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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Elycia Lee ☮

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Coincidences are great! So happy you met!

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That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

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Annette in France

a day ago
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After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

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Elisa: Middle Aged Stik

a day ago
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Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

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Kåre Enga in Montana

Sep 25, 2024 at 7:53pm
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

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Annette in France

Sep 25, 2024 at 3:20pm
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

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Brandiwyn🎶

Sep 25, 2024 at 2:49pm
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Love it when stars align!

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Sep 24, 2024 at 5:15pm
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I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

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Sep 24, 2024 at 11:30am
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That street art is definitely a step up.

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August 14, 2019 at 12:17am

#964173

Semicolonoscopy

Continuing yesterday's linguistic theme, today's link is about my personal favorite punctuation mark: the semicolon.

https://www.theparisreview.org/blog/2019/08/01/the-birth-of-the-semicolon/

The Birth of the Semicolon

That sounds... painful.

The semicolon was born in Venice in 1494. It was meant to signify a pause of a length somewhere between that of the comma and that of the colon, and this heritage was reflected in its form, which combines half of each of those marks. It was born into a time period of writerly experimentation and invention, a time when there were no punctuation rules, and readers created and discarded novel punctuation marks regularly.

I imagine it would be hard to experiment with punctuation symbols these days; the computer keyboard and the extended ASCII set of characters are pretty well fixed. Dammit, there I go using a semicolon.

The modern equivalent would probably be some godawful new emoji.

I've seen other articles promoting ideas for new (or possibly resurrected) punctuation marks, such as the interrobang

(‽), which is what you get when an exclamation point (known in geekery as a "bang") fucks a question mark. No one uses the little bugger - it's rare that it's needed, and when it is it's much easier to type two characters than to look it up. I think it even has an ASCII representation; I don't know what it might be, because I just did a copy/paste to get it to show up there. Goddammit. Another semicolon.

Nearly as soon as the ink was dry on those first semicolons, they began to proliferate, and newly cut font families began to include them as a matter of course. The Bembo typeface’s tall semicolon was the original that appeared in De Aetna, with its comma-half tensely coiled, tail thorn-sharp beneath the perfect orb thrown high above it. The semicolon in Poliphilus, relaxed and fuzzy, looks casual in comparison, like a Keith Haring character taking a break from buzzing. Garamond’s semicolon is watchful, aggressive, and elegant, its lower half a cobra’s head arced back to strike. Jenson’s is a simple shooting star.

Waxing eloquent about the shape of punctuation marks? I guess everyone's a geek about something. Who the hell is Keith Haring, anyway? Can't be arsed to Google the name.

(For the postmodernist writer Donald Barthelme, none of these punch-cut disguises could ever conceal the semicolon’s innate hideousness: to him it was “ugly, ugly as a tick on a dog’s belly.”)

Yet another reason why postmodernists should never be taken seriously.

The rhetorical question mark, on the other hand, faltered and then fizzled out completely. This isn’t too surprising: does anyone really need a special punctuation mark to know when a question is rhetorical?

Oh ho! I see what you did there, asking a rhetorical question and ending the sentence with an actual question mark.

Still, a few cranky complainants notwithstanding, most humanists believed that each writer should work out his punctuation for himself, rather than employing a predetermined set of rules. A writer or an annotating reader was to exercise his own taste and judgment.

As if writers can be said to have those qualities. Also, that way lies anarchy.

Anyway, just a fun bit of punctuation history. I probably overuse semicolons, myself; some would say any use of one is overuse. (DAMMIT) They're technically unnecessary, because a semicolon separates two related, but independent, clauses, and you can do the same thing by simply starting a new sentence or using a conjunction.

Still, I'd hate for any key on my keyboard to wither away from disuse - even though that's the ultimate fate of any ~/` key for me. Hell, it nearly didn't work just then.

Semicolons are, to be sure, hard to get right, but they do adjust the flow of your writing. Without them. You sound. Like Shatner. Or Hemingway.

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