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:

Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

July 17, 2023 at 11:27am

#1052714

The Step-Parent of Invention

They say necessity is the mother of invention. I say laziness is the milkman.

Leave it to Cracked to point out a third option: comedy.

4 Very Important Ideas That Started as Jokes

Have you ever had a moment where you’re, say, shopping for pants, and you find a barren pocket field where the pockets no longer grow. So you say, “I guess I’ll just stick my phone up my butt.” Someone overhears, and the next thing you know, intra-anal wallets are a billion-dollar business?

No, because I'm a man, and our trousers have pockets. Hell, I won't even buy sweats that don't have pockets.

Still, that idea would go a long way toward reducing pickpocketing.

4. Schrödinger’s Cat

I always had a problem with this thought experiment, and no, that problem wasn't "that poor cat." It was, after all, only a thought experiment. Austrian or not, torture wasn't Erwin Schrödinger’s thing. No, my problem is that it gives primary focus to the role of humans as observers, when there is one sentient observer who knows if he's alive; to wit, the cat.

Physics in the 1930s was a wild west, or at least as wild as a bunch of nerds can get. There was all this quantum shit going around, things that can be nowhere and everywhere until you look at them, and not everyone was on board.

For a dick joke site, this is a remarkably thorough but succinct summary of the situation in physics in the 30s.

In response, he developed the “Cat Paradox,” which was supposed to illustrate what Schrödinger regarded as a flaw in the theory in the most ridiculous way possible. Obviously, a cat can only be either alive or dead, not both, and it doesn’t particularly matter who’s looking at it. Any cat owner can tell you they couldn’t give less of a shit about the actions of humans.

That, too.

But the joke was on Schrödinger. Quantum mechanics is now a pretty uncontroversial theory, and we’ve differentiated the behavior of quantum particles and non-quantum, catty objects, but that hasn’t stopped physicists from taking Schrödinger’s supposed paradox as a challenge.

But still haven't actually involved real cats, to the best of my knowledge. Anyway, QM is indeed uncontroversial in its description of effects, but there's still debate about interpretations thereof. But yeah, it's all because one physicist was trying to get cute with a gedankexperiment.

On a somewhat related note, the term "Big Bang" was coined by someone who didn't accept the idea of an origin for the universe.

3. America’s (Possibly the World’s) First Female Mayor

To be fair, lots of politicians are jokes, regardless of gender. Or party.

In 1887, when women were only kind of considered people, the ladies of Argonia, Kansas had just won the right to vote in local elections but still really only had the power to be mad about drunk dudes.

I'm not going to blame the evils of Prohibition on females alone. The Eighteenth Amendment (Prohibition) was passed before the Nineteenth (chicks can vote). But seriously, it's shameful how long it took for women to be able to vote even locally.

...they did underestimate how much local Republican Party officials disliked election tampering.

Contrast that to today, when that party has it as an official platform plank.

2. Pickering’s Harem

Unfortunate name for the surprising discovery that women are people and can do jobs.

Pickering was well aware that Williamina Fleming, the immigrant single mother he’d recently hired as a maid, was brilliant. His wife, the daughter of a former Harvard president, had even previously told him he should hire her to do more than dust. He eventually reached a point where doing so seemed advantageous on a number of levels. For one thing, women could be paid a lot less than men, but as an added benefit, her success would humiliate all those guys he just fired.

How... progressive.

It turned out Fleming and the other women on the team she oversaw, grossly referred to as “Pickering’s Harem,” really were much better than the men they’d replaced. They were only supposed to do tedious clerical and computation work based on photos of the night sky, “but they were very bright, so they drew their own conclusions and made several important discoveries."

Imagine that.

1. Trickle-Down Economics

I can't even see that phrase without imagining Reagan and Thatcher pissing down on the general public. Or without getting spitting mad about it.

...but the man who articulated it best was mostly in the business of dancing around in silly cowboy costumes. No, not Ronald Reagan — humorist Will Rogers.

Like I said, comedy.

Like many of the comedy greats, Rogers was mostly talking out of his ass. He was a vaudeville performer with a 10th-grade education, not an economist. But he turned out to be right: Trickle-down economics has been a disaster for the American economy.

No, it hasn't, not for the ones doing the trickling.

Despite my blinding rage at the entire concept, it is a concept, and it started (unintentionally) with a comedian.

Yet another reason we really should watch what we say.

4 Like • Comment • View 2 comments

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