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

August 8, 2023 at 8:59am

#1053806

Those Are Words

Yes, it's time once again to have a look at word origins.

The Interesting Etymologies of 71 Everyday Words

I don't necessarily take all these as absolute truth. Etymology, as the article itself notes, is littered with guesswork and outright falsehoods. Consider it a starting point.

Also, no, I'm not listing all 71 words here. That would be cumbersome Which, it should be noted, is a word based on the archaic word "cumber," which itself seems to be from "encumber," which, though older, is still in use today. English is weird. An even older origin places "encumber" as a derivative of yet another French loan word that changed its original meaning. So that's bonus etymology #72.

1. Vaccine

Yeah, this one's seen a lot of use lately.

The word vaccine derives indirectly from the Latin for cow, vacca. The story goes that, just before the turn of the 19th century, a British doctor named Edward Jenner observed that milkmaids who had contracted cowpox, or variolae vaccinae, were much less likely to contract smallpox, which could otherwise devastate entire communities.

The article lists a bunch of Romance languages whose word for cow is obviously related to vacca. But it fails to mention the French version, which is vache.

4. Shampoo

"Why do people need shampoo when there's so much of the real stuff around?"

Okay, that's a pun, not an etymology. See also: sham-pain

6. Chortle

Lewis Carroll’s poem “Jabberwocky” has a character chortle in joy. It seems Carroll combined the words chuckling and snorting to build a new, intuitively understood verb.

Carroll was remarkably good at making up words that sounded right.

16., 17., and 18. Sinister, Dexterity, and Ambidextrous

These are, I think, pretty obvious. Or at least, I've known about their origins for a long time, though perhaps only because at one point, I studied Latin.

I'm only including them because there's a parallel in French. Even though it's a Latin-derived language, their words for left and right aren't related to sinister and dexter, respectively. But it's almost as bad: left is "gauche," which has a negative meaning in English; right is "droite," which is very similar to that other right, meaning correct, which is "droit." (We have words based on that, too: adroit and maladroit.)

Southpaws just can't catch a break in any language, it seems.

24. Pundit

In contrast, this is one I'd always wondered about, though never enough to be arsed to look it up.

Pundit comes to us from a Sanskrit word that has been transliterated as pundit, pandit, or pandita (पण्डित). It originally referred to someone who had committed to memory a significant amount of the Hindu religious texts known as the Vedas. It came to refer more generally to something like “a learned man” or “philosopher” by the 19th century, and today that meaning has expanded to include people who like to yell at one another on cable news.

They call it the Indo-European language group for good reason.

27. Tycoon

And yes, English has more than its share of non-Indo-European origin words, like "taboo" or this one.

Tycoon comes to the United States via the Japanese taikun, a word whose Chinese roots mean “great ruler.”

And it's definitely got its origins in US English. I don't know if they use it in other Anglophone countries or not.

29. Nimrod

This is an example of a word whose modern usage origin is subject to some misinformation.

Whatever the origin, it’s often stated that Bugs Bunny helped popularize the derogatory meaning of the word when he lobbed it at everyone’s favorite hapless hunter, Elmer Fudd. But no one ever actually mentions which ‘toon that was in; we found that Daffy actually calls Elmer “nimrod” in 1948’s What Makes Daffy Duck. Bugs used it in reference to Yosemite Sam three years later in Rabbit Every Monday.

Never underestimate the cultural power of old Warner Brothers cartoons.

33. Orange

And orange comes from, well … oranges. Its predecessors include the Middle French orange and the Sanskrit naranga-s, but what’s really interesting to note is that English didn’t really have a word for this color for centuries.

I've heard it said that "orange" came from "an orange," which in turn was slurred from "a naranj." I don't know how true any of this is, but it seems to me that the fruit named the color, rather than vice-versa.

38. Gerrymandering

Come on; I learned the origin of this one in, like, eighth-grade social studies, complete with a memorable image of an early American political cartoon.

42. Silhouette

I did, like, a whole entry on this one a while back.

58. Window

Another one I've wondered about. "Windshield" is obviously a descriptive name for a window-like pane in the front of a car, one which bears similarity to a window, so did "window" have its origins in "wind?"

Window comes from the Old Norse vindauga, or “wind eye.” It originally referred to a hole in the roof, but over the years made the jump down to glazed glass usually found on a wall.

Probably, yes.

63. and 64. Flower and Flour

Not just a rich source of puns, these words actually have the same origin, it seems:

Flower comes from the French fleur, and so does flour. The part of a plant used to make flour—the kind that you mill—was considered the “flower of the grain,” the best part.

Interestingly, the French word for "flower" is still "fleur," but their word for "flour" is "farina."

67. Walrus

Despite its Latin second-declension noun ending, "walrus" probably didn't come from Latin. Goo goo g'joob.

69. Nice

I've written about this one here before, too. I'm just including it because I'll bet $69 that they purposely made it #69 just so they could write "69. Nice."

And finally, one of my favorite topics:

71. Science

Last, but definitely not least.

The Latin word nescius comes from the prefix ne-, for not, and a form of the Latin verb scire, “to know.” That verb also eventually gave us science, which even today isn’t far from its mid-14th century meaning, according to the Online Etymology Dictionary: “What is known, knowledge (of something) acquired by study; information.”

While science is more of a process than a strict body of knowledge, the word origin here is less disputable than most.

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