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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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.

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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').

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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.

December 5, 2022 at 12:01am

#1041403

In Loo Of

Today, some in-depth historical investigation from Cracked.

A Brief Timeline Of The Toilet

It wasn't even a month ago that I linked an article from the same source about pooping on the Enterprise: "Captain's Log"

. Instead of the (fictional) future, though, we're contemplating the past here.

Maybe one of the greatest drivers of any sort of modern innovation is convenience. Convenience, of course, being a more polite way of saying “not requiring a bunch of annoying bulls**t.”

I'm saying this again because I'm quite proud of coming up with it: if necessity is the mother of invention, laziness is the milkman.

There is one modern convenience, however, that, despite being only about 2 centuries old, is massively underappreciated. That, of course, is the existence of a modern toilet that’s never more than a room or two away.

It really is quite remarkable when you think about how long we've been pooping. Which, of course, other animals do without the need for porcelain. Maybe a sandbox, though.

The dirtiest dive bar toilet with a kicked-off stall door would seem medically sterile compared to some of the solutions the human race utilized for its waste in the past.

Having used more than my share of those, I concur.

3500 BC - First Pit Toilet

Mesopotamia is the site of one of the first known examples of a pit toilet, from around 3500 BC.

I had to look it up, but apparently that was Sumer, not Babylon. You know, the same culture that invented beer. I can't help but think the two were related.

The article implies that this was the first time we had a dedicated place for taking shits (or, well, technically, leaving shits), but it might well have happened earlier and we just haven't found evidence for it yet. Or, more accurately, Cracked didn't bother looking for evidence for it.

Incidentally, analyzing (heh heh he said anal) prehistoric shit is common enough that shit-fossils have a name: coprolites.

2000 BC - The First Flush

Versus a cesspit, which would have to be emptied by (a very gross) hand, the first “flush” toilet has been traced back to the ancient city of Knossos, with similar systems showing up in the Indus Valley (now Pakistan and India).

Again, not necessarily the first.

This exists with a latrine that had channels where water could be manually poured to clear the, uh, waste, down into the sewers.

The invention of the sewer is the important part, here. No matter how primitive or advanced the toilet, it relies on a sewer system. Okay, maybe as a civil engineer I'm biased here.

1300 BC - Chamber Pots

The next development comes from a battle with one of mankind’s oldest and most relatable battles: the desire not to get out of bed when you have to pee. We may have to trudge only a couple feet to relieve ourselves nowadays, but when toilets were massively less common and probably also included going outside? Well, you come up with a solution. A horrible, horrible solution.

Still a better solution than pissing the bed. Or, worse, trudging to an outhouse in the snow.

750 BC - Rome’s Public Toilets

Rome is a massively important location and empire in history, responsible for many developments and advancements that contribute to our modern lives.

That may be a massively Eurocentric perspective.

These public latrines, usually located in bathhouses, had rows of marble seats with holes where you could do your business directly into the running water of the sewer systems below.

But there's little doubt that they at least made massive improvements to water delivery and sewage disposal. Hell, they even had a goddess dedicated to the sewer: Cloacina. When you get a god involved, you know your culture takes something seriously.

1100s - Garderobes

Garderobes aren’t the most complicated form of toilet, but they might be the funniest. Garderobes would emerge mostly in medieval castles and some manor houses, and were basically little rooms that let you s**t off the side of your house.

Ah, yes, those rooms hanging off castles and manors. Peasants had no access to them, and some probably had to clean them. This was likely the origin of the idea of "trickle-down economics."

1596 - First Modern Flusher

The first toilet that resembles anything we might use today was invented in 1596 by Sir John Harington. It used water stored in a cistern above to flush away waste.

Like Leonardo, Harington was a man ahead of his time.

1770s - Water Closets

It would take until the 1770s for flush toilets to actually catch on. Central to this was Alexander Cumming’s 1775 patent for a flush toilet which included one of the most important developments, the S-Trap.

See, the central problem with crapping into an open sewer like the Romans did is it probably smelled worse than a porta-potty at a beer festival.

This design exists in modified form today, after many advancements, including the U-Trap, often attributed to, I’m not kidding, Thomas Crapper.

Some have attributed the invention of the modern toilet to Crapper himself, but I think that's more of a desire for history to make sense in a punny sort of way. And no, the word "crap" didn't come from his name. "Crap" had different meanings in the past, and the surname was probably one of the many surnames based on a family's historical profession, like Weaver or Hunter or Smith.

1850s - Indoor Plumbing

With all these advancements, it would still take until the 1850s for toilets to be generally sanitary and, well, not completely disgusting enough to be brought into regular rooms of an indoor home.

Not surprisingly, this roughly (pun intended) coincided with the invention of toilet paper in 1857.

Truly, that was the actual beginning of the modern age.

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