Complex Numbers (Book)

Blog Calendar

◄ September ► 2022
S	M	T	W	T	F	S
				1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28	29	30
Archive				RSS

About This Author

Contact Me

Come closer.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

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

September 16, 2022 at 12:05am

#1037744

Dream Weavers

You know, I used to think it would be cool to be able to somehow record your dreams in the same way a video camera records reality.

Why and How Do We Dream?
Dreams are subjective, but there are ways to peer into the minds of people while they are dreaming. Steven Strogatz speaks with sleep researcher Antonio Zadra about how new experimental methods have changed our understanding of dreams.

Dreams are so personal, subjective and fleeting, they might seem impossible to study directly and with scientific objectivity. But in recent decades, laboratories around the world have developed sophisticated techniques for getting into the minds of people while they are dreaming.

SCIENCE!

In the process, they are learning more about why we need these strange nightly experiences and how our brains generate them.

For some of us, they're not just nightly. How about a little acknowledgement of different circadian rhythms?

In this episode, Steven Strogatz speaks with sleep researcher Antonio Zadra of the University of Montreal about how new experimental methods have changed our understanding of dreams.

Ugh. Podcast. No thanks; I prefer reading. Fortunately, this article is a transcript.

In this episode, we’re going to be talking about dreams. What are dreams exactly? What purpose do they serve? And why are they often so bizarre? We’ve all had this experience: You’re dreaming about something fantastical, some kind of crazy story with a narrative arc that didn’t actually happen, with people we don’t necessarily know, in places we may have never even been. Is this just the brain trying to make sense of random neural firing? Or is there some evolutionary reason for dreaming?

While they can't answer all those questions, at least not fully, it seems (from the article) that they're making some progress at them.

The bulk of the transcript is an interview between the author and a Dr. Antonio Zadra from Montréal, and that person's name would be excellent for a supervillain. But he's not. Apparently.

Even when dreams are studied in the laboratory, you can look at what’s going on in the brain or body while the person is dreaming — for instance, in REM sleep — but what they are dreaming about at that moment, we usually can only know once we wake up the individual, and he or she tells us about the dream they were experiencing. So dreams are a private, subjective experience.

Well, that's something of a relief.

Why, after I said I wanted to find a way to have them recorded?

Because we've seen what capitalism does to web browsing. Hell, one time I was just having lunch with a friend, on the patio of a local restaurant. We were talking about how much we'd like to go back to Vegas. After about a minute of this, our phones beeped nearly simultaneously (I'm Android and he's Apple, so they're different). We checked.

It was a message from Caesar's Palace.

Now imagine what marketers will do when they get access to our actual dreams.

But we know for instance, if we take the most vivid dreams, those that tend to occur in REM sleep, well, we know that the secondary visual areas are activated. And that makes sense because dreams are highly visual experiences. So the primary visual areas aren’t activated for the simple reason that your eyes are closed, there’s no visual input entering through your retina. So your brain is creating this. We also know that your motor cortex, the part of your brain that controls motor movement is activated. And that probably is one of the things that helps give us the impression that we are moving through a real three-dimensional physical world in our dreams. We know that the limbic system is also activated, and the amygdala, which probably helps explain why many dreams contain various degrees of emotions, so we are emotionally engaged in them. And we know that parts of the prefrontal cortex, the part of your brain that sits about an inch or so above your eyes, is deactivated. And so this also explains why these areas of the brain are important for what we call executive functions, judgment, critical thinking, planning, things that are usually absent in our dreams.

This might also explain why, most of the time, we don't "know" we're dreaming. Some people try different techniques to induce lucid dreaming, which sounds interesting to me, but too much work; without that, for me at least, once you figure out you're dreaming, the dream generally ends or takes on a different form.

Now, admittedly, the article is pretty long (one reason I prefer reading to listening/viewing is that I can skim if I want, and I did). And there's a lot more at the link; more than I'd care to sift through for this entry. But here are a few choice quotes that I found especially enlightening:

For every two hours we spend awake, it appears that the brain needs to shut off all external input for an hour to make sense of what we’ve experienced. And that is what sleep is in part.

That might be the most concise explanation for why sleep is a thing that I've ever seen.

And even in phenomena such as lucid dreams, dreams in which you know that you’re dreaming, you have little idea of what happens next in your dream. Your brain is keeping this information from you. So in a lucid dream, you might make a dream character appear, for instance, but then if you ask them a question — Who are you? What are you doing in my dream? What is the most important thing I should remember out of this? — You have no idea what the character is going to say. But your brain does. Your brain is what is creating this character.

Which should be especially pertinent to us writers.

And so when people say, “Oh, you can do anything in your dream,” or “You are the producer and main actor of your dreams,” I don’t think that’s correct. You’re not at the wheel of the dream construction process; your brain is. And your brain intentionally keeps much of the information of what’s going to be happening next, and how things unfold, away from you.

Here, though, our supervillain makes a distinction between "you" and "your brain," which I find distasteful. Fortunately, the interviewer calls him out on it.

Anyway. There's a long bit about lucid dreaming, dream communication with the outside world, and the potentially Orwellian "dream engineering," which is even more scary to think about than dream tracking for marketing:

And I wouldn’t want my great grandchildren to have to pay $10 a month to opt out of advertisement in their dreams, Scott.

But I'm certain it's coming. Only it'll probably be $100 a month. Inflation, you know.

2 Like • Comment

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.