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 8 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 10 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').

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.

March 2, 2024 at 6:33am

#1065370

Thought Bubble

Hope you're not prone to paranoia.

Who Controls Your Thoughts?
Our minds are being coerced in covert ways.

After yesterday's article, clearly, the answer to the headline question is "space aliens."

In 2017, Simon McCarthy-Jones wrote an article about schizophrenia for The Conversation. The piece, he jokes, got read by more than two people, which, as an academic—he’s an associate professor of clinical psychology at Trinity College Dublin—was a thrill.

Except that, when I read this lede, I thought he was making a schizophrenia joke, like the old "I'm schizophrenic and so am I" one, which perpetuates the conflation of schizophrenia with MPD. Which I might be able to get away with, but not a psych professor. Anyway, no, that wasn't the intent. As far as I can tell.

Shortly thereafter, however, he found himself “just gripped by the iron claws of Facebook,” looking over and over again to see who had liked his article, who had commented on it.

Isn't there a psych disorder where you attribute your actions to some outside force, instead of taking responsibility for them? I don't mean that in a "we have no free will" way, but in a "the Devil made me do it" way.

Was his thinking being covertly, coercively controlled by external forces (in this case, a big tech company)? The experience got him wondering just what “free thought” actually was. And so he started wading into the murky waters of the psychological, philosophical, cultural, and legal assumptions about what constitutes thought—and how it could remain truly free.

His intellectual quest has exited his head, as much thought eventually does, and now exists in the form of a new book: Freethinking: Protecting Freedom of Thought Amidst the New Battle for the Mind.

Of course it's a book ad. Everything on the internet is an ad, or it's behind a paywall. Well, except for this blog, of course.

In any case, the rest of the ad is an interview.

We might want to say: No, we’re independent, autonomous thinkers. But I think we have to recognize that in front of a persuasive AI, we are in deep trouble.

In my experience, the people who shout "I think for myself!" the loudest are the most likely to follow the herd.

The first was that the right to free thought is an absolute right, based on the United Nations’ Universal Declaration of Human Rights. In the United States, it’s as close to an absolute right as there is in the Constitution. And that’s quite exciting because it means that nobody can interfere with your freedom of thought. There are instances where you can limit someone’s speech if it’s defamatory or false advertising or fighting words. But thought is unimpeachable, you can create absolute protection for people’s minds.

Well, okay, except that it's (at least so far) easy to say "you have an absolute right to free thought" when there is no possible way to read minds. Introduce a mind-reading device, though, and watch how quickly that right disappears. Sure, we can sometimes take an educated guess as to what's in someone's mind. Like if you're a shoe clerk, and some guy is in there looking at the Nike selection, you'd probably be right in assuming the guy's thinking "I might be interested in buying athletic shoes." But only "probably." After all, he might be idly gazing at footwear while wondering how he's going to trick some lady into bed that night.

There’s a quote by Daniel Kahneman, the Nobel laureate, who said: “Human beings are to independent thinking as cats are to swimming. They can do it, but they prefer not to.”

Honestly, I mostly saved this article so I could share that quote.

Astute readers may note that a lot of this is in opposition to my assertion that we don't really have free will. And it is. But lack of free will doesn't necessarily imply that it's other people pulling on our strings. Sometimes it's just the universe conspiring to keep us complacent.

3 Like • Comment • View 2 comments

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