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:

Brandiwyn🎶

a day ago
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').

1 Like

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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

March 23, 2021 at 12:08am

#1006902

Denial is More than Just a River in Egypt

Seems to me I encountered the author of this article somewhere before. Perhaps a different article, or a YouTube video. He seems to be quite prolific, so it's hard to tell. Anyway, the background is Sean B. Carroll here is an evolutionary biologist; I seem to run into them a lot even though biology isn't my main scientific interest.

But the article isn't about biology; if anything, it's about sociology or, as I like to put it, how some people love to wallow in ignorance.

The Denialist Playbook
On vaccines, evolution, and more, rejection of science has followed a familiar pattern

The article is dated last November, so it was quite timely when it came out, considering the other denial that ran rampant at the time.

I won't quote from the beginning, which goes into the history of polio and the vaccine developed to prevent it -- as well as the denial that such a vaccine even worked.

One group in particular did not welcome the vaccine as a breakthrough. Chiropractors actively opposed the vaccination campaign that followed Salk’s triumph. Many practitioners dismissed the role of contagious pathogens and adhered to the founding principle of chiropractic that all disease originated in the spine.

If it ducks like a quack...

Look, I'm not ragging on chiropractic in general here. There are things it helps with. I had a good friend who's a chiropractor, and have benefited from their treatments myself -- for back and neck pain. In general, they have to be pretty smart and go through intensive, long training in anatomy and technique. The point is they're not usually what I'd call ignorant -- and yet, this happens.

Opposition to the polio vaccine and to vaccination in general continued in the ranks such that even four decades later, long after polio had been eradicated from the United States, as many as one third of chiropractors still believed that there was no scientific proof that vaccination prevents any disease, including polio.

Often, once a person believes a certain thing, no amount of "proof" will convince them otherwise; they will instead dig in and fortify their position, rather than admit being wrong.

The vaccine is widely viewed as one of medicine’s greatest success stories: Why would anyone have opposed it? My shock turned into excitement, however, when I began to recognize the chiropractors’ pattern of arguments was uncannily similar to those I was familiar with from creationists who deny evolutionary science. And once I perceived those parallels, my excitement became an epiphany when I realized that the same general pattern of arguments—a denialist playbook—has been deployed to reject other scientific consensuses from the health effects of tobacco to the existence and causes of climate change. The same playbook is now being used to deny facts concerning the COVID-19 pandemic.

This is where we get into the meat of the article. The rest of it summarizes, and then expands on, the six pillars of the "denialist playbook," which I won't reproduce here; the article is there to read or skim.

If we hope to find any cure for (or vaccine against) science denialism, scientists, journalists and the public need to be able recognize, understand and anticipate these plays.

And it's worth reading because, as I said, it's timely and pertinent to a lot of the stuff that's going on today. There are other issues people have raised regarding the Covid vaccines besides the general denialism, usually based on fear instead of actual evidence. For example, there was some concern about blood clotting issues from one of the vaccines. This was studied, and determined to be Not An Issue. But once the "blood clotting" thing got in peoples' heads, fear took over.

My take on it? No treatment is, or can be, perfect. But people generally suck at risk management. Yes, it is possible, in general, to die or suffer ill effects from a vaccine. But if the chance of a negative outcome is 0.0001%, while the chance of a negative outcome from the disease itself is (just pulling a number out of my ass here) 10%, that's a huge difference, and I'll take my chances with the vaccine, thank you very much.

I see that sort of thing as an Applied Trolley Problem. You know the Trolley Problem; it's been spoofed enough by now, but the basic conundrum applies: If you do nothing, five people will die. If you do something, one person will die. No, you're not allowed to think outside the box on this one; stop trying to find loopholes right now. Those are the choices, period. Only with vaccines, it's maybe a hundred thousand people on the "do nothing" track and one on the "do something" track. The ethical logic is clear, at least to me. Fucking do something.

That said, this article coming up today (the chance was about 1 in 50) is serendipitous; I got notified yesterday that I finally have an appointment to get my Trump Mumps vaccine. True to form, the universe had a small chuckle at my expense; the only time I could do it was Thursday. And I've had oral surgery scheduled for Thursday afternoon for a month now.

Sure, I was able to schedule the vaccine for the morning while the surgery is in the afternoon. And it's the first dose, so it probably won't kick my ass. But now I have to go get my shot in the morning and then sit through agonizing dentistry for most of the afternoon.

All of which is to say that either a) the vaccine will go fine and I won't have a problem getting my mouth cut up, which will then knock me on my ass for a while; or b) I'll have an adverse reaction to the vaccine, which will knock me on my ass for a while, and won't be able to get the surgery done, thus forcing me to wait another month or more for the next appointment and pay a no-show fee..Consequently, either way, I expect to be utterly useless (moreso than usual) on Thursday night, so don't expect the usual blog entry first thing after midnight Friday.

Still.

It's worth it.

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