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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 4 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

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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.

December 18, 2021 at 12:01am

#1023333

Persistence

Things continue to improve, so I thought I'd once again tackle something from my queue. This time, without glasses on.

What do near-death experiences mean, and why do they fascinate us?
Psychiatrist Bruce Greyson has spent decades talking to people about near-death experiences. His work raises questions about what happens when we die, and how we ought to choose to live.

I can take a stab at the answer to the second question in the headline: "Why do they fascinate us?" Because they're the Unknown, and the Unknown fascinates humans. It's how we got to where we are now, for good or ill.

The meeting had been organised by Bruce Greyson, now a professor emeritus in psychiatry at the University of Virginia.

Part of the reason I saved this link was because of the link to my hometown and alma mater. But to my knowledge, I've never met the guy. If I had, I probably would have made a "Bruce Wayne / Dick Grayson" joke, because I'm an asshole. It wouldn't matter to me that the last name is spelled differently. I have to make a conscious effort to remember that "gray" is more common in the US, and "grey" in the UK. But that's irrelevant to names.

Here's where I note that the link above is from the Guardian, which uses British spellings and other conventions, and I make no attempt to Americaniz(s)e them here.

A month into his psychiatric training, in the 1960s, he had been “confronted by a patient who claimed to have left her body” while unconscious on a hospital bed, and who later provided an accurate description of events that had taken place “in a different room”. This made no sense to him. “I was raised in a scientific household,” he says, over Zoom. “My father was a chemist. Growing up, the physical world was all there was.”

I can understand that, as my father was a chemist too (sailor by profession, chemistry degree). I like to think I got my fascination with science from him.

The thing about science, though, is that its whole purpose is to investigate things that we don't understand. To me, that includes subjects that are traditionally in the realm of mysticism, spirituality, and, well, the Unknown. Not all scientists agree, and so you get people scoffing at research into parapsychology.

UVA has, or at least had, a dedicated parapsychology department. Not as famous as Duke's, maybe, and they had a rough time of it right after Ghostbusters came out, but personally I think applying scientific methods to study the things we don't understand is overall a good thing -- so long as the researchers can try to be objective.

Greyson presents his research in a new book, After, which is bound by a series of case studies.

I do seem to find a lot of articles that are basically book ads. As usual, I don't have a problem with this as long as the article is interesting.

Greyson says. Some people recall out-of-body experiences, or report travelling through a long tunnel; others meet entities they think of as God or Allah or long-dead family members; some feel time bend and warp, as though it were elastic.

Leaving aside for the moment that these sound a lot like dreams -- which could be the mind's way of trying to make sense of misfiring neurons or whatever -- the "long tunnel" thing has been a mainstay of NDE descriptions since at least the time when Moody's book came out (it's mentioned in the article) back in the 70s. At the time, I was an impressionable child, and I read it, like many people, hoping to make sense of the Unknown of death.

One thing about the tunnel imagery always bugged me: some people use it as evidence for reincarnation, saying it's reminiscent of the process of being born, from the point of view of the baby. I'll leave the myriad problems with this "hypothesis" up to the reader.

Given that near-death experiences happen with limited warning, they are almost impossible to test.

I have a vague memory of a movie, maybe back in the 70s or 80s, that dealt with testing this. But it was only a movie, a work of fiction.

At the University of Kentucky, the neurologist Kevin Nelson, who, like Greyson, has spent years recording NDEs as a kind of academic side-gig, thinks of the experiences as “a blending of two states of consciousness – wakefulness and REM sleep – during a time of great physical or emotional danger,” and argues that many NDEs are “dream-like”, existing in a neurological “borderland”.

Like I said, similar to dreams.

In After, Greyson writes: “I take seriously the possibility that NDEs may be brought on by physical changes in the brain,” though he also accepts that the mind might be able to function “independent” of it. There have been reports of people experiencing near-death episodes while their brains are inactive, he says, and “yet that’s when they say they have the most vivid experience of their lives.” This doesn’t make sense to him. Partway though our conversation, he asks: “Are these the final moments of consciousness? Or the beginning moments of the afterlife?”

I think we all know I'm a materialist. But if actual evidence is produced, I'm willing to accept it. This isn't evidence. This is speculation.

When I ask him what his current logical understanding is, he looks resigned. “It seems most likely to me that the mind is somehow separate to the brain,” he says, “and, if that’s true, maybe it can function when the brain dies.” Then he adds, “But if the mind is not there in the brain, where is it? And what is it?”

Despite the subject matter, I feel these are legitimate questions to ask. But they need to be addressed with scientific rigor, not biased to what someone believes to be true on faith alone.

To Greyson, the impact near-death experiences have on people’s lives has been his most surprising discovery.

Whatever they are, whatever the mechanisms behind them, I have little doubt that NDEs are real, any more than I doubt that some people have seen strange phenomena in the sky. But as with making the immediate cognitive leap to "must be space aliens" in the latter case, assuming that NDEs are proof of some afterlife is a massive leap to conclusions. Having a mind separate from the brain makes no sense with our current understanding of science.

Which doesn't mean our current understanding is completely correct, and that's why research is needed. You never know when a paradigm shift can occur.

Well, I've banged on long enough. Most people have already drawn their own conclusions, and that's fine. Me? I'm not making a conclusion, but do consider this: if the mind is separate from the brain, then why do physical changes in the brain (injury, trauma, chemical imbalance, Alzheimer's, etc.) get reflected in the individual's personality and thought processes?

There's no doubt, though, that there is much we don't yet understand. And I'd hope that there will always be things we don't understand.

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