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:

tj-Merry Mischief Maker

about 11 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

July 3, 2024 at 7:45am

#1073489

A Bridge Too Far

Given the proliferation of confidence schemes these days—business, political, religious, and otherwise—I wanted to save and share this article about the con artist to whom every other grifter is compared.

‘I’ve Got a Bridge to Sell You’: The Con Artist Who Peddled the Brooklyn Bridge
Dean Jobb on John McCarthy, the "last of the old-time crooks."

It's kind of a long read, and I won't be quoting much.

The New York police considered him “an aristocrat of crookdom.” In the press, he was crowned the “dean of confidence men” and “the biggest of the big-time” swindlers. The Brooklyn Daily Eagle hailed his signature fraud as “the epic of the confidence world.”

His real name appears to have been John McCarthy. And he was the con man who sold the Brooklyn Bridge.

Well, not really, as the article goes on to point out. I'd always wondered about that. No, he sold fractional shares of the thing (at the time, it was a toll bridge), like a stock trader or banker selling collateralized debt obligations—the latter of which may not technically have been fraud, but turned out to be a Bad Idea anyway.

It was hailed as the Eighth Wonder of the World. The Brooklyn Bridge was “the greatest work wrought by the hand of man” in the nineteenth century, Brooklyn’s Daily Eagle proclaimed on the day it officially opened in May 1883, a “monument to human ingenuity, mechanical genius and engineering skill.”

As an engineer myself, I was always fascinated by bridges. Of course, by the time I hit the scene, many bridges had surpassed its superlatives, but as with other massive public works projects, we had to start somewhere.

The bridge was also a magnet for crime. Allegations of fraud, graft, and political corruption dogged the massive project during the construction phase.

Come on, now, that's just New York.

New York police estimated McCarthy reaped as much as a million dollars during his swindling career.

And this was back when, as Tom Waits put it, "a million was a million."

McCarthy’s swindle has become the ultimate con, synonymous with gullibility and blind trust. The catchphrase, “if you believe that, I’ve got a bridge to sell you” has been around almost as long as the bridge has been standing.

Did you know "gullibility" isn't in the dictionary?

And if you just went to look it up, I have some land in Florida for you to buy.

When New Yorkers celebrated the bridge’s one hundredth birthday in 1983 with parades and fireworks, the structure went on sale for real. Bits of wooden walkway and slivers of cable salvaged during repair work were sold as souvenirs, and ten dollars bought a certificate purporting to be the deed to the bridge, signed by Sid E. Slicker.

I was there for part of that year, though I didn't attend any of the celebrations. But the selling of souvenirs from it strikes me as something also ripe for cons, like churches having a saint's knucklebones or a sliver of the True Cross. I used to have a chunk of the Berlin Wall, sent to me by a friend in the Army stationed there at the time. Just a hunk of concrete, really. Could have come from anywhere. At least I didn't pay for it.

Mostly, though, I find the name "Sid E. Slicker" disproportionately hilarious. (Of course it's made-up, but it harks back to the many aliases used by McCarthy.)

Unlike some people, I don't really find this sort of thing fascinating. But it did occur to me that con artists serve an important function in society: they keep us from trusting too much. Skepticism is good. Would I prefer to live in a society where there isn't a thief or scammer on every corner? Of course. But as it is, at least it makes me know better than to buy naming rights to a star, or an extended warranty for my car.

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