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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Brandiwyn🎶

about 22 hours ago
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.

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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.

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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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

July 11, 2024 at 11:41am

#1073824

We'll Always Have Paris

An entry for "Journalistic Intentions"

[18+] that is somewhat relevant to my current situation...

Paris 1924

Even the math-challenged might recognize that 1924 was exactly 100 years ago; and, given the hype surrounding the Olympics, to which even I, who avoids all sports-related news like I avoid the not-so-great outdoors, was subjected, by now you probably know that this year's Summer Olympics will be hosted by Paris. Again.

If nothing else, I'm pretty sure I've mentioned it here a few times in the context of "I want to visit France, but absolutely not during the Olympics."

Well, I'll provide a travel plan update on Saturday, hopefully, but right now we're looking at early autumn—comfortably after the crowd nonsense.

Meanwhile, though, the hype is in Seine.

(Again, no, I will not get tired of making that pun.)

Specifically, there's been a good bit of buzz surrounding the suitability, or lack thereof, of the River Seine for hosting the swimming events.

Paris is a big, old city, a statement that somehow sounds better in French: Paris est une grande vieille ville. Thus, Paris is full of shit (that probably sounds better in French, too).

This is not to single them out; all big, old cities are full of shit. That's not the cities' fault. Cities are full of people, pretty much by definition, and people are remarkably good at producing shit.

The removal of waste products (as well as the distribution of water) has a long history. Some credit the Romans for having the civil engineering acumen to create aqueducts and sewers. They even dug a massive tunnel under the city for people to shit into; this was called the Cloaca Maxima, and, reportedly, it even had its own goddess, Cloacina. I don't know if they were really the first; much of the history we learn here is Eurocentric. Doesn't much matter for my point here. Before then, and even later in smaller cities and towns, the waste would just run off down the street and hopefully get washed out with the next rain.

But even the Cloaca Maxima had to discharge somewhere; goddess or not, it wasn't magically transported to some faraway Land of Shit (Secaucus, perhaps). In that case, it flowed into the Tiber and thence out to sea.

Other cities followed suit, hiding waste flows underground and directing it into the river. For Paris, that was the Seine.

People eventually figured out that this wasn't a great practice, and designed sewer systems to be separate from storm drainage, but retrofitting older cities like that is a civil engineer's nightmare (and job security).

On top of that, the Industrial Revolution came along and started discharging stuff even worse than shit into the rivers.

All of this was an issue (pun absolutely intended) in 1924. One source,

though it's a British one, says swimming has been banned in the river from 1923, which you might recognize as the year before those Olympic games.

From what I can find, the swimming events that year were held at an indoor pool, Piscine des Tourelles,

which apparently still exists (albeit with a name change).

But what's really interesting is that that France and/or Paris (I'm a bit unclear on who's responsible) has been making Herculean efforts to clean up the Seine. That's a good thing. Or at least, from an outsider's perspective, it's a good thing. But according to that Guardian article I linked above, "The €1.4bn (£1.2bn) state-backed plan has entailed several years of work on wastewater management, treatment plants, filtering stations and storm basins to lower the river’s bacterial contamination from faecal waste."

Converting euros and/or pounds to US dollars or whatever your pitiful local currency is, I'll leave up to you. Call it a billion and a half bucks; close enough.

On the plus side, like I said: job security for civil engineers.

French persons will protest anything, though, including massive expenditures for river cleanup, especially when said expenditures aren't necessarily having the advertised effect.

For whatever reasons (some humans are simply inexplicable), some Parisians have been protesting by shitting directly into the Seine, thus bypassing any attempts to redirect human waste into treatment plants, or whatever it is they've decided to build. Last I heard, the Mayor of Paris

volunteered to go for a swim in the river herself to "prove" that it's clean. This resulted in the protesters producing a map of exactly where to shit upstream and at what time so that it would all hit her at the same time.

If she dies from e-coli, I wonder if anyone will get arrested for merde-er.

(Why yes, this entire entry has been the buildup to that multilingual pun, and I'm exceedingly proud of myself for it.)

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