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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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

August 10, 2023 at 9:53am

#1053910

In Retrospect, These Were Bad Ideas

Today's list, from Cracked, is way too short. It fails, for example, to tell the story of "Mad" Mike Hughes, the flat-earth rocket scientist who, in an effort to see for himself the flatness of the planet, built a steam-powered rocket in the wilderness of Southern California, and promptly died in a lithobraking maneuver. I wrote about him, last year, here: "Bad Advice"

So here's more of that sort of thing, albeit not nearly as steam-powered:

4 Stubborn Idiots Who Were So Wrong That They Died

Stubbornness isn’t a particularly delightful trait to interact with.

It is also not a virtue, no matter how stubbornly people insist it is.

Dealing with a bullheaded friend or acquaintance is likely to end with a prominent forehead vein and a conversation that concludes in “okay, whatever, I don’t even care anymore.”

When a Gen-X meets a Taurus...

Combine that same stubbornness with an incorrect belief that’s considerably higher stakes, and you’ve got a recipe for disaster.

Like crashing your homemade rocket in the California desert. With you inside. Or like these items:

4. Franz Reichelt

For whatever reason, I've seen this guy from multiple sources recently.

Franz Reichelt was highly interested in the human power of flight, a hobby that, throughout history, has a pretty abysmal survival rate. Reichelt, however, lived at a time where airplanes already did exist and functioned — mostly. His interest took a slightly different slant, which was the development of a parachute suit that would allow pilots, or thrill-seekers, to jump from a great height and float down safely to the ground. To give you some idea of how this went, I can assure you that Reichelt is absolutely not the inventor of the parachute.

Or, for that matter, the wingsuit.

After what had to be an incredibly irritating back and forth with French police, Reichelt got permission to toss one of his dummies off the top of the Eiffel Tower. When he arrived at the top, though, he committed one of history’s most ill-advised and disastrous switcheroos, deciding he would wear the suit and jump himself.

And that, mes amis, is why today we call it the I-Fell Tower.

3. Herman Cain

As far as jokes about stubbornness go, politics is rather low-hanging fruit. But this guy was an entire branch of low-hanging fruits.

No person became more of a grim figurehead of the health risks of being a huge asshole than ex-presidential candidate Herman Cain.

I guess he wasn't Abel to let go of his stubbornness.

2. William J. Bailey

Throughout history, humans have come up with no shortage of wackadoo medical treatments and various snake oils.

Snake oil, it turns out, has legitimate uses. Most of the things we label "snake oil" are just useless. Some might actually do something, but no one's done studies. They don't make up for the ones that, you know, kill people.

In the 1920s, for example, radium was all the rage. Yes, the radioactive kind. Looking back, you can find it casually included in all sorts of consumer products like makeup or the glowing numerals on a watch face. Maybe the most straightforward, and incredibly stupid application was invented by a man named William J. Bailey, who created and sold Radithor, an “energy tonic” that was created from the simple recipe of water plus radium.

I'm not an expert on radioactivity (there are different kinds of radioactivity, and some of them are relatively harmless if their sources stay outside your body), but my understanding is that ingesting radium is particularly harmful because the body's chemical processes think it's calcium. So you end up with radioactive bones.

Suddenly, those glow-in-the-dark skeletons you see around Halloween aren't so cute, are they?

1. The Titanic

This one's so famous that I don't know why they even bothered including it on this list.

It might not be a single person, but I don’t think we could really end this article without acknowledging one of the greatest acts of human overconfidence in history.

This is true, but few of the people involved were directly, corporeally affected.

The submersible that went to look at the wreckage earlier this year, though...

7 Like • Comment • View 2 comments

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