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.

2 Like

October 16, 2021 at 12:02am

#1019445

We're All Mad Here

I don't know how I found this article from 2013, but it's unlikely that the age of it makes a difference.

What Neuroscience Says About The Link Between Creativity And Madness
New research sheds more light on the strong ties between an original mind and a troubled one.

Creativity is a thing that eludes me. Madness? Well. I wouldn't know, would I?

The idea that very creative people are also a little crazy has been around since humanity’s earliest days.

Maybe the article was written before we were supposed to stop using "crazy," which is retarded.

In ancient Greece, Plato noted the eccentricities of poets and playwrights, and Aristotle saw that some creative types were also depressives. In modern times, that connection has persisted, from Robert Schumann hearing voices guide his music to Sylvia Plath sticking her head in an oven to Van Gogh cutting off his ear to Michael Jackson … being Michael Jackson.

And there you have it, folks: Michael Jackson in the same paragraph as Plato.

Today the link between creativity and mental illness is firmly embedded in the public conscience. Unlike some supposed cultural wisdoms, however, there’s a good bit of scientific evidence behind this one.

First you'd have to define "crazy." I mean, sure, there are modern psychological definitions of all kinds of mental disorders, but it seems to me that by the common definition of "crazy," in order to be creative, one has to be crazy. Because the word describes someone whose thoughts and actions are outside the "normal" (whatever that is). So all creative people are crazy, but that doesn't mean (sorry, crazy people) that all crazy people are creative.

The article, though, mentions particular recognized mental disorders, and I won't copy all of the examples here.

The new work enhances a theory by Shelley Carson, a Harvard psychologist and author of the book Your Creative Brain, which says that creativity and mental illness share a process called “cognitive disinhibition.” The term is a mouthful, but essentially cognitive disinhibition describes a failure to keep useless data, images, or ideas out of conscious awareness.

There is no such thing as useless data. There is only data to be stored away for later use.

“[Y]ou have more information in conscious awareness that could be combined and recombined in novel and original ways to come up with creative ideas,” Carson tells Co.Design.

Which kind of touches on how I view creativity: it is, at least in part, the ability to make connections that others might miss. I think of it as living a metaphor.

That’s why not all creative people are a bit crazy and why not every mentally ill person is especially creative. “It’s not a one-on-one correspondence,” says Carson. In fact, she says, most creative people don’t exhibit severe mental problems at all; rather, the notable examples stick in our minds.

Sure, if you use the DSM to diagnose "crazy." My point above still stands in opposition to this assertion: that creative people are crazy by definition.

Which brings us back to our list of eccentric artists through the ages. Perhaps genes contributing to mental problems have persisted across humanity in part because they also contribute to superior creativity.

That is not how evolution works. Genes persist if the people possessing them live long enough to reproduce, and then reproduce. Having "crazy genes" (the existence of which I doubt in the first place) doesn't necessarily keep someone from finding a mate and making little crazy people. I mean, look around if you need evidence for that.

“Even though we know mental illness in and of itself is not conducive to survival of the individual, there may be aspects of mental illness that promote survival in the overall species,” Carson says.

Sure, profound mental illness may preclude passing on one's genes, but it doesn't have to be profound to exist. Also, at least some mental illnesses are environmentally-driven.

Anyway, my point in sharing this isn't to rag on their view of evolution, but because a few craz- er, I mean, creative people read this blog, and you should feel better about being just a little bit whacked.

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