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:

tj-Merry Mischief Maker

about 11 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

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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