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!

4 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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

May 10, 2024 at 9:59am

#1070838

Meditation Clause

Woke up with a case of gamer finger this morning. You know, when you sprain your finger from intermittently pushing the key that makes you go in the game? No? Damn, just me then. So I'd like to keep this short, but considering what I landed on today, I don't know if I'll be able to.

Are we morally obligated to meditate?
A growing body of neuroscience research shows that meditation can make us better to each other.

Just remember, my default response to headline questions is "no."

Starting in 2005, Harvard neuroscientist Sara Lazar began to publish some mind-blowing findings: Meditation can literally change the structure of your brain, thickening key areas of the cortex that help you control your attention and emotions.

Setting aside for the moment the ridiculous idea that "you" are some entity separate from your brain that can control it like a puppet, that finding, if true, is remarkable in itself. It'd be the brain equivalent of weight-lifting. The article does go on to mention follow-up studies, so there may be something to it.

Your brain — and possibly, by extension, your behavior — can reap the benefits if you practice meditation for half an hour a day over eight weeks.

Which might not be possible for everyone. As with exercise or recipe timing, the duration is misleadingly short. It takes time to prepare for these things, and more time to come down after. So you're looking at a 45-minute chunk of the day, maybe. Now, what are you willing to give up to get those 45 minutes? Sleep? If you do, you'll fall asleep trying to meditate (or, I don't know, maybe that's just me, too). Time with family? Dog-walking? Commute? Work? Lunch?

A host of other studies showed that meditation can also change your neural circuitry in ways that make you more compassionate, as well as more inclined to have positive feelings toward a victim of suffering and to see things from their perspective.

But what if you don't want to do all that?

Still skeptical, I fell down an internet rabbit hole and soon found many more neuroscientific studies. Looking closely at them, I did find that a fair number are methodologically flawed (more on that below). But there were many others that seemed sound.

See, this is what I look for in articles like this: some indication that the author didn't just accept whatever bullshit gets strewn about.

If it takes such a small amount of time and effort to get better at regulating my emotions, paying attention to other people, seeing things from their point of view, and acting altruistically, then … well … am I not morally obligated to do it?

That's not a science question. That's a philosophy question. But the whole next section of the article goes back to the science bit, which is worth reading, but I'm not quoting it here. So, provisionally accepting the science findings, I'm skipping to the philosophy section.

After I started wondering if we’re morally obligated to meditate, I soon realized that’s a very Western and Judeo-Christian way of thinking about it.

I would very much like for that hyphenated adjective to stop being used.

But Eastern traditions like Buddhism or Confucianism aren’t grounded in commandments that come from a divine being.

Neither are Western traditions. We've been conditioned to believe that the commandments came from a divine being. Which, from a practical perspective, I suppose produces the same results.

Plus, whereas the language of oughts and obligations suggests a prescriptive or proselytizing attitude, Buddhist tradition has generally been more interested in inviting people to try meditation and discover its benefits for themselves, rather than in mandating adherence.

Therein lurks an important difference between East and West.

I'd say this bit's important to note, too:

But there’s a caveat: For a small minority of people, meditation can actually provoke adverse effects, like intense mental distress or impaired physical functioning. Brown University psychologist Willoughby Britton is studying these cases in a project called “Varieties of Contemplative Experience.” More research is still needed, but given that meditation practices might precipitate or exacerbate challenging conditions in some people, it would be wrong to say that absolutely everyone would do well to meditate.

This is analogous to people urging us to get out and socialize more. That may be good advice for the extroverted majority, but we introverts would quickly spiral into doom. Or, to be even more simplistic in my analogies, saying that it's a good idea to always use your right hand to write, because studies show that it helps 90% of us to do so.

Scientists are publishing more and more studies on meditation each year. But many of these studies are beset by methodological flaws, leading to overhyped results. Davidson calls this “neuromythology.”

Yeah, we need to be really careful about that crap. It's like all those "quantum consciousness" bullshit books.

So, is the answer to the headline question still "no?" Well, you'll have to figure that out for yourself. Philosophy can guide science, and science can inform philosophy, but they're not the same sport. For me, I've spent too many years trying to figure out how to become more compassionate, but meditation just isn't my thing. I find it more practical to accept myself the way I am right now, flaws and all.

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