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🎶

a day 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.

3 Like

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.

2 Like

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.

1 Like

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.

1 Like

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.

3 Like

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

1 Like

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

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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