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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.
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Sure, let's take writing style advice from the only publication I know of that still uses diareses.
I've found hyphen use interesting ever since I was researching the history of a local sportsball stadium so we could do some work on it. It was originally built around the time large mechanical equipment was first introduced, and the description of its construction included something like, "the construction crew used 'bull-dozers' to move earth on the site."
Not only was it hyphenated, but in quotes, indicating -- to me, anyway -- that this now-common name for large earth-moving equipment was, at the time, brand new and had to be explained to the general public.
Among the many books about punctuation, precious few are devoted to a single mark. There’s “On the Dot,” by the Brothers Humez, which celebrates the period, or full stop; “Semicolon,” a thoughtful treatise by Cecelia Watson; and “Fucking Apostrophes,” a jewel of a book by Simon Griffin.
Personally, I don't get the confusion surrounding apostrophes. To me, they're simple as breathing. But I continue to see the grocer's apostrophe on signs, and people keep confusing its and it's, which causes me to pop a blood vessel in my forehead every time.
I'm not immune to making the occasional error, but when I see people consistently misuse that simple punctuation mark, I have to assume that they're either willfully ignorant, or trolling; whichever it is, I feel free to ignore their writing from that point on.
But I digress. We were talking about the hyphen.
The hyphen, which may not technically qualify as a punctuation mark, because it operates at the level of the word rather than the sentence—it doesn’t make you pause (though it may give you pause)—has inspired not one great book but two: “Meet Mr. Hyphen (And Put Him in His Place),” a classic by Edward N. Teall, published in 1937, and “Hyphen,” by Pardis Mahdavi, which came out in 2021.
Guess which one this is probably a stealth ad for. Also, it seems like TNY uses quotes instead of italics for book titles, making the rest of their style advice highly suspect.
The invention of the hyphen has been credited to Dionysius Thrax, a Greek grammarian who worked at the Library of Alexandria in the second century B.C.
Just... just take a moment to appreciate that name. Dionysius Thrax. Unlike some of the great names of modern literature, such as Stephenson's Hiro Protagonist, that one was apparently real. Dionysius Thrax. Hell, from now on, in honor of such an amazing moniker, I will refer, in my mind, to the hyphen, or the dash, as the "thrax."
The hyphen continues to serve a dual purpose: it both connects and separates.
Well, that makes things crystal clear.
In justified text, it divides into appropriate syllables a word that lands on a line break, a task that machines have not yet mastered...
Oh, I don't know; Word does a fair job of this. But for the most part, I find it more visually appealing if text justification is done without resorting to breaking up a word. It's still proper to use a hyphen sometimes, of course, and often, when you're dealing with a narrow column it makes sense. But usually, you're better off not orphaning a syllable (or two). Especially because the rules for breaking up words into syllables can be difficult to remember -- more difficult than those for apostrophes.
...and it is instrumental in the formation of compounds, where it is famously subject to erosion.
That's what happened to "bulldozer," and this article provides other examples:
Yesteryear’s “ball-point pen” became the “ballpoint,” “wild-flowers” evolved into “wildflowers,” and “teen-age” found acceptance as “teenage” in most outlets (but not in this one).
In 2019, the Associated Press dropped the hyphen from such terms as “German American” and “Chinese American,” gladdening the heart of Henry Fuhrmann, formerly a copy editor for the Los Angeles Times, who had long felt hobbled by the hyphen, believing it made people feel less than American. The New Yorker went along with the dropping of the hyphen in these constructions, on the principle that you call people what they want to be called.
And also because most news outlets follow the AP style guide, or they are subjected to ridicule.
No one seems to have a problem with hyphenating surnames, though. I'd often wondered, however, if those who hyphenate their surname think beyond their own generation. Say Smith marries Jones and they both take the surname Smith-Jones, which is then passed on to their sprogs. Said sprogs grow up and marry someone else with a hyphenated name -- let's say, Brown-Johnson. Do they then take the name Smith-Jones-Brown-Johnson? This convention has been around for a few generations now, and I have yet to see more than two surnames hyphenated together, so I suppose they've figured it out.
Now, some of you are probably wondering about the arrested state of development apparent at The New Yorker in the persistence of the hyphen in the word “teen-age.” It is the most unpopular hyphen we have. Without being able to go back in time and divine the reasons for it, the only explanation I have found is that “teenage,” without a hyphen, is listed in Webster’s Second Unabridged (1934) with a different meaning: “Brushwood used for fences and hedges,” from a variant of “tine,” to enclose.
And there you have it, folks. Mistrustful of their own readers, probably because they deliberately market to people who think they're upper class, to confuse a common word for an obsolete one and, moreover, to be undiscerning of context -- for I can not think of a single instance where a sentence including one definition of "teenage" could possibly be confused for another, excepting maybe something referring to The Who's lyric "teenage wasteland," which is, after all, not even the name of the song, which is called Baba O'Riley for reasons that are obscure even to this fan of classic Britrock -- TNY continues to use the archaic "teen-age."
Yes, that was a run-on sentence. Deal.
Still, for a TNY article, this one's mercifully short, and without the usual distractions, detours, and personal anecdotes that make me annoyed with that magazine. And, mostly, it's legitimately educational.
And since I mentioned the song, I can't let this go without linking to it. I should mention that, in addition to being just an awesome tune, it holds special meaning for me because of one warm summer night as a teenager when I heard it wafting across the water from someone's sailboat party.
Don't cry
Don't raise your eye
It's only teenage wasteland
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