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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 9 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 11 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.

June 12, 2021 at 12:02am

#1011710

Inevitable

"’Tis impossible to be sure of any thing but Death and Taxes."
-Christopher Bullock, 1716

The Notorious Legends and Dubious Stories of 10 Literary Deaths
Whose clothes were those, Poe?

The above quote about "death and taxes" is probably more famously ascribed to Ben Franklin, who died at the age of 84 and whose last words were reportedly, "a dying man can do nothing easy."

84 is a good run now, let alone in the 18th century. But whether 84 or 24, most deaths pass unremarked by the general public... unless the corpse was a famous figure in life.

Most of us don't choose the time and place of our demise, with notable exceptions such as Hunter S. Thompson. But the article linked above details some of the odd passings experienced by famous writers.

Fair warning: the best ones are from the Greeks. Obviously!

But yes, Poe is in there.

At least one source claims that Evelyn Waugh died “after attending an upsettingly modern Easter Mass,” but others suggest it was just the opposite...

I had to look up who Waugh was, because I'd never even heard of him. More interesting than his death was that he married a woman who was also named Evelyn.

The death of Poe is a long-standing mystery.

A lot of people didn't like the movie The Raven, the 2012 one with John Cusack. This is a real shame, because I think most viewers missed the point -- which was to make Poe's death (the events of the movie were clearly a dream or hallucination) something heroic. Still, it's fiction -- as is appropriate, I suppose.

They promised us Greeks, so here's Aeschylus:

He went outside the walls of the Sicilian city in which he was staying and sat down in a sunny spot. An eagle flying over with a tortoise in its talons mistook his shiny bald head for a stone. It dropped the tortoise on Aeschylus’s head, so that it could break its shell and eat its flesh. By that blow the source and beginning of tragedy in its more powerful form was extinguished.

Makes you wonder if, had he lived, he'd have invented gravity long before Newton.

[Francis] Bacon died of pneumonia in 1626, but according to John Aubrey’s Brief Lives, his illness was the result of a bizarre experiment to find out if “flesh might not be preserved in snow, as in salt.”

Appropriate for someone named Bacon.

Camus died in a car crash. Simple enough, right? ...Apparently, Camus once said that the most absurd way to die was in a car accident.

I can think of far more absurd ways to die, but most of them involve alcohol and maybe prostitutes.

As the story goes, Li [Bai] was drunkenly boating on the Yangtze River, and became so enamored of the moon’s reflection on the water that he tried to embrace it—and promptly drowned. A fitting death for a poet, amirite?

Or, you know... that. Or getting beaned by a turtle.

Common knowledge has it that Christopher Marlowe died in a bar fight—stabbed in the eye after a dispute over the tab, no less. But is that really what happened?

I mean, really, what's this one doing here?

We all know that Tennessee Williams choked to death on a bottle cap, but perhaps like me, you’ve always wondered: well, how?

Hey, now, we've all had that drunken escapade wherein we forget to take the cap off a bottle of beer before slugging it down. Most of us haven't died from that, though. Yet.

Less than a month before his 30th birthday, the lyric poet and husband of Mary Shelley drowned when his sailboat was caught in a storm in the Gulf of Spezia.

I like how he's taking second fiddle to his wife here.

But the really good legend is this: that after Shelley’s body washed up on shore, it was cremated—but his heart refused to burn. The story goes that Edward Trelawny plucked the unburnt heart from the pyre and took it home to Mary Shelley, and it was eventually buried with their son.

At least she didn't take her own literature too seriously, or we'd have Frankenpercy right now.

Here’s another tragedian of ancient Greece who supposedly died in an extravagant manner—legend has it that Euripides was torn apart by wild dogs.

Really? Only the second Greek on the list and it's a probably apocryphal story about wild dogs? Come on, there have to be better literary death stories than this. Which means it's time for a...

Merit Badge Mini-Contest!

Know any good -- by which I mean bad -- death stories? You don't have to limit it to famous writers; anyone moderately-to-well-known will do. The one I like best (which will probably be the most ironic one, or the funniest, and yes, death can be funny) will earn the commenter a Merit Badge. So no ordinary ones or depressing ones like the many entertainers who committed suicide, please. Deadline, as always, is the end of the day today, Saturday June 12, WDC time

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