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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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').

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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.

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

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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

April 15, 2023 at 10:36am

#1048245

The Less You Know...

I've been wondering what to say about this one.

Cognitive neuroscientist explains why stupidity is an existential threat to America

While the article emphasizes the US, it's been clear to me, thanks to the international reach of the internet, that this is a human problem, not just an American one. But as I live in the US, I'll run with what it says.

It may sound like an insensitive statement, but the cold hard truth is that there are a lot of stupid people in the world, and their stupidity presents a constant danger to others.

That depends on how you define "stupid," I suppose. I think they mean it in the sense of "stubbornly and willfully ignorant," because I fail to see how someone with a legitimate learning disability (Down's Syndrome, e.g.) presents a constant danger to others.

It would not be a stretch to say that at this point in time, stupidity presents an existential threat to America because, in some circles, it is being celebrated.

Which fits "stubbornly and willfully ignorant." Also any politician (of any party) who runs on a "common sense" platform.

Although the term "stupidity" may seem derogatory or insulting, it is actually a scientific concept that refers to a specific type of cognitive failure. It is important to realize that stupidity is not simply a lack of intelligence or knowledge, but rather a failure to use one's cognitive abilities effectively.

Okay, so a better definition than I had.

To demonstrate that stupidity does not mean having a low IQ, consider the case of Richard Branson, the billionaire CEO of Virgin Airlines, who is one of the world’s most successful businessmen.

Clearly, this article was written before Virgin Galactic fell into a black hole.

Still, business failure can't always be attributed to stupidity (as defined here).

Branson has said that he was seen as the dumbest person in school, and has admitted to having dyslexia, a learning disability that affects one’s ability to read and correctly interpret written language.

I've known several people with dyslexia, and this may be some sort of selection bias on my part, but as a whole, they seem to have an above-average intelligence. Mostly, they just have trouble with words and spelling. While I pride myself on my spelling, it's not proof of my intelligence.

Branson’s smarts come from his ability to recognize his own limitations, and to know when to defer to others on topics or tasks where he lacks sufficient knowledge or skill.

I hope his rocket science competition here in the US is taking notes.

Stupidity is a consequence of a failure to be aware of one’s own limitations, and this type of cognitive failure has a scientific name: the Dunning-Kruger effect.

Ironically, people throw around "Dunning-Kruger effect" like they know what they're talking about, even if they don't. Shit, I'm guilty of that, myself.

It is easy to think of examples in which failing to recognize one’s own ignorance can become dangerous. Take for example when people with no medical training try to provide medical advice. It doesn’t take much Internet searching to find some nutritionist from the “alternative medicine” world who is claiming that some herbal ingredient has the power to cure cancer. Some of these people are scam artists, but many of them truly believe that they have a superior understanding of health and physiology.

I'd reverse that. Many of these people are scam artists, but some of them truly believe they have a superior understanding.

There are many people who trust these self-proclaimed experts, and there is no doubt that some have paid their lives for it.

My Platonic ideal of the successful businessperson is not Richard Branson, but Steve Jobs. No, I'm not an Apple fanboi, but consider: he started a company with a couple of other geeks in his garage, and lived just long enough to see it become the most valuable company in the world (by market capitalization), exceeding far older corporations such as ExxonMobil or IBM. Sure, he was, by all accounts, a massive dick, but that's irrelevant to being a successful businessman.

And yet, he might have lived longer if he hadn't fallen for the alternative medicine scams.

What’s particularly disturbing about the Dunning-Kruger effect is that people are attracted to confident leaders, so politicians are incentivized to be overconfident in their beliefs and opinions, and to overstate their expertise.

"The best lack all conviction, while the worst / Are full of passionate intensity." -W.B. Yeats

It is only when we try to become an expert on some complex topic that we truly realize how complicated it is, and how much more there is to learn about it.

As a dedicated dilettante, interested in a broad range of topics yet not willing to delve too deeply into any of them, this is something I have to guard against, myself.

What we are dealing with here is an epidemic of stupidity that will only get worse as divisions continue to increase. This should motivate all of us to do what we can to ease the political division.

Sure, it should. But it won't.

We are all victims of the Dunning-Kruger effect to some degree. An inability to accurately assess our own competency and wisdom is something we see in both liberals and conservatives. While being more educated typically decreases our Dunning-Kruger tendencies, it does not eliminate them entirely. That takes constant cognitive effort in the form of self-awareness, continual curiosity, and a healthy amount of skepticism.

Clearly, I'm better at this than anyone else.

That's a joke. I'm deliberately overstating my competence to demonstrate the D-K effect.

The article is optimistic about being able to fix this. But there's only one thing more dangerous than willful stupidity, and that's optimism.

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