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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.
August 15, 2021 at 12:15am August 15, 2021 at 12:15am
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Hey, it's an article about writing style.
Like many writing "rules," there's no black-and-white answer. The ability to use passive constructions exists for a reason. Some people are still confused about what it actually means, though; it's more than just using linking verbs (such as forms of "be").
Even when you think you’ve got a handle on grammar, you might not realize how easy it is easy to unwittingly venture into lexical territory that would rile your freshman year English teacher. Certainly back in my college days, I had a few teacher’s assistants who would get on my case for using the passive voice.
I think that in many cases, the problem isn't using passive voice; it's overusing it, or using it unthinkingly. Teachers tend to drill rules and standards into our heads (rules and standards are drilled into our heads by teachers), but true wisdom lies in knowing when and how to break the rules (the rules can be broken by careful writers).
Alas, I had to learn that the passive voice is abhorred by a select few grammar snobs.
What they did there is seen by me.
In order to understand the passive voice we first need to dive into the more commonly used active voice. Basically, the active voice is being used when the subject of a sentence performs the actions denoted by a verb.
This is simple, straightforward English. I'd only add that while technically, verbs are all "action" words, some verbs are more active than others. The article uses the example, "The boy really loved soccer." Without context, that's a perfectly acceptable sentence. The boy - subject. Really - adverb. Loved - verb, past tense. Soccer - direct object, tells you what it was that the boy really loved.
If I came across that particular sentence in a story outside of dialogue, though, I'd cringe. It's a prime example of telling, not showing. And I still adhere to King's advice to slash every damn -ly adverb out of one's fiction, even though I still find myself using them, and also even though Rowling apparently never heard that advice and yet she became a world-famous author. Better would be to describe the exuberance of "the boy" in a paragraph or two where he's playing association football. That's longer, yes, but it tends to stick in peoples' minds better.
Still, for the purposes of the arguments presented in this article, like I said, it's an example of active voice.
But so is "The boy is a soccer player." The point being that active voice doesn't require an active verb, while style prefers the active verb, for example, "The boy plays soccer." In either case, the subject comes before the verb.
I should also mention that the verb "to be" isn't equivalent to the equals sign in math. You can say A is B and B is A, but "A soccer player is the boy" has a different meaning than "The boy is a soccer player." Neither, however, is passive voice.
Purdue’s definition gets at what most writing instructors are trying to convey when they’re teaching you to write. Using the active voice is straightforward and simple, but that doesn’t necessarily mean the passive voice is to be avoided at all costs.
Yes, there exist situations where the passive voice should be used.
A passive construction occurs when you make the object of an action into the subject of a sentence. That is, whoever or whatever is performing the action is not the grammatical subject of the sentence.
Yes, this can get tricky. The subject of a verb is the actor doing the action; the object is, well, the object receiving the action. The article provides the passive-voice example: “The ball was kicked by the boy.”
If you were going to write this in the active voice, you’d just go with “The boy kicked the ball,” a sentence that clearly delineates that the subject of the sentence—the boy—did the kicking.
Now, despite the above, I'm not a grammar expert. So I get confused defining the subject and object of a passive-voice sentence. In the end, though, strict definitions don't matter; the question I ask myself as a writer is: "Why am I using passive voice in this particular instance?" Without wading into the quagmire of grammar labels.
In any case, it's abundantly clear to me with "The ball was kicked by the boy" that "the boy" was the one doing the kicking. It's just -- usually -- too convoluted a construction to go with under normal circumstances.
It isn’t always crucial (or pleasurable for the reader) to emphasize the main actor of a sentence. As Wolfson explains, “there are many examples where we either cannot or do not want to emphasize the actor, particularly if there is an element of mystery involved.”
The article provides but one example of an appropriate passive-voice sentence: "My car was stolen on Sunday night."
I find this interesting because, while that's acceptable English, in French, one might say, for that sentence, "On a volé ma voiture dimanche soir," which, if you translated that literally, word for word, would be more like "We(informal) stole my car (on) Sunday night/evening." But I suppose a less literal, though more descriptive, translation would be "Someone stole my car on Sunday night." Either way, though, it's what we Anglophones would consider active voice.
French tenses can get weird. There's no difference in French, for example, between "The boy kicks the ball" and "The boy is kicking the ball," both of which would require the French translation of "kicks the ball" which is -- and I'm not kidding here -- donne un coup de pied au ballon.
Okay, there's a shorter way of saying that, but that's the first construction I learned and it's glorious.
But I digress. The point is, if French has a passive voice, I haven't learned it yet. Which means, probably, the grammar snob's aversion to the passive voice is similar to their finger-wagging at split infinitives: as with Latin, it's impossible to split an infinitive in French, because while the English infinitive is always "to [verb]," in French and Latin it's just [verb](with particular suffix). Snobs have always considered other languages superior to English, and so forced their "rules" on the much more pliable and forgiving English grammar structures.
Don't give them control. Allow yourself to boldly split infinitives when appropriate. Or, hell, allow infinitives to be boldly split by you. Just know it's going to throw some people into a snob rage.
Sometimes, that's the whole point.
Lots of horribly annoying sounds in yesterday's comments. I definitely sympathize with all, or at least most, of them. But the one I most agree with was from Write_Mikey_Write!  -- for the mosquito buzz. I think I've lost that range of hearing with age (because I know mosquitos still exist), but I have vivid memories of trying to swat those little bastards when they were whirring around my head while I was trying to sleep.
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