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.

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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.

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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.

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.

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June 8, 2024 at 9:18am

#1072332

Terminator

This piece, a recent one from The Guardian, is about communication, so it may be of interest to writers here.

The big idea: the simple trick that can sabotage your critical thinking
Influencers and politicians use snappy cliches to get you on side – but you can fight fire with fire

Ha! I see what you did there, headline writer.

Since the moment I learned about the concept of the “thought-terminating cliche” I’ve been seeing them everywhere I look: in televised political debates, in flouncily stencilled motivational posters, in the hashtag wisdom that clogs my social media feeds.

No, you've always been seeing them. Now, you recognize them for what they are.

Coined in 1961 by psychiatrist Robert Jay Lifton, the phrase describes a catchy platitude aimed at shutting down or bypassing independent thinking and questioning.

These days, when I see or hear "platitude," I picture an angry platypus (one with an attitude). Look, it helps me, okay? Apparently, the word derives from the French plat, meaning flat or dull (I guess in the same sense that we use "flat" for non-glossy paints). But in that same language, plat also means dish—in the same way that, in English, "dish" can refer to both a plate (a word obviously also derived from plat) and the food you put on it to eat. This makes a platitude feel like something easily prepared and consumed, which is really damned appropriate.

Also, I'm pretty sure it was Ogden Nash who came up with the phrase "duck-billed platitude."

But I digress, as per usual.

In his book Thought Reform and the Psychology of Totalism, Lifton wrote that these semantic stop signs compress “the most far-reaching and complex of human problems … into brief, highly selective, definitive-sounding phrases, easily memorized and easily expressed. They become the start and finish of any ideological analysis.”

That's a whole lot of words to express what I call "bumper-sticker philosophy," which I suppose is my own thought-terminating cliché.

As the article points out, though, TTCs (do not expect me to type the phrase every damn time) aren't necessarily bad. Like any word or phrase, they can be used for good or evil.

Unfortunately, mere awareness of such tricks is not always enough to help us resist their influence. For this, we can blame the “illusory truth effect” – a cognitive bias defined by the unconscious yet pervasive tendency to trust a statement simply because we have heard it multiple times.

This gets weaponized a lot, too. Keep repeating things like "greed is good," and people start to believe it. Or, like someone we all know of, continue to lie and your minions lap it up as divine Truth, even though it's objectively a lie.

But what if we could do that with the actual truth?

To compete in the marketplace of thought-terminating cliches, then, our best bet might be to take what we know about illusory truth and harness it to spread accurate information.

Like that.

Beyond repetition, studies show that people perceive statements as more believable when presented in easy-to-read fonts or easy-to-understand speech styles, such as rhyme.

Or, you know, alliteration like in my example above.

This, of course, is where writers come in.

And a 2021 study showed that humour is among the qualities that make information more memorable and shareable. A titbit is “just more likely to spread if it’s funny”, says Scheirer.

Did you know that we here in America call small morsels of food or information tidbits instead of titbits because our Puritanical sensibilities required the change? Yeah? Then you were subjected to misinformation.

Probably. I mean, it felt right to me, too, because I'm not without bias, myself. But apparently, "tit" has at least two entirely separate etymologies.

It doesn’t only have to be shameless disinformers who exploit the power of repetition, rhyme, pleasing graphics and funny memes. “Remember, it’s OK to repeat true information,” says Fazio.

The trick, of course, is to know what's true, and in a postmodern world, that's becoming increasingly difficult. Even facts that should be universally accepted, such as the approximate shape or age of the Earth, are subject to strong opposition.

As the article implies, though, this doesn't mean we shouldn't make the attempt.

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