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

June 1, 2023 at 11:41am

#1050375

Stop Making Sense

First, a plug for the June round of

Journalistic Intentions

(18+)
This is for the journal keeping types that come to PLAY! New round starts July 1!
#2213121 by Turkey DrumStik

Just do eight entries (chosen from 16 prompts) in a month.

I'll be participating, myself... but not today.

Today, we try to talk sense about senses.

Beyond the Five Senses
Telepathy, echolocation, and the future of perception

I've noted in the past that we really have more than the classical five senses, but I don't think I'd seen this article, which is from 2017. It's talking about different "extra" senses, anyway.

The world we experience is not the real world.

I think several philosophers would agree. Several other philosophers would vehemently disagree. As for me, first you'd have to define "real," and good luck with that.

It’s a mental construction, filtered through our physical senses.

Yesterday, I banged my shin on a mental construction.

How would our world change if we had new and different senses? Could they expand our universe?

We already have new and different senses, courtesy of technology. They translate input into output discernible by one of the Five. A Geiger counter, for example, detects ionizing radiation (which we have no sense for) and converts it to auditory (the clicking sound) and visual (a gauge), which our brains can interpret. The article also treats writing as sense-substitution, trading the auditory (language) for the visual (writing on a page).

More recently, researchers in the emerging field of “sensory enhancement” have begun developing tools to give people additional senses—ones that imitate those of other animals, or that add capabilities nature never imagined. Here’s how such devices could work, and how they might change what it means to be human.

Some animals can detect electrical fields. I can't imagine what that would feel (or look, smell, sound, or taste) like.

Which gives me an opportunity to once again trot out my great epiphany about the Classic Five: They are all, at base, touch. Even vision, which relies on the impact of photons on our retinas.

Researchers are working on other technologies that could restore sight or touch to those who lack it. For the blind, cameras could trigger electrodes on the retina, on the optic nerve, or in the brain. For the paralyzed or people with prosthetic limbs, pressure pads on real or robotic hands could send touch feedback to the brain or to nerves in the arm.

All of which sounds pretty cool, and used to be relegated to science fiction. Well, yesterday's science fiction often becomes science.

Autistic people might even gain a stronger social sense.

Careful, there. I know several autistic people with strong social sense. They tend to have different perspectives, and those perspectives are often more useful. Don't fix stuff that ain't broke.

Last year, MIT researchers revealed the EQ-Radio, a device that bounces signals off people to detect their heart rate and breathing patterns.

Finally, a device that can tell you if someone's into you.

We can also substitute one sense for another. The brain is surprisingly adept at taking advantage of any pertinent information it receives, and can be trained to, for instance, “hear” images or “feel” sound.

Sounds like induced synesthesia to me.

Scientists are also exploring ways to add senses found elsewhere in the animal kingdom.

The article talks about things like direction sense and echolocation, but echolocation is basically hearing. I've read that some humans, especially blind ones, have learned this trick without technological aid.

In the future, cochlear implants could be tuned to pick up really low frequencies, such as those used by elephants, or really high ones, such as those used by dolphins. Bionic eyes could be built to allow humans to see ultraviolet rays (as butterflies, reindeer, dogs, and other animals can) and infrared light (as certain snakes, fish, and mosquitoes can).

As cool as all that would be, all it would do is extend our existing classical senses.

Some researchers think we may eventually install a port in our brains that would allow us to swap in different sensors when we need them.

Because that always ends well in SF stories.

We might also gain senses that no other animal has. The vibrating vest Eagleman created can be programmed to receive any input, not just sound. He says it could be used to monitor the stock market, or sentiment on Twitter, or the pitch and yaw of a drone, or one’s own vital signs.

Ah, yes, 2017, when Twatter was still marginally relevant.

“You can do whatever you want,” says Neil Harbisson, a “cyborg artist” who’s originally from Spain. “You can design a unique sense that is related to your interests or to your curiosity.”

Well, I know some people who could benefit from a sense of humor implant.

Also... "cyborg artist?" Come on.

Perhaps we’ll even achieve that so-called sixth sense: ESP.

That's not as farfetched as the article implies. It's not much of a stretch to imagine electrodes that pick up signals from one brain and transfer them to the electrodes of another, either through implants or induction. Doubtful that we'd be able to "read" the minds of unwilling subjects anytime soon, though.

But last I heard, we still haven't figured out how to keep the brain from outright rejecting any implanted electrodes. Yeah, yeah, Neuralink (also mentioned in the article; it was new at the time). As with most Muskmelon projects, that's more hype than anything else.

Exactly how all this tinkering will change us remains to be seen.

And this is what science fiction is for.

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