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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Show Recent Comments

10 Most Recent Blog Comments:

Wordsmitty ✍️

about 2 hours ago
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?

1 Like

🌖 HuntersMoon

about 5 hours ago
In response to "Gray Matters"

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

1 Like

tj-Merry Mischief Maker

a day ago
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

a day ago
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

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

June 28, 2024 at 9:33am

#1073277

It Ain't Wrong

Hey, look, an article about writing. Sort of. From thesaurus.com:

Why Is “Ain’t” Such A Controversial Word?

I still remember a rhyme from elementary school: Don't say ain't, or your mother will faint, and your father will fall in a bucket of paint.

The image was so compelling that I spent the next few days saying "ain't" every chance I got; sadly, the promised amusing scenario never presented itself.

Look, there are myriad reasons I never wanted kids, but high on the list was that they might be like me.

What’s all the fuss over ain’t about?

I always figured it was kind of like "y'all," which I did a whole entry on a while back: basically, an attempt to squelch Black vernacular.

Ain’t is a contraction that can mean am not, are not, and is not. It can also mean have not, has not, do not, does not, or did not. We ain’t joking: ain’t is incredibly versatile, a kind of one-stop-shop for saying something isn’t, didn’t, or wasn’t.

That ain't confusing at all.

Ain’t apparently begins as amn’t, a contraction for am not, which you can still hear in Ireland and Scotland today.

I also vaguely remember wondering why we didn't have "amn't" when we had contractions like aren't and weren't.

Ain’t is a perfectly valid word, but today, ain’t is considered nonstandard. At worst, it gets stigmatized for being “ignorant” or “low-class.” At best, it’s considered a no-no in formal writing.

Lots of things are frowned upon in formal writing that we can get away with in, say, texts or blog entries. But "ignorant" is how I'd describe elitist snobs who are out of touch with common ways of communication.

While a lot of people consider ain’t improper, it’s a very regular and legitimate part of many forms of English, including in Black English (AAVE). Be mindful that judging someone’s use of ain’t as “wrong” can be a very socially loaded act, to say the least.

So I was at least partially right. It's also more acceptable in the South here, though it's not as quintessentially Southern as "y'all."

Outside of regional and dialectical forms of English, ain’t also appears in a great many expressions (not to mention in important titles, from “Ain’t I a Woman?” to “Ain’t That a Shame”). In fact, it’s hard to imagine these sayings without ain’t. Try replacing them with their formal counterpoint and see if they still keep their charm.

At least it's more acceptable now than when I was a kid. I wouldn't expect to see it in business correspondence or scientific research papers (apart from those that study language, that is), but for everyday speech, there ain't nothin' wrong with it.

Maybe one of these days I'll do an entry on how a double negative is an intensifier, rather than mathematically turning a sentence a positive. But this ain't the time.

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