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!

about 8 hours 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

about 10 hours 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').

1 Like

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.

March 28, 2024 at 9:35am

#1067075

What're Y'all Doin'?

I'm aware that language changes over time. But some things are just always wrong. For example, "alot" is not, never has been, and never will be a word, as in "I have alot of problems with spelling." One does not "loose" one's keys if they're misplaced; one can only "lose" them. "It's" is never possessive, and is always a contraction of "it is." But, being Southern, this one especially annoys me:

What's The Difference Between Y'all And Ya'll?
There's only one correct way to spell it, y'all.

The South is known for its laundry list of unique, quirky, cultural sayings, like "Bless your heart," "Too big for his britches," and "Well, I s'wanee," to name a few.

Okay, I've never heard that last one until now. I don't know everything, but damn if I'm not working on it.

But the best-known word in the Southern vernacular is probably our most-loved pronoun: y'all.

And make no mistake, it is absolutely a word. Modern English lacks a universally-agreed-upon second person plural, and the other candidates, with the possible exception of the very British "you lot," just don't work outside their regions.

The only proper way to spell the contraction of "you" and "all" is "y'all."

And that should be the final word on the subject, but I guarantee you, sometime soon, I'll see someone's typed it as "ya'll" and rage will boil up once again.

The article goes into why. Basically, an apostrophe serves two major purposes in English: possessives (for words that are NOT pronouns such as "it"), and to replace missing letters. There are a few edge cases, but they mostly involve borrowed words from other languages. "Y'all" is obviously not possessive, so the apostrophe replaces missing letters. The missing letters are "ou." As in "you all" contracts to "y'all."

I did say the apostrophe is not used for pronoun possessives, but here, "y'all" is an exception. If you're writing about something that belongs to y'all, it's "y'all's," as in "Where's [where is] y'all's [belonging or relating to all of you] tater tots?" (Never mind that it should be "Where are your tater tots?"; you gotta cut people a bit of slack.) For context, one might ask a helpful grocery store employee this question, usually while standing right next to the freezer containing the tater tots.

Though, to be fair, Wikipedia

asserts: "The possessive form of y'all has not been standardized; numerous forms can be found, including y'alls, y'all's, y'alls's, you all's, your all's, and all of y'all's."

Unlike French, German, and Spanish languages, the English language does not have a designed second-person plural pronoun.

I think they mean "designated," because English was far from "designed." But yes, it does. It's "y'all." Anyway, the French second-person plural doubles as the formal second-person singular, and you have to figure out the difference from context.

What we desperately need is another second-person plural. Right now, if I said, "We are going to a party tonight," it's ambiguous whether "we" refers to "I and you" or "I and my closest friends, of which you are not one." Call it first-person plural inclusive and first-person plural exclusive. I suggest "ze" for the latter, but no one cares.

While we're on the subject of poor spelling, what the hell is up with "Opps!"? Look. I've never, ever heard someone exclaim "opps," which, by that spelling, would be like in "drops." No, I've only ever heard "Oops," though the oo there can be pronounced either like book or like boob.

Speaking of boobs, I've been seeing more and more people writing something like "a women" as in "There was a women waiting for the store to open." No. There is no such thing as "a women." There's a woman, or there's several women.

Look, I'm not saying I never make mistakes. That would be a mistake. And by Waltz's Second Law of the Internet, because this post discusses spelling mistakes, it probably contains an inadvertent spelling mistake. But come on... at least try.

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