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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Write_Mikey_Write!

a day ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

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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.

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.

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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