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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 22 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

a day ago
In response to "Revisited: "Exercise""

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

Elisa the Bunny Stik

Apr 27, 2024 at 10:33am
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 11, 2019 at 12:26am

#964037

Inalienable

Oh, here we go again.

https://undark.org/article/pitfalls-of-searching-for-alien-life/

The Pitfalls of Searching for Alien Life
Scientists looking for evidence of extraterrestrials can draw media attention but also cynical, even hostile, reactions from their colleagues.

This just in: scientists are humans (as far as we know), and humans have opinions.

In October 2017, a telescope operated by the University of Hawaii picked up a strange cigar-shaped object (artist rendering in top image), which had slingshotted past the sun at a more-than-brisk top speed of 196,000 miles per hour. Scientists at the university dubbed it ‘Oumuamua, Hawaiian for scout, and at first labeled it an asteroid, then a comet, but agreed that it came from another solar system.

Let me get this out of the way, since it's in the lede: it's remarkably easy to eject a perfectly natural, orbiting thing from a solar system. All you need is the thing you're ejecting, and an enormous mass to overcome the gravity of the primary. Hell, Jupiter could do it - and we've found many, many objects more massive than Jupiter orbiting other stars. Eventually the objects in their orbits line up just right, and *sproing*.

Consequently,

...Avi Loeb, the chair of Harvard University’s astronomy department, published a paper in The Astrophysical Journal Letters the following year theorizing that the object could be artificial.

Hey, it's great to speculate, isn't it? "Could be artificial" translates to "can't rule it out," but as I noted above, it's unnecessary to postulate an artificial origin. Occam's Razor does the rest.

I've gone into my argument against a proliferation of intelligent alien life before. The short version: I like Star Trek as much as anyone and more than most, but... no.

The long version: It comes down to one term in the infamous Drake Equation.

Specifically, f_i, the fraction of planets with life that go on to develop intelligent life (the definition in the popnote above will suffice for this). Naturally, all we have for that is one data point: us. This tells us nothing about the probability of a planet producing intelligent life. As an analogy, think about if you won the lottery, but you have no idea what the odds were of winning the lottery - all you know is that no one you know has won the lottery. You don't even know how many of those people have played the lottery. The chance could have been one in a thousand or one in 10¹⁰⁰; it doesn't matter, because your chance of having won the lottery is exactly unity, once you've won it.

We do know that it didn't take long for life to develop on Earth, and that it took a comparatively very long time for "intelligent" life to emerge. And we've only been building spaceships for half a century, compared to the 4 billion + years life has existed, or even the 1 million + years that humans have existed. Meanwhile, other species are thriving just fine on our planet, having evolved for exactly the same amount of time that we have (they would probably be doing better if we weren't around, but that's another issue entirely), so there's zero evidence that evolution must produce that kind of intelligence. All we know for sure is that it can, and we know that because we define ourselves, as a species, as "intelligent."

The article I linked, though, seems to make the usual mistake I see in journalism on this subject: it conflates "life" and "intelligent life." Throw a dart at the timeline of Earth thus far, and you're almost certain to hit a time containing "life," and almost entirely unlikely to hit a time containing "intelligent life."

“I’m really surprised they get blowback,” said physicist Richard Bower of Durham University in England. He’s never gotten flak for his research in cosmology, which entails making computer simulations of possible parallel universes. He’s concluded that life elsewhere could be quite common, and others in his field back him up. “We used to say that life is incredibly rare and we’re lucky to live on a habitable planet,’’ he said. “But we’ve now observed so many planets that are plausible habitats. It seems, based on scientific evidence, there’s no reason to think that planets like the Earth are rare.”

I'm not arguing that. I'm arguing that Earth may well be a galactic outlier in terms of producing a species that has space travel capabilities (as rudimentary as they are right now). We also don't know how common any kind of life is in the rest of the universe - but as it's a big universe, it's probably safe to say it exists, somewhere.

It's also important to draw distinction between scientific inquiry and fantasy, though. The search for extraterrestrial life is important. Finding so much as a microbe, even just a strand of proto-RNA or its equivalent on another world, well, that would have a huge impact on our understanding. But the topic has been forever tarnished by fringe UFO conspiracies and popular imagination, to the point where even writing serious papers about it invites speculation about flying saucers, death rays, the JFK assassination, and whatnot.

Hence the blowback when serious scientists start speculating about alien intelligence. Such speculation is not without merit - it needs to be done. It's just that when being reported on, it gets all wrapped up in popular imagination and cultural zeitgeist.

Now, I'm not trying to change anyone's mind, here. You want to believe in little green men, or storm Area 51 next month, hey, that's your choice. And I could be wrong. I actually hope I am. All I'm doing is critiquing the way these things are presented in media. So when you see things like "this might be artificial in origin" or the ever-present "astronomers discovered an Earthlike exoplanet," just remember to take the news with a grain of salt. After all, to an astronomer, Mercury and Venus are Earthlike - and the last so-called "earthlike" planet I heard of was six times the size of our little world.

Imagine trying to escape that gravity well.

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