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').

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.

August 2, 2019 at 12:02am

#963583

Speakeasy

Grab a drink. Sit back. Relax. It's time for Adventures in Prohibition.

https://www.citylab.com/life/2017/04/the-secret-lines-of-speakeasies/521865/

The Secret Lives of Speakeasies

Decades before Prohibition, the unlicensed saloons of Pittsburgh flouted state liquor laws, fomented social movements, and started a national trend.

Dark days in our country's history, to be sure. I mean, it's not exactly on the abyssal level of slavery or Native genocide, but still... dark.

The story goes like this: Hester owned a saloon in McKeesport, just southeast of the city, that sold booze in defiance of a state law that upped the costs of licenses for alcohol so much that it was nearly prohibited. When customers got too rowdy, Hester would hush customers with “speak-easy, boys!” to avoid attracting the attention of authorities; the expression soon spread to the city, and the nation.

Which would have attracted the attention of authorities, so I'm calling bullshit. Sounds like every other fauxtymology. I just made that word up, so if it takes off, just remember it was me and not some Instawhore.

Still, I don't doubt the saloon existed. And I'm pretty sure McKeesport exists, because I've been there.

While the Volstead Act, which prohibited the production, sale, and transport of liquors, would not be enacted until 1920 in anticipation of the Eighteenth Amendment’s passage, the temperance movement in the U.S. began to get dry laws on the books in the late 19th century.

It's a cliché in time travel stories: "Let's go back and kill Hitler!" Doctor Who even had an episode titled similarly. The idea is, think about the worst person in history and stop that person from doing what they did before they did it.

Leaving aside the potential paradoxes and/or impossibility of time travel, I, of course, have thought of whose career I'd nip right in the bud, and that's Carrie Nation

. (My assumption is that someone else can do Hitler.)

I'm not sure offing that smug bitch would keep Prohibition from happening, but it's worth a try.

The saturation of the booze-slinging market suggests the importance of these public spaces in 19th-century cities. Christine Sismondo, author of America Walks into a Bar, says that saloons and speakeasies provided hubs for networking and space to build social and political movements, especially among immigrant newcomers.

It goes deeper into the past than that. The American Revolution was planned, largely, in taverns, alehouses, and inns: in short, it was booze-fueled. Our country was literally founded on booze; turning our backs on that was like abandoning our parents at a shitty nursing home and then "forgetting" to pay the bill.

From those connections, political organizing can trace a long tradition back to the American tavern. This was especially true during the Progressive Era from the 1890s to the 1920s, Sismondo says, when bars served as “centers of resistance.” That’s why so many Gilded Age industrialists wanted to shut them down: Margaret “Mother” Finch, the proprietor of the Rolling Mill House saloon on Fourth Avenue in Pittsburgh, was one of the key organizers of the 1892 Homestead Strike, battling Andrew Carnegie and Henry Clay Frick’s union busting.

In our current condition of late-stage capitalism, we find ourselves with an abundance of choices once again: bars, breweries, wineries... places for people to gather and, one hopes, foment the next revolution.

So of course I expect some Puritanical monster to try to shut them down any day now. I must remain vigilant.

While the drinks seemingly never stopped, Prohibition did bring about one big change: It brought men and women together in drinking establishments for the first time.

I choose to trace the modern search for gender equality to that. I suppose even in the darkest of nights, some light must glimmer.

I ask Schalcosky what he thinks is behind the renewed allure of speakeasies today, given the absence of legal prohibitions on alcohol. “I think it has to do with trying to connect with our communal past,” he says. “People appreciate a good tale and location to go along with it. That, and our timeless love for alcohol.”

It's important to remember the past. Especially if we can learn from the mistakes it made. And Prohibition was a huge one.

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