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:

Kåre เลียม Enga

about 5 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

about 22 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

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.

5 Like • Comment • View 2 comments

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