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

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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 13, 2021 at 12:01am

#1017312

Past Lies

Today's article is a long one, and I couldn't blame anyone for not reading the whole thing, because I started skimming it after a while, myself.

Meet the People Who Believe They’ve Traveled to a Past Life
Christopher was an ancient Egyptian prisoner. Stephanie's dating the man who had her murdered. They and many others swear by the controversial benefits of past-life regression.

I also couldn't blame anyone for going right now, "Waltz, why are you even giving this bullshit oxygen?"

Bear with me on this one. No, I don't "believe" in past lives. But we also don't understand how the mind works, and we have to start somewhere.

The unsettling visions and sensations Benjamin experienced while imprisoned thousands of years ago were part of what he thinks may have been a past life. His mind traveled to that time and place during a session of past-life regression, a practice in which a person, under hypnosis or in a meditative state, experiences a memory that they believe is from a time when their soul inhabited another body.

I get unsettling visions and sensations sometimes, too. They're commonly known as bad dreams. We may not know much about the mind, but we do know it's capable of giving us sensations with little to no connection to external reality.

The American Psychological Association is deeply skeptical of past-life regression’s viability, and there are serious questions about the ethics of using it as a treatment.

Skepticism is good. Outright rejection can be counterproductive.

But, Bliliuos says, “In hypnosis, you go always to the most important memory you’ve experienced,” whether that’s in this life or perhaps a previous one.

I feel obligated to point out -- in case you skim the part where the article mentions it -- that another thing we do know is that hypnosis is very, very good at installing false memories.

Notions of reincarnation are diverse and nuanced, but for past-life regression advocates like Eli Bliliuos, the New York hypnotist, “The basic belief is that we are souls; we choose to incarnate these bodies for purposes of learning from experience, growing from experience.”

I want to respect peoples' beliefs, but at the same time, how about some real evidence? Or, at the very least, a plausible mechanism whereby memories can be transferred? And what about other hypotheses about what's going on here? Like, for instance, as I noted above, false memories or dreams?

Full disclosure, though, I've had dreams that seemed to be about past lives, also. That doesn't mean that it's any more than "seemed to be." I've forgotten, on a conscious level, almost everything I've ever read or heard, but who's to say that some tidbit from history or a novel set in the past didn't lurk in my subconscious? As support for this, I also read a lot of science fiction, so one might expect unconscious visions of the future, spaceships or other worlds in such instances, and behold, yeah, I get those too.

Once a traditional psychotherapist, Weiss... has written that he was a past-life regression skeptic at first. But a hypnotized patient of his, whom he called “Catherine” in one of his books, recounted past-life memories that were so precisely outlined and, as it turned out, historically accurate, that he felt it was impossible she could have invented them.

One way to verify reincarnation would, of course, be to find details that fit historical facts. But the question would always remain: how can you tell the difference between that, and someone's vivid but subconscious memory of reading history books or memoirs?

Bliliuos and other advocates of past-life regression say that you don’t have to believe in reincarnation to benefit from the experience. He recalls one session with a client who told him that they thought their past-life regression was a figment of their imagination.

“That’s perfectly fine,” Bliliuos responded, but he asked them to at least consider whether the vision might be a message from their unconscious. If it had some relevance to their life now, then the important element of the experience was the lesson they took away from it.

“Who cares if it’s ‘real’?” he adds.

For me, though, this is the important bit. The fact of a phenomenon remains even if none of the explanations for it fit (I have another article in the wings about UFOs, or UAPs as they're trying to get us to call them, and I had many of the same thoughts reading that). I look at the past-life thing as a kind of a metaphor, and humans can be better at relating to metaphors than to reality. If it helps one deal with one's problems, then what's the harm?

Well, for one thing, the harm is that if you start believing bullshit, there's no end to the bullshit you could believe.

I once had a conversation with Kid Me. We were chatting back at the place where I grew up, a sunny summer day, everything golden, green and still, with the oppressive humidity that anyone in the general vicinity of Washington, DC knows all too well. Not to mention the clouds of gnats that we both swatted at during our discussion. Sight, smell, sound... all there, all achingly real. Going into the conversation, I had no memory of speaking to my older self. What we actually talked about, I don't remember.

Of course it was a dream. But I woke from the dream with the strange sensation of remembering the conversation from both points of view: Kid Me and Decrepit Me. More, though, I was at peace about... something. Damned if I remember what, but I vividly remember the sensation of being calm and content. It was as if the conversation had settled something that had been bugging the back of my head (like those goddamned gnats) for all those years.

I'm under absolutely no illusion that I "actually" went back in time and talked to my younger self. It was a dream, a construct of a sleeping mind. But the dream, or hallucination, or mental exercise, whatever you want to call it, had a real effect.

At least for a while. Then I had to go do something or other and it was back to the daily annoyances. But the point is that something about it changed my mental state, at least temporarily.

So, similarly, I wouldn't jump to the conclusion that "I" (however that is defined if there's no continuity of consciousness) experienced an actual past life, but if there's something in the story, metaphor, or memory that helps me come to grips with something... well, that's okay. But will it, though? That's the real question. As the article points out, it can also lead to harm.

I just wish people wouldn't attribute extraordinarily unlikely things to the activities of one's mind. Sure, I can't rule out reincarnation, any more than I could rule out unicorns or underground lizard people or Bigfoot or space aliens. But I need more than just "I experienced X, so Y *must* be true." It doesn't follow, and there's no plausible mechanism for it.

“My conclusion, then,” Andrade states, “is that it is better to play it safe.” He advocates that people seek out more evidence-based forms of treatment instead.

Absolutely agree with this. But at the same time, we (by which I mean, people who train for this sort of thing) should be working on finding out more about how the mind works. More evidence, as it were.

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