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

October 16, 2022 at 12:02am

#1039277

Hall of Fame

I don't really have a lot to say about today's article. It was just interesting to me, and something I'd never heard about before.

'The Limping Lady': Virginia Hall, the One-Legged World War II Superspy

I'd heard of one-armed bandits, but not one-legged superspies.

To plow forward through 50 miles of dangerous hiking on foot would be arduous in the extreme. But if she remained, she’d almost certainly be captured by the Nazis, who now considered her their most feared Allied spy.

As much as I hate doing outdoors stuff, if the alternative is to be captured by Nazis, I'd take the hike.

Cuthbert was what Hall had named her wooden leg. It was going to be a long journey.

I was curious why an American would name her wooden leg "Cuthbert." The only Cuthbert I knew of was a monk at Lindisfarne (which, incidentally, if you're ever in England, is absolutely worth the trip—do it before sea level rise cuts off the only land access).

So I looked up St. Cuthbert, and what's the first thing I see on his Wiki page?

An image with the caption "Cuthbert discovers a piece of timber, from a 12th-century manuscript of Bede's Life of St. Cuthbert." Like, seriously, someone in the 1100s found this event so transcendentally important that they painted it.

Why it matters that Cuthbert discovered a piece of timber, so thrilling that Bede illustrated it, shall remain a mystery. was England deforested in his era? Was it a Norse timber and they were about to be invaded (again)? I can't be arsed to research it any further than Wikipedia. But I found it amusing that Hall named her wooden leg after someone apparently famous for discovering a hunk of wood.

Though it would be decades before the world knew the full extent of Hall’s efforts during the war, it was clear from a young age that she seemed destined for an exceptional life. Born on April 6, 1906, in Baltimore, Maryland, to parents Edwin and Barbara Hall, Virginia enjoyed a comfortable upbringing and could have easily settled into a sedentary existence.

I also found it amusing that a couple of Baltimorons would name their daughter after a neighboring and rival state.

In school, she picked up several languages and appeared disinterested in conforming to the societal expectations of the time. She happily accepted parts in plays intended for boys and enjoyed being slightly provocative, once shocking classmates by showing up at school with a “bracelet” made of live snakes around her wrist.

I think I would have liked her.

During a bird hunting expedition in 1933, she discharged her firearm into her foot while climbing over a fence. The blast from the 12-gauge shotgun caused severe injury, and the resulting gangrene forced doctors to amputate half of her left leg below the knee.

I think for most people, literally shooting yourself in the foot would end an adventuring career.

Hall’s greatest disguise, though, was achieved by taking advantage of chauvinism. Few men believed women could be effective spies—particularly one with a wooden leg.

Nowadays, of course, everyone expects the spy to be female. Due to some unfortunate choices by Hollywood, we also expect them to exclusively run honey traps.

Still, I imagine that spycraft involves doing stuff that the enemy wouldn't expect.

She connected with a brothel in the city of Lyon, France, where she was able to gather intelligence from prostitutes that had met with German troops.

So you don't have to run the honey trap yourself.

Her contributions grew so significant that the Gestapo began searching France for la dame qui boite, or "the lady with a limp."

Seems like they finally figured it out, though. The Nazis were a lot of things, but stupid wasn't one of them.

When Hall arrived in Spain, she was promptly arrested for not having a passport. It was better than facing a horde of angry Nazis.

Also better than hiking one-legged through the Pyrénées.

Despite her reputation in Nazi-occupied France, she insisted on returning, adding grey to her hair, drawing wrinkles on her face, and even having her teeth ground down to alter her appearance, according to author Sonia Purnell, who wrote a book on Hall titled A Woman of No Importance.

"We found a woman with a fake leg, but she has grey hair. Can't be the same spy." Okay, maybe some Nazis were stupid. Or maybe Hall was just that good.

Anyway, like I said, not a lot of snark here; just something I thought I'd share. At least her name wasn't Eileen; then I would make some more puns like the horrible one in the title..

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