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Complex Numbers

by Waltz Invictus

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

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:

Annette

about 16 hours ago
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

a day ago
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

a day ago
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

🌖 HuntersMoon

Dec 19, 2024 at 8:53am
In response to "Five Seconds"

2 Like

Aiva Raine

Dec 17, 2024 at 7:39pm
In response to "Need for Speed"

Even with the actual explanation- it still left me going- sooooo what's the difference between the special expensive "speed tape" they use for "temporary fixes" and duct tape? LOL- sure sounds like duct tape to me and they are using it for exactly the same reasons you would use duct tape. I'm guessing that the only difference between it and duct tape is the "speed tape" is slightly stronger and handles temperature differences better. It still comes down holding the plane together with tape which doesn't exactly fill me with confidence. Especially if it's a Boeing plane.

1 Like

tj-Merry Mischief Maker

Dec 17, 2024 at 6:14pm
In response to "Need for Speed"

I suppose they call it speed tape to keep people calm. If they called it what it is, temporary repair tape, they may lose passengers.

2 Like

Waltz Invictus

Dec 17, 2024 at 9:43am
In response to "Need for Speed"

Ichabod Crane-- Mince Pie

, I've heard that speed tape is extremely expensive. I can only assume the maintenance guys at Delta got a five-finger discount. But if you're going to steal from work, might as well go big and not just lift a box of paper clips.

1 Like

May 10, 2024 at 9:59am

#1070838

Meditation Clause

Woke up with a case of gamer finger this morning. You know, when you sprain your finger from intermittently pushing the key that makes you go in the game? No? Damn, just me then. So I'd like to keep this short, but considering what I landed on today, I don't know if I'll be able to.

Are we morally obligated to meditate?
A growing body of neuroscience research shows that meditation can make us better to each other.

Just remember, my default response to headline questions is "no."

Starting in 2005, Harvard neuroscientist Sara Lazar began to publish some mind-blowing findings: Meditation can literally change the structure of your brain, thickening key areas of the cortex that help you control your attention and emotions.

Setting aside for the moment the ridiculous idea that "you" are some entity separate from your brain that can control it like a puppet, that finding, if true, is remarkable in itself. It'd be the brain equivalent of weight-lifting. The article does go on to mention follow-up studies, so there may be something to it.

Your brain — and possibly, by extension, your behavior — can reap the benefits if you practice meditation for half an hour a day over eight weeks.

Which might not be possible for everyone. As with exercise or recipe timing, the duration is misleadingly short. It takes time to prepare for these things, and more time to come down after. So you're looking at a 45-minute chunk of the day, maybe. Now, what are you willing to give up to get those 45 minutes? Sleep? If you do, you'll fall asleep trying to meditate (or, I don't know, maybe that's just me, too). Time with family? Dog-walking? Commute? Work? Lunch?

A host of other studies showed that meditation can also change your neural circuitry in ways that make you more compassionate, as well as more inclined to have positive feelings toward a victim of suffering and to see things from their perspective.

But what if you don't want to do all that?

Still skeptical, I fell down an internet rabbit hole and soon found many more neuroscientific studies. Looking closely at them, I did find that a fair number are methodologically flawed (more on that below). But there were many others that seemed sound.

See, this is what I look for in articles like this: some indication that the author didn't just accept whatever bullshit gets strewn about.

If it takes such a small amount of time and effort to get better at regulating my emotions, paying attention to other people, seeing things from their point of view, and acting altruistically, then … well … am I not morally obligated to do it?

That's not a science question. That's a philosophy question. But the whole next section of the article goes back to the science bit, which is worth reading, but I'm not quoting it here. So, provisionally accepting the science findings, I'm skipping to the philosophy section.

After I started wondering if we’re morally obligated to meditate, I soon realized that’s a very Western and Judeo-Christian way of thinking about it.

I would very much like for that hyphenated adjective to stop being used.

But Eastern traditions like Buddhism or Confucianism aren’t grounded in commandments that come from a divine being.

Neither are Western traditions. We've been conditioned to believe that the commandments came from a divine being. Which, from a practical perspective, I suppose produces the same results.

Plus, whereas the language of oughts and obligations suggests a prescriptive or proselytizing attitude, Buddhist tradition has generally been more interested in inviting people to try meditation and discover its benefits for themselves, rather than in mandating adherence.

Therein lurks an important difference between East and West.

I'd say this bit's important to note, too:

But there’s a caveat: For a small minority of people, meditation can actually provoke adverse effects, like intense mental distress or impaired physical functioning. Brown University psychologist Willoughby Britton is studying these cases in a project called “Varieties of Contemplative Experience.” More research is still needed, but given that meditation practices might precipitate or exacerbate challenging conditions in some people, it would be wrong to say that absolutely everyone would do well to meditate.

This is analogous to people urging us to get out and socialize more. That may be good advice for the extroverted majority, but we introverts would quickly spiral into doom. Or, to be even more simplistic in my analogies, saying that it's a good idea to always use your right hand to write, because studies show that it helps 90% of us to do so.

Scientists are publishing more and more studies on meditation each year. But many of these studies are beset by methodological flaws, leading to overhyped results. Davidson calls this “neuromythology.”

Yeah, we need to be really careful about that crap. It's like all those "quantum consciousness" bullshit books.

So, is the answer to the headline question still "no?" Well, you'll have to figure that out for yourself. Philosophy can guide science, and science can inform philosophy, but they're not the same sport. For me, I've spent too many years trying to figure out how to become more compassionate, but meditation just isn't my thing. I find it more practical to accept myself the way I am right now, flaws and all.

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