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.

January 3, 2023 at 12:01am

#1042554

Wake Up Call

Time to wake up?

The forgotten medieval habit of 'two sleeps'
For millennia, people slept in two shifts – once in the evening, and once in the morning. But why? And how did the habit disappear?

I can take a stab at that second question: the Industrial Revolution with its artificial lighting and the replacement of serfdom with wage slavery and social control.

It was around 23:00 on 13 April 1699, in a small village in the north of England. Nine-year-old Jane Rowth blinked her eyes open and squinted out into the moody evening shadows. She and her mother had just awoken from a short sleep.

Evening shadows at 23:00? Even the north of England in April would have been full darkness. Authors and their poetic license, I'm telling you.

Nearly 300 years later, in the early 1990s, the historian Roger Ekirch walked through the arched entranceway to the Public Record Office in London – an imposing gothic building that housed the UK's National Archives from 1838 until 2003. There, among the endless rows of ancient vellum papers and manuscripts, he found Jane's testimony. And something about it struck him as odd.

Having access to historical documents like that would be cool.

Originally, Ekirch had been researching a book about the history of night-time, and at the time he had been looking through records that spanned the era between the early Middle Ages and the Industrial Revolution. He was dreading writing the chapter on sleep, thinking that it was not only a universal necessity – but a biological constant. He was sceptical that he'd find anything new.

So far, he had found court depositions particularly illuminating. "They're a wonderful source for social historians," says Ekirch, a professor at Virginia Tech, US. "They comment upon activity that's oftentimes unrelated to the crime itself."

I know I'm contractually obligated to rag on Tech, but I admit they sometimes do good work.

But as he read through Jane's criminal deposition, two words seemed to carry an echo of a particularly tantalising detail of life in the 17th Century, which he had never encountered before – "first sleep".

Apparently unrelated to the first and second breakfasts for Hobbits.

In her testimony, Jane describes how just before the men arrived at their home, she and her mother had arisen from their first sleep of the evening. There was no further explanation – the interrupted sleep was just stated matter-of-factly, as if it were entirely unremarkable. "She referred to it as though it was utterly normal," says Ekirch.

A first sleep implies a second sleep – a night divided into two halves. Was this just a familial quirk, or something more?

That's gotta be tricky for historians. Things taken for granted at the time change, and we lose the cultural referents for them. No context.

Over the coming months, Ekirch scoured the archives and found many more references to this mysterious phenomenon of double sleeping, or "biphasic sleep" as he later called it.

So was he the one who came up with that term? I use it myself.

Some were fairly banal, such as the mention by the weaver Jon Cokburne, who simply dropped it into his testimony incidentally. But others were darker, such as that of Luke Atkinson of the East Riding of Yorkshire. He managed to squeeze in an early morning murder between his sleeps one night – and according to his wife, often used the time to frequent other people's houses for sinister deeds.

But does that imply that not everyone followed this pattern? Or that the time between sleeps was different for everyone?

Biphasic sleep was not unique to England, either – it was widely practised throughout the preindustrial world. In France, the initial sleep was the "premier somme"; in Italy, it was "primo sonno".

Thanks for limiting the "preindustrial world" to Europe. Also, it's not like those other terms were any more creative; they both translate (based on my limited understanding anyway) to "first sleep."

In fact, Eckirch found evidence of the habit in locations as distant as Africa, South and Southeast Asia, Australia, South America and the Middle East.

Okay, that's better.

And far from being a peculiarity of the Middle Ages, Ekirch began to suspect that the method had been the dominant way of sleeping for millennia – an ancient default that we inherited from our prehistoric ancestors. The first record Ekirch found was from the 8th Century BC, in the 12,109-line Greek epic The Odyssey, while the last hints of its existence dated to the early 20th Century, before it somehow slipped into oblivion.

"Somehow," my lazy ass. The article does go into this later, but the Industrial Revolution saw people start to be treated like cogs in a machine, an appropriate metaphor for the time. To keep the wheels of industry moving, it was important that everyone follow the same schedule of work, leisure, sleep, and there was no room for individual idiosyncrasies or the maintenance of agricultural-era schedules.

At the time, most people slept communally, and often found themselves snuggled up with a cosy assortment of bedbugs, fleas, lice, family members, friends, servants and – if they were travelling – total strangers.

Lest you think I'm straying into the territory of longing for an idyllic past... I'm totally not.

But humans aren't the only animals to discover the benefits of dividing up sleep – it's widespread in the natural world, with many species resting in two or even several separate stretches.

Anyone with cats knows this. Also dogs, to some extent. The word "catnap" exists for a reason, unrelated to the similar word "kidnap." Lovely language we have.

One example is the ring-tailed lemur. These iconic Madagascan primates, with their spooky red eyes and upright black-and-white tails, have remarkably similar sleeping patterns to preindustrial humans – they're "cathemeral", meaning they're up at night and during the day.

And that's my new word for the day. Actually it was my new word on whatever day I first found this article, but whatever.

As it turns out, biphasic sleep never vanished entirely – it lives on in pockets of the world today.

Including among some lazy retirees.

Collectively, this research has also given Ekirch the explanation he had been craving for why much of humanity abandoned the two-sleep system, starting from the early 19th Century. As with other recent shifts in our behaviour, such as a move towards depending on clock-time, the answer was the Industrial Revolution.

I'm not above saying I told you so.

However, though people weren't going to bed at 21:00 anymore, they still had to wake up at the same time in the morning – so their rest was truncated. Ekirch believes that this made their sleep deeper, because it was compressed.

Forgive me if I don't take the word of an historian at face value for that. Plausible hypothesis, but I'd want to see studies supporting or disproving it. Not that I can be arsed to find them tonight.

One major side-effect of much of humanity's shift in sleeping habits has been a change in attitudes. For one thing, we quickly began shaming those who oversleep, and developed a preoccupation with the link between waking up early and being productive.

Few things piss me off more than sleep-shaming. It's a blatant effort in the direction of universal conformity. Everyone has their own natural sleep patterns, which can be changed by effort, but I'm not convinced it's healthy to do so.

Despite near-constant headlines about the prevalence of sleep problems, Ekirch has previously argued that, in some ways, the 21st Century is a golden age for sleep – a time when most of us no longer have to worry about being murdered in our beds, freezing to death, or flicking off lice, when we can slumber without pain, the threat of fire, or having strangers snuggled up next to us.

Unless you go camping, of course. Which is one major reason I don't go camping.

As usual, there's a lot I'm leaving out here, so go check the article if you're interested. And if I ever hear "look who finally decided to join us" or "welcome to the land of the living" or other such attempts at sleep-shaming after waking up late in the morning again, I'll shove this article in the offender's face.

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