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

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 26, 2021 at 12:02am

#1007070

Dream Territory

I mentioned dreaming, mostly as an aside, yesterday, but today's article is entirely about some research into dreaming.

Why Do We Dream? A New Theory on How It Protects Our Brains

Not, of course, interpretation of dreams, but the process itself.

When he was two years old, Ben stopped seeing out of his left eye. His mother took him to the doctor and soon discovered he had retinal cancer in both eyes.

And so we start with a nightmare. Well, not a nightmare; worse, purportedly an actual thing that happened. Don't worry; it's not the rambling anecdotal "style" of The New Yorker; there's a point to this. I think.

But by the time he was seven years old, he had devised a technique for decoding the world around him: he clicked with his mouth and listened for the returning echoes.

Batboy is real! Still waiting for the point...

How could blindness give rise to the stunning ability to understand the surroundings with one’s ears? The answer lies in a gift bestowed on the brain by evolution: tremendous adaptability.

Ah, there it is. Or at least we begin to see a glimmer of a point.

What does brain flexibility and rapid cortical takeover have to do with dreaming? Perhaps more than previously thought. Ben clearly benefited from the redistribution of his visual cortex to other senses because he had permanently lost his eyes, but what about the participants in the blindfold experiments? If our loss of a sense is only temporary, then the rapid conquest of brain territory may not be so helpful.

And this, we propose, is why we dream.

Determining the "why" of something is, to me, a bit slippery. Why do we have four fingers with an opposable thumb? So we can grasp tools? I mean, that's the result, but a distant ancestor might have said it was so we could climb trees.

Evolution doesn't have a "why;" only a "how."

But okay, I'll grant that one result of dreaming could be to help make new neural connections.

We suggest that the brain preserves the territory of the visual cortex by keeping it active at night. In our “defensive activation theory,” dream sleep exists to keep neurons in the visual cortex active, thereby combating a takeover by the neighboring senses.

However, I'm not sure framing it in terms of competition is useful. I mean, yeah, the popular view of evolution is one of a process driven by competition. I think it's at least as much about cooperation. If you think about everything in terms of competition, you can start thinking that everything's a win/lose scenario. This is not borne out by evidence. I have another article in my queue that talks about this sort of thing; I'll probably get to it eventually.

REM sleep is triggered by a specialized set of neurons that pump activity straight into the brain’s visual cortex, causing us to experience vision even though our eyes are closed. This activity in the visual cortex is presumably why dreams are pictorial and filmic. (The dream-stoking circuitry also paralyzes your muscles during REM sleep so that your brain can simulate a visual experience without moving the body at the same time.)

Yeah, and in some of us, that mechanism can work too well (sleep paralysis) or not well enough (sleepwalking).

For example, because brain flexibility diminishes with age, the fraction of sleep spent in REM should also decrease across the lifespan. And that’s exactly what happens: in humans, REM accounts for half of an infant’s sleep time, but the percentage decreases steadily to about 18% in the elderly. REM sleep appears to become less necessary as the brain becomes less flexible.

I can't be arsed to delve deeper into this right now (because at the moment, it feels like the only active neurons in my head are the ones responsible for reminding me about how bad pain can be), but could they have switched cause and effect there? That is, instead of "REM decreases because our brain becomes less flexible with age," maybe it's "our brain becomes less flexible with age because REM decreases." I mean, that would be more in line with the authors' hypothesis here, wouldn't it?

Dream circuitry is so fundamentally important that it is found even in people who are born blind. However, those who are born blind (or who become blind early in life) don’t experience visual imagery in their dreams; instead, they have other sensory experiences, such as feeling their way around a rearranged living room or hearing strange dogs barking.

Oddly, if this is the case, my own experience is that dreams can be, but are not always, visual. I have clear memories of dreams that took place in darkness, with my dream-self's knowledge of the outlines of objects coming from -- I don't know where. Some other dream-sense besides sight.

Since the dawn of communication, dreams have perplexed philosophers, priests, and poets. What do dreams mean? Do they portend the future? In recent decades, dreams have come under the gaze of neuroscientists as one of the field’s central unsolved mysteries. Do they serve a more practical, functional purpose? We suggest that dream sleep exists, at least in part, to prevent the other senses from taking over the brain’s visual cortex when it goes unused. Dreams are the counterbalance against too much flexibility.

Though I'm convinced that science is the best tool we have for understanding the universe (including ourselves), there are probably some things best left to philosophy. What function dreaming has is likely a reasonable scientific pursuit. What they "mean," however, is far more subjective.

For a long time, now, I've made puns in my dreams (and sometimes even remembered them). Lately, French, which I continue to practice daily, has crept in. Perhaps they're helping me learn the language.

In any case, these authors' research is worth noting, I think -- though as with all new science, I suggest we don't take it as authoritative fact, but rather as a different way of looking at things -- until such time as the hypothesis is disproven.

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