know that articles like “the” should precede nouns, as should prepositions like “of.” These and other patterns, sometimes referred to as “rules” as though they represented inviolable edicts voted on by a committee, help to give English sentences a predictable ordering of words. It is this predictable ordering that is usually referred to when linguists talk about a language’s syntax.

As I understand it, some languages do have inviolable edicts voted on by a committee. L'Académie française, par exemple.

They have a Sisyphean task, but they persevere.

Without syntax, it would seem, statements could not be understood, because they would be transferred from speaker to hearer in a jumbled mess of words. This is, it turns out, a bit of an oversimplification since a number of the world’s languages do not have rule-governed word order to the extent that English does.

I don't remember all that much about Latin, because it's been a very long time since I actively studied it. But one thing I do remember is that word order wasn't nearly as important there. The language is, I guess the adjectival phrase is "highly inflected," where the purpose of a word - whether it's a subject, an object, the target of a preposition, or whatever - is determined more by its suffix than its position in the sentence.

But even Latin evolved over time, and eventually became Italian, French, Spanish, etc., in which word order became important. Like how, in French, some adjectives have to come before the nouns they modify, while others have to come after.

My point being that anyone who thought syntax was universal in human languages probably needed to defend that hypothesis vigorously.

Syntactic conventions can be exceedingly complex, and any given language contains so many of them that linguists have long wondered how individuals can learn them.

Some people never do.

An increasing number of linguists now think this “dictionary and grammar” model of language was misguided. According to them there is no real distinction between words and sentences, as odd as that claim may seem, and no material distinction between a dictionary and grammar.

For full context there, it would be necessary to read the article. Basically, if I'm understanding this correctly, at first, linguists (such as Chomsky) considered words and sentences independently: dictionary and grammar, respectively. This new viewpoint doesn't make that separation. I can't say I fully understand it, but it tracks with the more holistic view other fields of study have pursued more recently.

Chomsky and others suggested that the ability to recursively combine clauses like these is at the core of human speech, implying that it was a key characteristic shared by all human languages. Countless studies have been published on recursive phenomena like embedded clauses, as boring as that might sound.

Boring? Maybe. But so are most fields of study when you drill down into them. Those that pursue the study find it exciting, I'd imagine.

In 2009, linguists Stephen Levinson and Nick Evans pointed out that, judging from the data, syntactic recursion is not actually found in all languages. Part of the evidence they relied on comes from the famous case of the Pirahã, an Amazonian language I have already discussed. My father published a series of papers around fifteen years ago that described the absence of evidence for recursion in Pirahã (among other things), contradicting the claims of Chomsky and others regarding the proposed universality of recursion in the world’s languages.

The problem with any claim of universality is that all it takes is one counterexample to refute.

From this perspective, maybe we just have not come across recursion in the language yet despite the hundreds of hours of recordings. To date, anyhow, no clear evidence for recursion in Pirahã has been offered.

On the other side, it's exceedingly hard to prove the absence of something. We know that cats exist because we've seen them, but we don't know with any real certainty that Bigfoot doesn't exist, because we haven't looked everywhere.

As linguist and syntactician Geoffrey Pullum has noted, part of what was lost in the discussion of Pirahã was the fact that it is not the only language that undermines the notion that recursion is a fundamental feature of syntax.

And that's another point: finding one exception could make that exception an outlier. The original claim may have lost its "universality," but one could still make a good claim for recursion being an essential feature of most languages. Finding other counterexamples, especially unrelated ones, blows that hypothesis out of the water.

Like many claims about universals in human psychology that may be called into question by examining very distinct populations worldwide, claims about universals in syntax tend to face challenges once a truly representative sample of the world’s languages is considered.

As with anything else, it's prudent to examine one's own biases. And that, not the intricacies and minutiae of high-level linguistics, is my real takeaway here. It may not be possible to completely eliminate them; one could consider that a human universal, if one wishes. But we can work towards that ideal.

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