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 13 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 15 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.

May 4, 2020 at 12:18am

#982704

Recipe for Disaster

I learned to cook in self-defense, but I don't really want to talk about cooking today; there are other things on my mind. Besides, talking about cooking makes me hungry, and I'm still working on losing weight. I've plateaued, but I don't want to backslide.

PROMPT May 4th

Describe your cooking or baking ability. What was the last thing you cooked/baked that you we’re proud of? Are you a recipe-follower or freestyler?

My mom was great in many ways, but she was a terrible cook. Her method of cooking pasta, for example, was to break the shit out of a bunch of spaghetti, throw it in a pot, cover it with cold water, and boil it for an hour.

If it weren't for TV commercials, I'd never have even known that pasta sauce was a thing that existed. So I guess TV commercials aren't all bad after all.

Now, my parents were old-school, so my dad only cooked on those occasions when my mom was sick or visiting family. Fortunately, I never absorbed sexist attitudes towards cooking (though I did and still do reserve the right to make jokes about it), which is a good thing since I've been single way more than I've lived with women.

As an aside, I have a friend out in Mormon country, and she recently perused ads for rentals. One such ad was put out by two guys who were specifically looking for a female housemate. "If I answer this ad," she asked me, "will I get raped and/or murdered?"

"No," I pointed out. "They want someone to do the cooking and cleaning."

She didn't answer the ad, but I suspect that they'd have been sorely disappointed. She'd be coming home from work all like, "Greetings, housemates. Hey, you need to pick up your Legos. Also, what's for dinner?"

Asides aside, like any other skill, cooking is something you work on or you lose, the way I lost my ability to read Latin. With it being just me -- my housemate and I don't usually share food -- I don't see the point, especially since with the weight-loss plan and all I don't eat large portions, and certainly not big bowls of pasta, or cakes or cookies.

When I do cook, though, I need a recipe to follow. I got shit for this once from a guest. "You're following a recipe? That's cheating!" She never got invited back. "I don't follow recipes" is code for "I lack the simple ability to follow instructions." As an engineer, I value instruction-following over creativity. Or maybe it's the other way around: I have zero creativity, so I pride myself in my ability to follow instructions. Also, I'm absolute rubbish at substitutions. "Oh, this recipe calls for baking powder. I don't have any baking powder, so I'll substitute cornstarch. It's about the same consistency, right?"

Since I lack creativity, I also don't have a good way to segué into the topic I actually want to discuss, so I'll just dive right in.

Today, May 4, 2020, is the 50th anniversary of the Kent State fascist murders of innocent college students.

Yeah, I'm biased.

Here's a less biased look at it,

from the New Yorker, a source I usually despise for its rambling, beat-around-the bush, deconstructionist style of "journalism." But somehow I read this article in its entirety.

This spring marks the fiftieth anniversary of the Kent State shootings, an occasion explored in Derf Backderf’s deeply researched and gut-wrenching graphic nonfiction novel, “Kent State: Four Dead in Ohio”

Incidentally, if I created a character named Derf Backderf, my (hypothetical) agent and publisher would crucify me.

Bill Schroeder, a sophomore, was an R.O.T.C. student. “He didn’t like Vietnam and Cambodia but if he had to go to Vietnam,” his roommate said later, “he would have gone.” Schroeder was walking to class when he was shot in the back. Jeff Miller, a junior from Plainview, Long Island, hated the war, and went out to join the protest; he was shot in the mouth. Sandy Scheuer had been training to become a speech therapist. Shot in the neck, she bled to death. Allison Krause, a freshman honor student from outside Pittsburgh, was about to transfer. She’d refused to join groups like Students for a Democratic Society, which, by 1969, had become increasingly violent. (Her father told a reporter that she had called them “a bunch of finks.”) But she became outraged when the National Guard occupied the campus.

I would normally quote more, but I don't feel like it. Read the article.

History doesn't really repeat itself. Nor, as some have suggested, does it rhyme. No, it echoes. The echoes of Kent State and the anti-war protests are still reverberating today, fifty years later. The players may have changed; to use the astrology-like vernacular popular today, those kids who were protesting fascism then were, and are, Baby Boomers. Now they're mostly on the other side, and fifty years from now the kids screaming for political shifts are going to be the targets of their grandchildren's protests.

Every revolution fights the Establishment. Every successful revolution becomes the Establishment.

I'm not old enough to be a Boomer, though not by much. By the way these stupid things are counted, I'm Gen-X, the generation that has largely been ignored in the latest culture war, which suits me just fine.

Reading that article, you might think, "Oh, but some of the protesters did violent things. They had to be put down." Be very careful with this attitude, I will urge. We have protestors these days, too, and some of them engaging in biological warfare. What else can you call deliberately infecting others with a potentially deadly virus? Should we kill them to save ourselves? And I strongly suspect that these protesters would invoke the Second Amendment in a heartbeat if they grow fed up with government overreach, real or perceived.

It's said that "violence never solves anything." This is demonstrably untrue. It may not be the best solution, but what if the North had engaged in sit-ins instead of fighting the Civil War? What if England had followed Chamberlain's policy of appeasement toward Nazi Germany?

There's no simple answer, I know. But in war, innocents suffer the most.

And to quote my second-favorite video game:

War.

War never changes.

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