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 6 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 8 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.

1 Like

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.

August 20, 2020 at 12:18am

#991199

Good Willpower Hunting

I couldn't resist posting today's article.

http://nautil.us/issue/45/power/against-willpower

Against Willpower
Willpower is a dangerous, old idea that needs to be scrapped.

They both succumbed to short-term temptations, and both didn’t keep their long-term goals.

I went to a dentist today. See, approximately 20 seconds after my state declared a lockdown, back in March, I got a toothache. It's been bearable up until about last week, whereupon the pain in my mouth exceeded my desire to avoid covfefe-19.

Few people enjoy going to the dentist, but most realize that the short-term discomfort involved is worth it to prevent problems in the long run. Not me, though. I've never been one to let long-term goals get in the way of immediate gratification.

Fortunately, the dentist told me what the problem (probably) is. Unfortunately, it involves redoing a root canal on a tooth that got a root canal many years ago. And root canals are the Platonic ideal of "things I don't want to deal with." That's something I have to schedule tomorrow, which means at some point soon, I'm going to have to visit a specialist's office, again, and keep my mouth wide open for two hours in the middle of a fucking pandemic.

So, naturally, afterwards, I walked over to a nearby brewery and drank. I Ubered home. See, I figured something like that would happen, so I had the foresight to not drive to the dentist in the first place. I can plan ahead, when it involves drinking.

Back to the article.

Ignoring the idea of willpower will sound absurd to most patients and therapists, but, as a practicing addiction psychiatrist and an assistant professor of clinical psychiatry, I’ve become increasingly skeptical about the very concept of willpower, and concerned by the self-help obsession that surrounds it.

I'm concerned by the self-help obsession, period.

More fundamentally, the common, monolithic definition of willpower distracts us from finer-grained dimensions of self-control and runs the danger of magnifying harmful myths—like the idea that willpower is finite and exhaustible. To borrow a phrase from the philosopher Ned Block, willpower is a mongrel concept, one that connotes a wide and often inconsistent range of cognitive functions. The closer we look, the more it appears to unravel. It’s time to get rid of it altogether.

Good idea. Do we have the will to do that?

The specific conception of “willpower,” however, didn’t emerge until the Victorian Era...

If ever there were a time for that concept, it would be then.

Self-control became a Victorian obsession, promoted by publications like the immensely popular 1859 book Self-Help, which preached the values of “self-denial” and untiring perseverance.

Sometimes I think I was unlucky to have lived during the declining period of American civilization (we peaked on July 20, 1969, and it's been downhill ever since). But then I read stuff like that and realize it's not so bad after all, pandemic be damned.

The earliest use of the word, in 1874 according to the Oxford English Dictionary, was in reference to moralistic worries about substance use: “The drunkard ... whose will-power and whose moral force have been conquered by degraded appetite.”

Always with the drunk-shaming. Stop it.

In the early 20th century, when psychiatry was striving to establish itself as a legitimate, scientifically based field, Freud developed the idea of a “superego.” The superego is the closest psychoanalytic cousin to willpower, representing the critical and moralizing part of the mind internalized from parents and society.

Since Freud proposed it, 96.5% chance it's wrong.

By mid-century, B.F. Skinner was proposing that there is no internally based freedom to control behavior. Academic psychology turned more toward behaviorism, and willpower was largely discarded by the profession.

Make that 98%.

That might have been it for willpower, were it not for an unexpected set of findings in recent decades which led to a resurgence of interest in the study of self-control.

Dammit.

These studies also set the stage for the modern definition of willpower, which is described in both the academic and popular press as the capacity for immediate self-control—the top-down squelching of momentary impulses and urges. Or, as the American Psychological Association defined it in a recent report, “the ability to resist short-term temptations in order to meet long-term goals.” This ability is usually portrayed as a discrete, limited resource, one that can be used up like a literal store of energy.

If I had a battery icon on my forehead for that, it would be nearly empty and have a slash through it. No, I'm not tempted to get that as a tattoo. But the thought did cross my mind.

Studies supporting the ego depletion effect were supposedly replicated dozens of times, spawning best-selling books (including Baumeister’s own, Willpower) and countless research programs. But a 2015 meta-analysis examining those findings more closely, along with previously unpublished research, found a good deal of publication bias and very little evidence that ego depletion is a real phenomenon.

And here we have another example of science correcting itself.

Related studies have shown that beliefs about willpower strongly influence self-control: Research subjects who believe in ego depletion (that willpower is a limited resource) show diminishing self-control over the course of an experiment, while people who don’t believe in ego depletion are steady throughout.

Funny how that works.

A paradigmatic example of reframing is the phenomenon of “temporal discounting,” in which people tend to discount future rewards in favor of smaller immediate payoffs. When offered $5 today versus $10 in a month, many people illogically choose immediate gratification.

"Illogically?" I don't think so. You tell me you'll give me $5 today or $10 in a month. I'll take the $5. Why? First, I might be dead in a month. Second, you might be dead in a month and your executor won't know you owe me $10. Third, $10 is couch change for me. And finally, if I'm in Vegas, I could put the $5 on 27 and maybe turn it into $150 -- or probably not, but the $5 was free to begin with and I won't miss it.

I'm obviously skipping a bit here but...

A conscientious reframing of a problem in this manner would certainly be an example of willpower, but it would not fall into the conventional understanding of the term. Rather than relying on an effortful fight against impulses, this kind of willpower has the individual completely reimagine the problem and avoid the need to fight in the first place.

Which, actually, is what I usually do when faced with temptation. Often I indulge anyway. But not always. Like today, when I woke up from my drunken stupor and there was pizza. I think I might have mumbled something about pizza before I passed out, and my housemate, either thoughtful or easily persuaded, had ordered some for delivery. I declined. Alcohol was enough extra calories for me today, and while my weight loss has plateaued, I don't want to backslide.

But now comes the important part of the article:

Notions of willpower are easily stigmatizing: It becomes OK to dismantle social safety nets if poverty is a problem of financial discipline, or if health is one of personal discipline. An extreme example is the punitive approach of our endless drug war, which dismisses substance use problems as primarily the result of individual choices. Unhealthy moralizing creeps into the most quotidian corners of society, too. When the United States started to get concerned about litter in the 1950s, the American Can Company and other corporations financed a “Keep America Beautiful” campaign to divert attention from the fact that they were manufacturing enormous quantities of cheap, disposable, and profitable packaging, putting the blame instead on individuals for being litterbugs. Willpower-based moral accusations are among the easiest to sling.

Which is kind of what I've been saying all along.

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