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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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

April 4, 2023 at 11:46am

#1047477

Only the Lonely

I saved this one a long time ago (relatively speaking), and I don't remember why. Let's find out together, shall we?

One Word That Fights Off Both Viruses and Loneliness
There's a way to limit your exposure to the dangers of illness and isolation.

"One word?" But "chicken soup" is two words. Also, what about limiting your exposure to the danger of being around people?

Have you ever gone to bed at night feeling sad or lonely?

Well, that's not a good question. I think you'd be hard-pressed to find someone over the age of, I dunno, zero, who has never gone to bed feeling sad or lonely.

Then, you woke up not feeling refreshed, but exhausted and on edge. You are not alone.

Yes. Yes, you are. Otherwise you wouldn't have been feeling lonely, would you?

We all know how Covid has produced higher levels of isolation. Wearing masks, as important as they are, is the equivalent of placing an emotional wall between yourself and others because you can't recognize facial expressions.

Oh, boo hoo. Most of my human interaction takes place on the internet. Facial expressions are entirely optional, with practice. What surprised me, with my partial face-blindness, was how I'd manage to recognize people even with masks on. I quickly came to realize that I don't rely on mouths at all, noses somewhat less, mostly hair and eyes. Speaking of which, eyes are very expressive, and they're generally not covered by N-95 masks.

Furthermore, the lack of daily contact caused by sitting in front of a computer screen (I'm certainly guilty of that) is no substitute for real, person-to-person contact and human touch.

Extrovert-like typing detected.

See? I can tell nuance just from phrasing.

But, okay. Most people are extroverts, so while very little of this applies to me, it's a window into that other world, the world of (shudder) people.

A study from the Proceedings of the National Academy of Sciences found that "Prior-day feelings of loneliness, sadness, threat, and lack of control were associated with a higher cortisol awakening response the next day." When cortisol floods the body and the brain, it has been shown to put the brakes on our immune system by reducing T-cells; cortisol has been shown to kill NK cells—important immune cells that help fight off viruses and even some kinds of tumors.

If I cared, I'd check that link and double-check some of the claims here. For our purposes, though, I'll take them, provisionally, at face value.

Imagine you had feelings of loneliness that lasted for days or weeks or longer.

I'm not a psychologist, but isn't that called "clinical depression?"

This could produce a chronic state of stress resulting in chronic health conditions. In fact, it's been reported that eight of 10 commonly prescribed medications are for symptoms of stress.

Or, and bear with me here, maybe people are stressed because they find themselves having to work long hours at a shitty job for subpar wages.

That's where mindfulness comes in.

Oh. That's why I saved this article.

To help reduce suffering, mindfulness teaches us how to experience the world through immersive inter-being with our surroundings. This means seeing ourselves mirrored in our human community, the air, the water, the planet, the plants, and all creatures large and small.

Too much work. And I've heard rumors that you can get such an experience through the use of certain pharmaceuticals. Oh, hey, maybe that's a reason why micro-doses of LSD are reported to cancel depression?

In our culture, it's commonplace to put a high value on our independence. This illusion of independence was lifted for me when I was in the monastery and had my first meal with the monks.

This came out of left field, but think "eastern," not "western."

If you haven't guessed by now, the "one word that fights off viruses and loneliness" is relationships.

There. Saved you a click.

The next evening that you feel lonely, remember that your body is listening in on your social and emotional experiences of the day. This is why that matters:

If you feel alone and lacking support, then your body will boost your cortisol response to prepare you for what it thinks will be a stressful day.

I'm not going to deny the link between mental and physical health, though it's probably way more complicated than a mere cortisol response. But I have a mantra for just such (rare) occasions: "Damn, it's good to be alone."

Alone doesn't mean lonely. Alone means you're not obligated to mitigate your actions for others' comfort or convenience. You don't have to watch your words or apologize later when you failed to. You don't have to concern yourself with whether or not someone else is hungry or tired. And best of all, you don't have to explain yourself.

Which is not to say I'm antisocial. I enjoy being around people, usually. Like on Saturday, when I went on a wine-tasting adventure with four other people (plus the very well-paid driver). Had a great time, found some delicious wines, saw some great views (it's a real privilege to live in Virginia).

Then everyone else went home, I passed out, and later, I got to nurse a hangover without worrying about how loud other people were being.

Back to the fluffy article:

Create mutually positive and satisfying relationships. This can take time, but the best mountain climber in the world can only take one step at a time. Start tonight by taking a simple, first step that connects you with another.

A "simple, first step" like "Hey, girl, I wonder if your software is compatible with my hardware?" My days of using pickup lines are long gone. My days of successfully using pickup lines are but a distant memory.

The article has a "conclusion" section, but I'm substituting my own:

Conclusion:

Thank you for your visit to our copium den. Buy our books, subscribe to our website! Because you aren't already financially overextended. And remember, kids, if modern society has you beaten down into a corner, it's your fault, not society's.

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