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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Lazy Writer est 4/24/2008

about 14 hours ago
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

about 18 hours ago
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

a day ago
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Udon Thani

Apr 25, 2024 at 2:58am
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

Apr 24, 2024 at 4:26pm
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

Apr 23, 2024 at 6:51pm
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

August 30, 2019 at 12:33am

#965240

Life As We Know It

http://nautil.us/issue/17/big-bangs/the-unique-merger-that-made-you-and-ewe-and-...

The Unique Merger That Made You (and Ewe, and Yew)
All sophisticated life on the planet Earth may owe its existence to one freakish event.

Really, I debated with myself over linking this article here. It's not that it's very technical - it's not, at all; there's some biology lingo, but no math or physics. It's just, on the surface, not very relevant to much outside of biology. I, of course, find it fascinating.

But there's some speculation toward the end that's worth looking at. First, some highlights:

The transition from the classic prokaryotic model to the deluxe eukaryotic one is arguably the most important event in the history of life on Earth. And in more than 3 billion years of existence, it happened exactly once.

Well... [citation needed], I suppose. If it happened more than once, and the resulting organisms were out-competed or otherwise died off, we might never know. But I'll grant that the writer knows more about biology than I do; that's not a very high bar to clear.

If this story is true, and there are still those who doubt it, then all eukaryotes—every flower and fungus, spider and sparrow, man and woman—descended from a sudden and breathtakingly improbable merger between two microbes. They were our great-great-great-great-...-great-grandparents, and by becoming one, they laid the groundwork for the life forms that seem to make our planet so special. The world as we see it (and the fact that we see it at all; eyes are a eukaryotic invention) was irrevocably changed by that fateful union—a union so unlikely that it very well might not have happened at all, leaving our world forever dominated by microbes, never to welcome sophisticated and amazing life like trees, mushrooms, caterpillars, and us.

Now, those who like to deny evolution might ask something like "but if the eukaryotes were so successful, then why are there still prokaryotes?" I would suggest they read up on some basic biology; I'm certainly not going to waste time on that argument or its ilk.

“[Endosymbiosis] was taboo,” says Bill Martin. “You had to sneak into a closet to whisper to yourself about it before coming out again.”

That's legitimately funny.

The article goes on to lay out the case for eukaryotic origins - again, fascinating stuff, and it highlights how science gets shit done, but not really anything worth commenting on here - until we get to the end, where we find stuff that intersects with another of my favorite topics: extraterrestrial life.

This improbability has implications for the search for alien life. On other worlds with the right chemical conditions, Lane believes that life would be sure to emerge. But without a fateful merger, it would be forever microbial. Perhaps this is the answer to the Fermi paradox—the puzzling contradiction between the high apparent odds that intelligent life would exist elsewhere among the billions of planets in the Milky Way, and our inability to find any signs of such intelligence. As Lane wrote in 2010, “The unavoidable conclusion is that the universe should be full of bacteria, but more complex life will be rare.” And if intelligent aliens did exist, they would probably have something like mitochondria, too.

The "improbability" mentioned there is the unlikelihood of the formation of eukaryotes. Again, I'll point out that however improbable something is, once it happens, the probability of it having happened is unity - and since we eukaryotes wouldn't be around without it, we can't argue that something won't happen just because it's improbable. If that makes any sense.

I've also pointed out before that the "Fermi paradox" isn't a paradox. I don't accept the "high apparent odds that intelligent life would exist elsewhere." There's not a single rule, law, or requirement of evolution that those qualities that we call "intelligence" (please, can the "no intelligent life on Earth either" jokes; they're really tiresome) must occur, any more than there's a rule stating that eukaryotes must emerge.

So what this information does, really, is lower the odds of complex life existing in another biosphere. We don't know what those odds are, admittedly, because we simply don't have enough data points (we have precisely one data point - Earth).

Not that this idea is new to me; I've been sold on the Rare Earth Hypothesis for some years now. But this article goes into more depth than I've seen in the past, and does it without getting, as I said, too technical about it.

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