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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Write_Mikey_Write!

2 minutes 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.

Adherennium Plotting something

about 2 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.

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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.

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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.

May 31, 2024 at 7:14am

#1071930

Pros and Consciousness

If you're not conscious, this article isn't for you.

The nature of consciousness, and how to enjoy it while you can
In his new book, Christof Koch views consciousness as a theorist and an aficionado.

Well, I guess he's solved the Hard Problem,

then.

Now, with AI systems behaving in strikingly conscious-looking ways, it is more important than ever to get a handle on who and what is capable of experiencing life on a conscious level.

Oh, that's an easy one. Only I experience life on a conscious level. The rest of you, including the AIs, only mimic actual consciousness.

Solipsism makes everything simpler.

Koch, a physicist, neuroscientist, and former president of the Allen Institute for Brain Science, has spent his career hunting for the seat of consciousness, scouring the brain for physical footprints of subjective experience.

So, okay, someone with credentials, and not just some guy (like me). Doesn't mean he's right, mind you (pun intended), but it does mean it catches my interest.

It turns out that the posterior hot zone...

Posterior hot zone? Seriously? That's what you geeks are going with? You're just begging for it, aren't you? Okay, I'll bite: "Scarlett Johansson has a gorgeous posterior hot zone."

(In reality, I don't find asses to be attractive. But that's never stopped me from making jokes.)

Seriously, though, shouldn't they have looked up those words in their handy Latin dictionaries and called it that, like they do with most chunks of anatomy? Google tells me it's "calidum zona," because I haven't had an actual Latin course in 40 years.

Moving on...

...a region in the back of the neocortex, is intricately connected to self-awareness and experiences of sound, sight, and touch.

This ties in with what I believed to be the reason for consciousness: the nervous system had to evolve in such a way as to integrate sensory experiences, and those mechanisms got hijacked into "awareness." But I'm just some guy, so you shouldn't take that seriously.

Dense networks of neocortical neurons in this area connect in a looped configuration; output signals feedback into input neurons, allowing the posterior hot zone...

Snort.

...to influence its own behavior. And herein, Koch claims, lies the key to consciousness.

Makes sense, sure, but has he, or anyone else, done the science to back that up?

This declaration matches the experimental evidence Koch presents in Chapter 6: Injuries to the cerebellum do not eliminate a person’s awareness of themselves in relation to the outside world.

Okay, so there is some support.

His impeccably written descriptions are peppered with references to philosophers, writers, musicians, and psychologists—Albert Camus, Viktor Frankl, Richard Wagner, and Lewis Carroll all make appearances, adding richness and relatability to the narrative.

I mean, that's probably good writing; I'm not sure it's good science. As this is basically a book ad, though, I can cope.

The takeaway from the first half of Then I Am Myself the World is that IIT might offer the best-fit explanation for consciousness—a view, it’s worth mentioning, that is highly contested by many other neuroscientists.

Good. That's how science gets done.

Koch discusses transformative states of consciousness in the second half of his book, including near-death, psychedelic, and mystical experiences.

Aaaaaand that's not.

He also discusses the expansive benefits of sustained exercise—drawing upon his personal experiences as a bicyclist and rock climber—through which a person can enter “the zone.”

The zone? You're telling us how to enter the posterior hot zone?

Koch suggests that exercise, meditation, and the occasional guided psychedelic might be beneficial to many people.

Jokes aside, he's not the first scientist to come up with that nugget. Timothy Leary comes to mind, though it's arguable whether psychologists are real scientists.

Oh, and no, he didn't solve the Hard Problem of anything except "how to market your book." Nevertheless, I found this review/ad interesting enough to share. Even if he is talking out of his posterior hot zone.

1 Like • Comment

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