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.

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').

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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.

3 Like

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.

2 Like

July 27, 2023 at 9:37am

#1053111

Basque in Its Glory

I mentioned the Basque language a couple of weeks ago, here: "Can I Have Two Words?"

. As sometimes (rarely, I hope) happens, I made a mistake there. I said, "Basque is, weirdly, unrelated to other Indo-European languages," which gives the impression that the language is itself of the Indo-European family... which it is (probably) not, and I knew that at the time; I just phrased it badly. I meant to point out that it's unrelated to other languages in Europe.

Editing previous blog entries isn't something I do.

The reason I'm issuing that correction at this time has to do with today's random article.

The unsolved mystery of Europe’s oldest language
For linguists, the uniqueness of the Basque language represents an unsolved mystery. For its native speakers, long oppressed, it is a source of pride.

Unless you grew up in Spain, chances are you first learned about Basque Country through Ernest Hemingway’s 1926 novel The Sun Also Rises, which follows a group of Lost Generationers traveling across Western Europe after the First World War.

Oddly enough, no. I didn't read that until college. It was through an in-depth study, in high school, of this Picasso painting,

which merits a mention in the article's second paragraph.

In Basque Country, politics is narrowly intertwined with language. This language, called Euskara, sounds nothing like Spanish (gracias = eskerrik asko), nor does it resemble French.

What I did not know was the name Euskara. So, another mistake in that previous blog entry, by calling the language "Basque."

In fact, Euskara is so fundamentally different from its neighbors that linguists doubt it even originated from Indo-European, the language family that gave rise to Icelandic, Russian, Hindi, and virtually everything in between.

I have to take the linguists' word for it (pun absolutely intended).

Many researchers have taken a crack at the Basque problem over the years, each coming up with a different solution.

Given that region's history, with its historically oppressed minority, I don't think I'd have phrased it as "the Basque problem."

Each researcher coming up with "a different solution" reminds me of the field of economy: ask four economists a question, get six answers.

It has been suggested that Euskara is a predecessor to and a survivor of Iberian, a non-Indo-European tongue spoken on the Iberian Peninsula before the Romans arrived. Euskara has also been linked to a number of languages spoken in the Caucasus, as well to the Saharan Berbers, a pre-Arab ethnic group from northern Africa.

Given my limited knowledge of the spread of humans across Europe, all of those origins seem reasonable to me.

While the independent spirit of the Basques undoubtedly contributed to their isolation, the defining factor seems to have been the geography of Basque Country itself. Protected by the Bay of Biscay and Pyrenees mountains, the rugged terrain wards off outsiders as easily as it prevents insiders from leaving.

Geographic isolation is one of the forces driving species evolution; there's no reason to believe it wouldn't also affect language evolution, which tends to be much faster.

Centuries of persecution have taken their toll on Euskara and Basque culture at large.

In many ways, this reminds me of the historical arc of the Welsh language. But this sort of linguistic persecution is hardly unique to Europe; we've done it here, too. It seems to be one way those in power try to erase the identities of those they deem inferior.

Basque literature, previously endangered to the point that a single collector — one Edward Spencer Dodgson — is credited with preserving an entire society’s literary canon, is currently experiencing a renaissance.

Curious, I looked him up. It's quite ironic to me that the only Wikipedia entry on him is written in French.

I'm at the point where I can muddle through that, but not fluent enough to provide an English translation for Wikipedia.

"Né en 1857 en Angleterre, il s'intéresse à la langue basque à partir de 1886 et se consacre à cette étude jusqu'à sa mort."

Born in 1857 in England, he became interested in the Basque language in 1886 and devoted himself to that study until his death.

Or something like that. The double irony is that it was the English who tried to pull the same kind of shit with the Welsh.

In retrospect, Franco’s attempt to destroy Euskara helped ensure its survival.

And we've heard that one before, too. Perhaps the most famous iteration of those attempts, at least on this side of the pond, was when the US government tried to eradicate Native American languages, only to turn to the Diné (Navajo) for help when they needed to send coded messages.

The surprising thing is that they provided it, thus claiming the moral high ground once and for all.

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