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 3 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 5 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.

April 15, 2021 at 12:52am

#1008535

Fringe

How about a trip out on a limb, today?

The Hard Science of Reincarnation
All over the world, scholars studying reincarnation are making findings even skeptics have difficulty explaining.

Let me preface this by saying: I don't believe in this stuff.

Nevertheless, just like with out-of-body experiences, hauntings, UFOs, cryptids, and other phenomena considered "fringe," something is going on. Some of it is hoax, some false perception, and a lot of attention-seeking behavior is involved; but even discounting those things, there's something there that, if it doesn't tell us anything about the world outside, can shed some light on how the mind works.

The trap is falling into the belief in these things wholeheartedly, finding "evidence" that supports said belief, and discounting that which does not. That's the worst kind of bias, and it's the sort of thing that divorces people from reality. For instance, the phenomenon of sleep paralysis is fairly well documented and partially understood. But it often tracks neatly with descriptions of alien abduction. Having experienced sleep paralysis myself, I can absolutely understand how some people can be convinced they were teleported up to the mothership and experimented upon.

It's also been established that certain frequencies of sound waves, too low to be perceived by our ears, can induce fear and hallucination. Hence, ghost stories.

While I'm absolutely on the side of science when it comes to these things, part of science is admitting that we don't know everything. Another part is trying to fix that. It's an ongoing process, hopefully one that never ends.

But the history of science is littered with the dashed hopes of people who wanted a thing to be true, a thing that turned out to not be the case. And I think "reincarnation" is going to end up being one of those things.

You'll notice I'm not quoting much from the article today. It's an interesting article, if a bit credulous, though it includes some of the skepticism involved. But one of the things it focuses on is my own former university's research into these phenomena, in a department I've known about since I started going there lo these many years ago. This was right around the time Ghostbusters came out, and afterwards, it was probably embarrassing for a major state university to have a Parapsychology Department, so I suspect that's why they ended up changing the name.

It's this quote from a skeptic, from the linked article, that I find most appealing:

“There could only be two possibilities. One is that there is something genuinely paranormal happening, and if that is true, that would be amazing,” he told me. “Or, alternatively—which is more the line that I do favor—it tells us something very interesting about human psychology. So either way, it's worth taking seriously.”

While I object to the characterization of "only two possibilities," I certainly can't think of others right now.

As for reincarnation itself, we know this: on a purely physical level, of course we're recycled. Whatever they do with your body after you kick it, those atoms don't just poof from existence; they get incorporated into the environment, perhaps even into other living things. You've almost certainly eaten carbon atoms that were once in the body of an ancestor of yours; you've definitely breathed air that has spent some time in other living beings. (People who fart in elevators cannot be trusted. Just saying.)

That's so obvious as to be trivial, which is why we don't think about it much.

What reincarnation purports to be, though, is a kind of continuation of consciousness, or at least of the memories that are incorporated into consciousness. And we don't understand consciousness, or even the way memories are formed and stored, so there's no way to say with certainty "this is utter bullshit." And yet, it's probably utter bullshit, or at least something else is going on that we don't fully understand.

Given consciousness's utter inability to conceive of its own nonexistence, the belief persists. So I'm all for the research. Just let's not be too trusting of our own conclusions about it.

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