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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10 Most Recent Blog Comments:

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.

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.

August 27, 2019 at 12:39am

#964888

We Don't Need No Steenkin' Rules

Oh, this should go over well on a writing site...

https://qz.com/1561426/grammar-expert-benjamin-dreyer-lists-three-rules-you-can-...

The three most useless English language “rules” you can ignore

I have nothing against rules. They’re indispensable when playing Monopoly or gin rummy, and their observance can go a long way toward improving a ride on the subway. The rule of law? Big fan.

I'm just going to take a moment here. Monopoly? Long ago, I made it my personal mission to learn the rules of Monopoly inside and out. Every time I played it, I'd insist on not keeping money on Free Parking, enforcing even-build rules, and so on. The result? I got really, really good at Monopoly and really, really bad at keeping friends. Seriously. Don't play Monopoly if you want friends.

The English language, though, is not so easily ruled and regulated. It developed without codification, sucking up new constructions and vocabulary every time some foreigner set foot on the British Isles—to say nothing of the mischief we Americans have wreaked on it these last few centuries—and continues to evolve anarchically.

I'll say it again: the Brits may have invented it, but we perfected it.

Also simply because, I swear to you, a well-constructed sentence sounds better. Literally sounds better.

Fhweeet! Flag on the play. Unnecessary begging of question. 15 yard penalty. Fourth down for using the word "literally," even if literally.

I say this begs the question because the reason a well-constructed sentence sounds better is that because the "rules" of English grew in part from a desire to make English sound better. So he's saying it's well-constructed because it's well-constructed. Or that it sounds better because it sounds better. But I suppose he's a linguist, not a logician.

A good sentence, I find myself saying frequently, is one that the reader can follow from beginning to end, no matter how long it is, without having to double back in confusion because the writer misused or omitted a key piece of punctuation, chose a vague or misleading pronoun, or in some other way engaged in inadvertent misdirection.

Show-off. Last I heard, though, the record for "longest sentence" goes not to the asshole who ran a car into a crowd two years ago in my hometown, but apparently to a 3687-word abomination

in James Joyce's Ulysses. Personally, I say this doesn't count, because it's Ulysses, and therefore only marginally related to the actual English language.

Why are they nonrules? So far as I’m concerned, because they’re largely unhelpful, pointlessly constricting, feckless, and useless. Also because they’re generally of dubious origin: devised out of thin air, then passed on till they’ve gained respectable solidity and, ultimately, have ossified. Language experts far more expert than I have, over the years, done their best to debunk them, yet these made-up strictures refuse to go away and have proven more durable than Keith Richards and Mick Jagger. Put together.

That's legitimately funny.

I’ll dispatch these reasonably succinctly...

Really? Because you haven't done anything succinctly thus far.

Remember, these are rules that shouldn't be rules:

1. Never begin a sentence with “And” or “But”

Agreed, as anyone who reads my shit here knows.

2. Never split an infinitive

This is one of those rules that was imposed upon English from above, rather than grown organically. For a long time, English was considered a vulgar language (and, dammit, it still is, but by "vulgar" I mean "common," not "rude'), and Latin, at least the Latin of Caesar's time or thereabouts, was considered the Perfect Language, the Platonic ideal of languages. In Latin, it is physically impossible to split an infinitive. We'd say "to be," but in Latin that was handled by one word: est, or something like it (I don't remember much of my Latin and can't be arsed to look it up). Most Latin verb infinitive forms ended in -are or -ere, as I recall, but the point is, they were all one word each, whereas in English we know an infinitive because it starts with the word "to" and ends with whatever root verb you're talking about.

Consequently, some asshole decided that because you can't split an infinitive in Latin, you can't split it in English either. But you can - it's two words. Whether you should or not is still up for debate, obviously, but I'm with the author here - you want to boldly, calmly, or preciously split infinitives? Go for it.

3. Never end a sentence with a preposition

What's that rule for?

Probably another Latin thing. I'm just guessing here, but as I recall, it was another thing you couldn't, or at least shouldn't, do in Latin.

This is the rule that invariably (and wearily) leads to a rehash of the celebrated remark by Winston Churchill that Winston Churchill, in reality, neither said nor wrote:

“This is the kind of arrant pedantry up with which I will not put.”

I suppose linguists have their tired jokes, much as astronomers grow weary of finding new pronunciations of Uranus that don't result in Beavis and/or Butthead snickering. Still, I was kinda hoping Churchill actually said that. That particular sentence reads so nicely in his posh accent.

I'm reminded of an old joke from my time at UVA:

A student from Virginia Tech was visiting the UVA grounds for a football game, and he got turned around. He goes up to a typical UVA snob and says, "Excuse me, could you tell me where the Rotunda's at?"

The UVA student looks down his nose at the other guy and says, "You know, you should never end a sentence with a preposition."

To which the Tech student responds, "Okay. Could you tell me where the Rotunda's at, asshole?"

I'm pretty sure there are 14 million versions of that same joke, but that's the one I heard, so it's the one I'll stick to.

See what I did there?

I'll add one of my own: the double negative. Some will say there ain't no use for a double negative, or that, as in math, a double negative makes a positive. Both of these are bogus. While they won't fly in formal writing, in speech or dialogue the purpose of a double negative is not redundancy or cancellation, but emphasis. To eschew the double negative is to deny the color of many of the major dialects of English.

In conclusion, the rules of English, like dictionary definitions, are descriptive, not prescriptive. That is, they reflect usage, not govern usage. One of the great things about English is that it's always evolving. Sometimes, it evolves in ways that piss me off, but it's really a very democratic process - anyone can change English. Really, the only "rule" is to promote clarity - and if you can find a better way to do that, great. Until then, the other rules are there to help.

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