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 5 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 7 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.

July 16, 2022 at 12:03am

#1035263

History of the Mundane

Today's article is from 2020, hence the COVID quarantine angle. But the interesting bits are historical, so feel free to ignore the COVID part just like you're ignoring COVID itself.

Ten Surprising Facts About Everyday Household Objects
While COVID-19 has us homebound, it’s a good time to reflect on the peculiar histories of housewares we take for granted

Throughout the world, from the humblest abode to the most lavish mansion, our homes have always been a respite from the world.

That is, until recently, when the internet brought the world inside our houses.

When we think of the technology that makes our homebound life bearable, we call to mind those electronic devices that allow us to remain connected to the outside world.

Nah, I think of beer. And modern beds. A few years ago, I sprung for a memory-foam mattress with an adjustable frame so I could sit up to read in it. Worth every penny, and it even came with a remote control. Now that's a fine use of modern technology.

But yeah, the internet has been competing with books, bed, and beer for my attention since the early 90s.

However, it might surprise us to know that, for our ancestors, many of the objects we take for granted, like napkins, forks and mattresses, were also once marvels of comfort and technology—available to only the few.

Well, not that surprising. Things like adjustable beds—or even nice comfortable mattresses—pretty much everything was once a luxury item, until it could be mass-produced enough to be accessible to us peons.

Our temperature-controlled homes filled with comfortable furniture and lights that turn on at the flick of a switch are luxuries unfathomable to the kings and queens of the past.

Yeah, but kings and queens had servants.

Perhaps, like me, you’ll find yourself grateful for our ancestors who suffered with stone or wooden headrests, stiff-backed chairs and cold nights before feather-stuffed pillows and fluffy duvets were part of everyday life (and appreciative of those who imagined that things could be better).

Non-homeless-like typing detected.

In The Elements of a Home: Curious Histories Behind Everyday Household Objects, from Pillows to Forks, I’ve uncovered the stories behind the objects that fill our homes and our lives.

And there it is, folks: the "Buy My Book" portion of the article. Look, the article's free; there's gotta be an ad of some sort. Apparently they're also getting paid by the History Channel (ugh) and of course they want you to subscribe to the magazine. For every great advance in technology and society, there's a price. Today, the price is goddamned ads. (Or, like me, you can install a script blocker. I had to open the article in another browser to see the actual ads.)

I'm not going to quote every section here. Just a few highlights.

The fork was once considered immoral, unhygienic and a tool of the devil.

This should not be surprising. There's a certain segment of the population that considers everything new to be a tool of the devil. It takes a generation or two, at least, for it to become mainstream (see also: Dungeons and Dragons). To be fair, though, a fork literally looks like a tool of the devil.

In 1004, Maria Argyropoulina, niece of the Byzantine emperors Basil II and Constantine VIII, was married to the son of the Doge of Venice.

Much nobility. Wow.

She brought with her a little case of two-pronged golden forks, which she used at her wedding feast. The Venetians were shocked, and when Maria died three years later of the plague, Saint Peter Damian proclaimed it was God’s punishment. And with that, Saint Peter Damian closed the book on the fork in Europe for the next four hundred years.

Okay, so what about the rest of the world? Europe was hardly the pinnacle of culture at the time.

Keys weren’t always pocket-sized.

Locks as we know them are a surprisingly modern invention. So are the doorknobs they generally inhabit.

However, the ones that opened the wooden locks of the massive marble and bronze doors of the Greek and Egyptians could be three-feet in length, and so heavy that they were commonly carried slung over the shoulder—a fact that is mentioned in the Bible.

I'm thinking Superman's neutronium key to the Fortress of Solitude here.

Ancient Romans, who lived extravagantly in most other aspects of their lives, were surprisingly spartan when it came to their bedrooms.

The poor slept on a straw mattress set in a simple wooden frame. If your purse allowed, the frame was cast in bronze or even silver, topped with a mattress stuffed with wool or down. The bed—and only the bed—resided in a room called a cubiculum (from which we get the word cubicle), a small space with tiny windows that let in little light.

And what do they tell us these days? "Use the bedroom only for sleeping and sex, and keep it dark. Especially for sex." Well, in that spirit, screw you; I read in there (see above). Still, in this respect (and a few more), the Romans were rather ahead of the curve.

Plates were once made out of bread.

Well, the dishwasher is a thoroughly modern invention, requiring running water and electricity. Without a dishwasher, I'd be eating off of bread too. Mine's broken right now. I mean, it works, but one of the rack rollers popped off. Fortunately, this is something I can fix myself once the parts I ordered come in. As opposed to my clothes washer, which finally died after 30 years and I had to buy a new one (so I got a new dryer while I was at it). Having to replace yet another critical appliance right now would... well, I could do it, but it would still be a massive pain in the ass.

Playing cards came from the only nation with the paper-making technology to pull it off: China.

The first known cards, developed in the ninth century A.D. were the size of dominoes.

Like I said, Europe wasn't exactly the pinnacle of culture around this time.

And yet, the standard cards we use now are based on French and English designs.

In 969 A.D., when Emperor Muzong of Liao capped off a 25-day drinking binge by playing cards with his empress, it’s doubtful he had any idea that his favorite pastime would travel the Silk Road through India and Persia before igniting a frenzy for the game in Europe.

Props to that guy for going on a 25-day drinking binge. That's impressive.

Eating on a bare table was once something only a peasant would do.

Yes, because unlike nobility, peasants didn't have servants to clean the damn tablecloths. I'm betting that having such things as tablecloths indicated that you were rich enough to have servants (in the days before washing machines), because it would have been just another thing to beat in the wash tub. It's kind of the same thing as, today, buying the most expensive car your credit score will let you.

If you could afford it (and maybe even if you couldn’t), the table would be covered by a white tablecloth, pleated for a little extra oompf. A colored cloth was thought to impair the appetite. (The exception to the white-only rule was in rural areas where the top cloth might be woven with colorful stripes, plaids or checks.)

Now that's interesting. If you want to indicate "country" these days, you show a picnic table covered with a checkered tablecloth.

I guess some fashions don't change as quickly as others. Like calling any innovation a "tool of the devil."

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