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!

about 5 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 7 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 27, 2024 at 10:14am

#1069877

Pleasure and Pain

Ever notice that it's possible for the same stimulus to evoke pleasure or pain, depending on context?

Pleasure or Pain? He Maps the Neural Circuits That Decide.
The work of the neuroscientist Ishmail Abdus-Saboor has opened up a world of insights into precisely how much pleasure and pain animals experience during different forms of touch.

My only hesitation here is that doing research on pain often means deliberately causing pain, and to me, that's ethically questionable at best. Especially when the subjects can't consent. But it was only a hesitation; I'm doing this entry anyway. As an omnivore, there's a limit to how much hypocrisy I'll tolerate in myself when it comes to this sort of thing.

Ishmail Abdus-Saboor has been fascinated by the variety of the natural world since he was a boy growing up in Philadelphia.

Probably because there's so much of it in Philadelphia.

Yes, I know I've pointed out before that, as we are part of the natural world, so is everything we build. But I never let the facts get in the way of a good joke. Or a bad one. Especially a bad one.

Today, he is an associate professor of biological sciences at the Mortimer B. Zuckerman Mind Brain Behavior Institute at Columbia University, where he studies how the brain determines whether a touch to the skin is painful or pleasurable.

From Philadelphia to Manhattan... hm. Anything to stay out of New Jersey, I suppose.

To find those clues, Abdus-Saboor probes the nervous system at every juncture along the skin-to-brain axis. He does not focus on skin alone or home in on only the brain as many others do. “We merge these two worlds,” he said.

This kind of ties in with other things I've been saying. People tend to draw a boundary between body and brain, probably because of old ideas about mind/body duality, but the brain is part of the body.

Abdus-Saboor has also pioneered a new quantitative measure of pain in mice, a tool he and his team adapted to gather evidence for the transgenerational inheritance of opioid addiction.

The addiction thing aside, I'm intrigued about the "quantitative measure of pain" thing. Pain in humans is often subjective; hence the classic 1-10 scale doctors ask you to rate your own pain on. Pain in other animals is generally inferred by their reactions, but it gets tricky; for a long time, people thought nonhuman animals didn't actually experience pain as such. The latter part of the article discusses this further.

Quanta spoke with Abdus-Saboor about his penchant for starting over in science, his zebra fish eureka moment and his hopes for a newly imported naked mole rat colony.

Thus, the rest of the article is in interview format. I won't quote more, but it's there at the link. I find it interesting on several levels, not least of which is the actual science involved, but also because, to me, it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

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