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 10, 2021 at 12:46am

#1006112

Predictive Anarchy

I don't have much to say today. Yesterday was the first time in months that the thermostat hit 70F and I was just so happy to feel it (anything below 70F is "cold" and anything below 55F is "freezing") that I exhausted myself playing video games outside.

The next few days should hit 70+ too, but then it's back to freezing for a while. Oh well. Such is March in Virginia. At least there hasn't been gale-force winds as per usual. Yet.

Speaking of predictions...

To Make Sense of the Present, Brains May Predict the Future
A controversial theory suggests that perception, motor control, memory and other brain functions all depend on comparisons between ongoing actual experiences and the brain’s modeled expectations.

Gosh, it's almost like "living in the present" is impossible, isn't it?

Okay, fine, it's not settled science, but it's a remarkable approach to an old problem -- and might even give us some insight into consciousness. And it might even explain compelling writing, as you'll see below.

I'm not going to quote a lot of it because, like I said, exhausted. But this bit is the idea in brief:

According to this “predictive coding” theory, at each level of a cognitive process, the brain generates models, or beliefs, about what information it should be receiving from the level below it. These beliefs get translated into predictions about what should be experienced in a given situation, providing the best explanation of what’s out there so that the experience will make sense. The predictions then get sent down as feedback to lower-level sensory regions of the brain. The brain compares its predictions with the actual sensory input it receives, “explaining away” whatever differences, or prediction errors, it can by using its internal models to determine likely causes for the discrepancies. (For instance, we might have an internal model of a table as a flat surface supported by four legs, but we can still identify an object as a table even if something else blocks half of it from view.)

This may sound all esoteric and shit, but I felt like some of this is relevant to me as a writer. See, I've long suspected that the key to interesting writing is to be unpredictable, to use words that don't quite dishrag. This is most obvious with comedy, but giving people unexpected combinations of words makes them stop and think.

The researchers observed a greater brain response when the study’s subjects came across the unexpected word “dog,” characterized by a specific pattern of electrical activity, known as the “N400 effect,” that peaked approximately 400 milliseconds after the word was revealed. But how to interpret it remained unclear. Was the brain reacting because the word’s meaning was nonsensical in the context of the sentence? Or might it have been reacting because the word was simply unanticipated, violating whatever predictions the brain had made about what to expect?

This may, in fact, explain the otherwise inexplicable attraction people have to James Joyce. Or even jazz, which delights in playing the exact wrong note next.

The rest of the article is long and somewhat technical, but between the connection to writing and the theory's apparent refutation of my hated "living in the present" philosophy, I found at least part of it relevant.

Perhaps I'll have more to say tomorrow, or perhaps cat.

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