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Tuesday, Apr. 30, 2024

4:48pm EDT

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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/

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Show Recent Comments

10 Most Recent Blog Comments:

GeminiGem of House Lannister

a day ago
In response to "Supplementary"

Working in health care for a zillion years, I've heard it all. The whole thing about something being natural, therefore it would be good for a person is very, very, very...very prevalent. An internist I worked with had a fantastic comeback to that line of thinking. She would say, "Arsenic is natural, are you going to start taking that?"

1 Like

Lazy Writer est 4/24/2008

Apr 28, 2024 at 2:14pm
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

Apr 28, 2024 at 10:43am
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

September 22, 2021 at 12:12am

#1017850

Out There

I've seen several UFOs.

‘What I saw that night was real’: is it time to take aliens more seriously?

But as usual, the answer to the headline question is "no."

In this case, a qualified "no."

The Pentagon has been quietly investigating unidentified flying objects since 2007. The fact that they think they might exist is good news to those who claim to have seen them

There's a difference between acknowledging that there are things in the sky that we can't identify, and jumping to the conclusion that they must be sentient alien life. UFO, as everyone knows, stands for Unidentified Flying Object. The Pentagon attempts to change this to UAP, or Unexplained Aerial Phenomenon, which sounds to me like they're trying to lump in things that don't qualify as "objects" while at the same time taking away the implication of "aliens" that has encrusted the term UFO like barnacles on a barge.

So when I say I've seen several UFOs, I mean I've seen shit in the sky that I, personally, couldn't identify (some of which I did, in fact, identify later on); that doesn't mean I saw flying saucers (which I didn't).

The unclassified version of the report (there was also a classified version seen only by US lawmakers) found “no clear indications that there is any non-terrestrial explanation” for the sightings. But neither did it rule it out.

Of course it didn't rule that out. I can't rule out the existence of vampire bats living in Saturn's rings, either; that doesn't mean that there is one there.

If the US military has been quietly and seriously investigating UFOs (or, as the Pentagon would have it, UAPs) since 2007, and if the Pentagon’s official report cannot rule out the existence of extraterrestrials, is it time we looked again at claims of close encounters and the people who have made them?

There is one fact that we can take as a certainty (or as close to a certainty as anything): People have reported UFOs.

Some of those reports were hoaxes. But there is no doubt in my mind that some people have seen something. I'm not denying the existence of UAPs; I'm just cautioning against making the cognitive leap to "it must be visitors from outer space."

Andrew Abeyta, professor of psychology at Rutgers University, co-authored We Are Not Alone, a study into why some of us want to believe in aliens. Abeyta explains that belief in aliens is akin to religiosity: unfounded beliefs in unfalsifiable ideas, which require a leap of faith.

The comparison of jumping to that conclusion to believing in some sort of god hasn't escaped me, either.

I tell Abeyta about an interview I carried out with a young man in Florida. The man, who did not want to be named, described an ambiguous close encounter that took place during his sleep.

My working hypothesis for any described encounter that takes place when a person is asleep is always "sleep paralysis." Because I've had similar experiences; the only difference is I recognized them as the effects of sleep paralysis. Again, that doesn't rule out other explanations, but you're going to have to give up more evidence than "I was asleep and suddenly I was being probed on a flying saucer."

However, now even the Pentagon has conceded there’s more to UFOs than that. In its nine-page report it states: “Most of the UAP reported probably do represent physical objects given that a majority of UAP were registered across multiple sensors, to include radar, infrared, electro-optical, weapon seekers and visual observation.” In other words, there was something out there and the images were not technical glitches.

Any mechanical/electronic recording device is a much more trustworthy source of information than the human memory. Again, I'm not denying there was something there.

Godfrey, a no-nonsense Yorkshireman born and raised in Oldham

I'm just leaving this sentence here to note that the Guardian is clearly not above repeating itself. (There is no such thing as a non-no-nonsense Yorkshireman.)

Is it time to start taking these stories more seriously? “I’m not saying that I believe it’s literally true that these are alien spaceships,” says Pope. “But at the very least, these people who were previously disbelieved and ridiculed should be listened to and given a hearing.

This, I can accept. If someone says they encountered something, they shouldn't be ridiculed for it. On the other hand, we should always keep in mind the possibility that they're perpetrating a hoax -- many Bigfoot sightings have been revealed to be hoaxes, for example.

Does Pope think ETs are among us? “I don’t know. I am certain that they are out there, but whether they’re down here or not? I don’t know. I think it’s much more likely that we’re dealing with unmanned probes.”

Insofar as it's possible to draw comparisons between humans and entirely speculative technological creatures from other worlds, it stands to reason that just as we sent robot probes out to other planets before sending humans there, any hypothetical technological civilization might easily do the same. That's still speculation, though.

Anyway, the article is worth reading even if, like me, you're a skeptic. I know I've talked about this sort of thing before, but I'll reiterate my basic thoughts on the subject:

1) Life almost certainly exists on other worlds;
2) There is no certainty that the evolution of said life produces a technological civilization;
3) Any technological civilization will be made of life that has been evolving for a very long time (it took Earth 4 billion years).
4) During that "very long time," the planet would have to avoid all major cataclysms and have comparatively few minor ones (such as the dinosaur-killer).

I'm calling the dinosaur-killer a minor cataclysm only because it didn't completely clear the planet of life. For major cataclysms, I mean like the planet breaking up or spiraling into a gravity well; the sort of thing that would wipe out all life.

Basically, I find it unlikely. Some say that to think we are unique, or alone, is hubris -- I say that the hubris is in thinking that evolution necessarily produces something like us. It does not. Evolution only selects for traits necessary for survival, and there are plenty of species that survive quite well without having developed big brains and the ability to produce spaceships; and only one that did so (on this planet).

Therefore, I'm not saying technological aliens don't exist; I'm saying they're probably exceedingly rare. (Also, please spare me the "we're a bunch of fucking assholes so why would anyone be interested in us" bullshit -- we study geese, and geese are assholes too.) And if you want me to believe that space aliens exist, show me one, or at least their devices.

So if some of these people aren't perpetrating hoaxes, if instruments detected something unidentified, if some of the stores are compelling -- okay. But none of that is positive evidence of alien visitation.

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