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 2 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 4 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.

January 17, 2023 at 12:01am

#1043234

Ice Empress

At some point, I think it was last year, I finally got around to watching Frozen. What can I say? Without kids, I was never forced to, but it's part of my Disney+ subscription, so at least I didn't have to pay more for it. (Still haven't seen the sequel.)

As usual, though, truth is stranger than fiction.

The Time a Russian Empress Built an Ice Palace and Forced Her Jester To Get Married In It
Cruel joke, power move, or both?

Now, the truth is, I don't have much to say about this beyond the Frozen reference above. I just thought it was—to understate things a bit—cool, so here it is, shared.

It was 1740 and one of the coldest winters St. Petersburg, Russia, had ever seen. But few residents mentioned the bitter winter in letters or accounts. They were, understandably, distracted. While the rest of Europe shivered through the deep freeze, Russians were busy—building a palace. On the orders of Empress Anna Ioannovna, numerous craftsmen were charged with constructing an elaborate, fairy tale–esque castle, one made entirely of ice.

On the plus side, I suppose the exertion kept them warm. And the fear of being executed for underperformance, but mostly, the exertion.

Rising 66 feet from the surface of the frozen Neva River and nearly 165 feet long, the Ice Palace was built “according to all the rules of the most current architecture,” noted Russian mathematician Georg Wolfgang Krafft.

I guess one of the rules wasn't "don't build your palace out of ice."

A steam bath, or bania, built from ice sat beside the palace. Decorative ice dolphins blew fire. A life-sized ice elephant’s raised trunk served as a fountain by day and as a stunning torch by night.

Gotta say I'm impressed by the violation of the laws of thermodynamics, which, to be fair, hadn't been formulated yet.

The palace was, by all accounts, a marvel. But the elaborate, temporary palace isn’t the strangest part of the story. That winter, Anna ordered a bizarre wedding to take place at the Ice Palace between the disgraced noble-turned-jester, Prince Mikhail Golitsyn, and a Kalmyk woman, Avdotia Buzheninova.

Okay, I had to look up "Kalmyk." Apparently a Mongol ethnic group in Russia. The article does explain this later, but I hadn't gotten to that part yet. I'm sure their story is fascinating, too, but I'm short on time tonight.

“The nobles choose Anna because, as a woman, they think she’ll be very easily manipulated,” says Russian historian Jacob Bell of the University of Illinois. The nobles imposed a list of conditions (creatively dubbed the “Conditions”) on Anna’s power and made her sign on the dotted line. She signed, but Anna was far smarter than they gave her credit for.

Within a couple of months, Anna solidified support from a group of nobles and the local guards’ regiments. “[She] then, very dramatically tears the conditions in half and declares she will rule in the way she wants to rule,” says Bell.

This is one of the most Russian things outside of depression and vodka.

Continuing many of the efforts begun by Peter the Great, she funded the Russian Academy of Science and encouraged Westernization. She founded the Cadet Corps, a premiere military training school, and maintained a brutal secret police known as the Secret Office of Investigation.

The Russianness intensifies.

Ahead of the nuptials, Anna decreed every Russian province to send a man and woman wearing traditional dress to attend the spectacle.

I suppose at least she didn't make them all wear ice?

With all of this sort of thing going on, it's surprising that it took over 150 years after the Ice Palace incident for Russians to rise up in revolt. I guess they might have started to, but got cold feet.

3 Like • Comment • View 3 comments

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