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 6 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 8 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.

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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').

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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.

July 2, 2022 at 12:01am

#1034573

Damson In Distress

Really short today. It's just one of those days. But there's still something interesting I want to share.

Resurrecting Virginia’s Forgotten ‘Caramel’ Pie
Beloved by Edna Lewis, this plum-based favorite deserves a resurgence.

Well, this is a thing local to me that I'd never heard of, so of course I had to check it out.

“I’ve looked in all my cookbooks for a caramel pie which my husband tells me is made with damson preserves. There just ‘ain’t no sech animal.’ Please see what you and your readers can do to solve this marital difficulty.” The letter was signed “CPJ, Richmond.”

And I'd never even heard of damson.

Given the Virginia angle, at first I thought that it was truly something hyperlocal. But no, it turns out (spoiler alert for the article) that the damson (which might not be really a plum) came from Europe and possibly West Asia.

Today, however, most people in the United States, and even Virginia, would be hard pressed to even tell you what a damson is, much less offer a recipe for its signature jam pie.

Yep.

There’s a reason that most recipes involve cooking or pickling the fruit. A raw damson, with its inky hue and silvery bloom, is less luscious than its looks might suggest: Bite into one and you’ll get a sour, astringent mouthful. Cooked gently with a quantity of sugar, however, the fruit’s flavor unfolds into something deeper and more beguiling.

Not meaning to be rude, but with enough sugar, almost anything becomes edible. Maybe even eggplant (I haven't tried).

In some European countries, damsons have remained an essential ingredient of spirits (such as Czech slivovice) and jams; in the U.S., not so much.

I've heard of slivovice (which I thought was spelled slivovitz, but whatevs), but never had any, and didn't know what it was made of.

Anyway, the whole thing is fascinating as a bit of culinary history, but this part struck me the most:

Gothie explains that because the fruits are so small, growing on brittle stems, they are hard to harvest mechanically and “not too great for commercial production.” In the early part of the 20th century, other fruits became cheaper to grow on a large scale, and fresh fruits could be imported out of season. Preserving fruit for one’s pantry was no longer a necessity, and damson caramel pie was, at its heart, a country pie born of that need.

Because it made me wonder what else we might have lost because it wasn't suited for mass production and mechanization.

I mean, that hardly stops a lot of harvesting. Coffee, for example, was always labor-intensive; fortunately for Western coffee drinkers, it mostly grows where labor is basically free

(and from what I understand now there's more mechanization).

Daniel Perry, proprietor of Jam According to Daniel in Charlottesville, Virginia, says that a taste of his damson preserves seems to trigger “a really Proustian thing … I get many great stories from customers about their memories.”

I probably should get out more. That's in my town and this is the first I'm hearing of it, as well. In my defense, it doesn't involve beer.

But like I said, not feeling any kind of in-depth analysis tonight. Still, I was hoping someone else might find this whole thing interesting.

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