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:

Write_Mikey_Write!

about 2 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 4 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.

August 17, 2022 at 12:04am

#1036589

Business Development

As far as I'm concerned, history ended in 1969; I can trace my first memories to the moon landing that year. After that, it's "lived experience."

Overexposed: A History of Fotomat

Part of my lived experience was being a photographer. I learned early on how to develop film—black and white film, that is; color was beyond my resources, if not expertise.

Usually I'd take my color rolls to a reputable photo shop.

Sometimes, I'd use Fotomat.

Fotomat was in the photography business, offering tiny huts situated in shopping plaza parking lots that were staffed by just one employee. Men were dubbed Fotomacs. Women were known as Fotomates, and management required them to wear short-shorts, or “hot pants,” in a nod to the strategy used for flight attendants at Pacific Southwest Airlines.

"But at least we're equal opportunity!"

Seriously, though, if you don't remember those kiosks, look at the picture in the article. How's anyone supposed to know if the "Fotomate" is wearing hot pants? Or pants, period?

Cars pulled up to the Fotomat location and dropped off film they wanted processed. After being shuttled via courier to a local photo lab, it would be ready for pick-up the following day.

Meanwhile, everyone involved got a good look at the naughty pictures you'd taken.

In the 1960s, Americans were fond of Kodak Instamatic cameras and film. People submitted the familiar yellow spools full of images from weddings, birthdays, trips, and other social events to photo processing labs, which might take days to return prints.

"...and this is a picture of my thumb... and a close-up of my thumb..."

You might expect me to rag on Instamatics. I will not do so. They had their place. One of the first lessons I learned while doing photography was, "It's not the camera; it's the photographer." This did not stop me from owning a Nikon and lusting after a Hasselblad.

The concept of a kiosk where people could easily drop off and pick up film that would be ready overnight originated in Florida, where Charles Brown opened the first location in 1965. After buying Brown's stock shares and arranging for a royalty, Fleet and Graham founded the Fotomat Corporation in 1967, with Graham president and Fleet vice-president.

I didn't know they were quite that old. But that stretches back into what I'm calling "history."

Charles Brown isn't exactly a rare name, though I question the morals of any Brown naming their kid Charlie after the mid-50s. Another famous Charles Brown ran AT&T for a while, and, if I recall, oversaw its breakup. Or maybe it was the same guy; I can't be arsed to research it. That Charlie Brown was one of the reasons I got into computers, but that's a story for another day.

While it was relatively easy to slot in a Fotomat hut in a parking lot, a business operating as an island surrounded by traffic had its problems. Remembering an old Fotomat in New Dorp on Staten Island, residents on Facebook recalled plowing into the kiosk or backing into it.

It had a bright yellow roof for a reason, but Staten Idiots are blind.

There was also the matter of bathrooms: They weren’t any. Employees often made arrangements to duck into local supermarkets or other stores when nature demanded it.

I guess Amazon followed their lead on a much larger scale.

But Graham’s controversial business practices made him a short-timer. In 1971, he was ousted from Fotomat over allegations he was misusing funds for his own personal gain, including his political interests—Graham was a supporter of both Richard Nixon and football player-turned-congressman Jack Kemp, who became an assistant to the president in the Fotomat corporation and referred football pros to become franchisees.

Now, see, if I'd have known that, I'd never have used them. Even though 1971 was long before I was able to drive up to one.

By the early 1980s, Fotomat—now minus Fleet, who had sold off his shares, and Graham—had opened over 4000 locations. That was both impressive and problematic. Fotomat had far overextended itself, sometimes opening kiosks so close to one another it cannibalized sales.

You'd think Radio Shack would have learned from that, but they did not.

The real death blow for Fotomat, however, wasn’t over-expansion. It was the emergence of the one-hour minilab.

Also known in the photography business as "Freakin' Sorcery."

The company tried to recalibrate, converting home movies to videotape and even offering VHS rental during the VCR boom of the 1980s, but it wasn’t successful.

Maybe if they'd offered a VHS-by-mail service?

Fleet, who had exited Fotomat years prior—the company had been sold to Konica—was no worse for the wear. Prior to his death in 1995, he authored a book, Hue and Cry, which called into question the authenticity of works attributed to William Shakespeare.

Questioning the authenticity of works attributed to William Shakespeare is a giant red flag that you're an elitist asshole. "How could a commoner have penned such art?"

Graham’s future after Fotomat was far more colorful. Promoting a bogus gold mining operation he named Au Magnetics, he promised he could turn sand into gold. Instead, he was accused of fleecing investors.

You can turn sand into gold. It would cost more than you'd get for the gold. That doesn't stop people who have a hard-on for gold.

As for the Fotomat locations themselves: Following the company’s collapse, many were repurposed into other businesses. Some became coffee shops; others morphed into watch repair kiosks, locksmith huts, windshield wiper dealers, or tailors. Presumably, none of the owners who took over mandated their employees wear hot pants.

It's been a while since I've seen one; I think all the old kiosks around here were completely removed. It's not like they were built to last. Only their memory remains, a faded photograph on a questionable website. I feel no nostalgia for them, however; they were a product of their time, and today's digital photography is far superior to that of the Instamatic era (though not to the late, lamented Kodachrome).

And on that note, this song is appropriate today:

They give us those nice bright colors
Give us the greens of summers
Makes you think all the world's a sunny day, oh yeah
I got a Nikon camera
I love to take a photograph
So mama, don't take my Kodachrome away

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