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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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

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

Show Recent Comments

10 Most Recent Blog Comments:

Kåre Enga in Udon Thani

a day ago
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Udon Thani

a day ago
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

Kåre Enga in Udon Thani

a day ago
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

a day ago
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

Apr 23, 2024 at 6:51pm
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:11pm
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:04pm
In response to "Revisited: "Hack This""

In my last psychology class, we discussed the evolution of the brain. It is now settled science that we evolved to have not only larger, but also vastly more powerful brains than most other animals because we learned how to cook (process) our food.
Although we are only about 1/4 to 1/3 of a normal gorilla, we eat half of the calories in a day. Not only that, we prepare the food in a way that we get to use more of the calories that we ingest.
The result is that our brain takes up more space in our noggin than a gorilla brain takes up in his noggin. Not only that, each of our brain cells is smaller, so we can fit exponentially more of those into our head.

Now, using all that goodness ...

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 2:58pm
In response to "Hope for Hops"

Wait! Why are beer and coffee affected by climate change? That wasn't supposed to happen. All we had to do was have a boat by the side of the house for when the sea levels rise and beg our property insurance to cover fire damage. How are we supposed to remember all of that if we don't get coffee to wake us up in the morning so that we can think and beer at night to dull our sense of doom?

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.

3 Like • Comment • View 5 comments

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