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:

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

JayNaNoOhNo

Nov 5, 2024 at 1:16pm
In response to "Anomalies"

*Toronto, Canada submits a sternly-worded petition and disputes the center of the universe claim.*

2 Like

February 14, 2021 at 12:35am

#1004420

Toxic Positivity

As I refuse to acknowledge what today is, I'll just post a random link.

Trying to Stay Optimistic Is Doing More Harm Than Good
No more FONO. How to recognize and break the habit of “toxic positivity.”

Fortunately, "trying to stay optimistic" isn't a trap I fall into. So, what's FONO? I guess I have to give them clicks to find out!

When her patient started talking about sick notes, neuropsychologist Judy Ho decided to intervene. Her client, a wildly successful entrepreneur, was rich, happily married, and well-regarded by his peers.

So, not someone anybody else is going to feel sorry for no matter what happens. Only in Bloomberg.

The problem was the days when he felt depressed and run-down but unable to admit it. The only way to address it, he felt, was to regress, like a schoolboy, and look for permission from a doctor to regroup. “He knew he wasn’t sick, but he’d go in and make something up,” she says, “just so he could take a day off and be OK with himself.”

Free business idea for you: sell miniature violins.

She recognized he was suffering from a surging contemporary malaise. “He always had to demonstrate his worth to people,” she continues. “He was thinking, ‘I must exude this image of success and a happy life that everybody has come to know about me, and I don’t want to ever change that image.’ That’s toxic positivity.”

Oh. Damn. I was hoping "toxic positivity" referred to the trend of people to exhort other people to always look on the bright side of life. Those people make me grumpy. I mean, good for them, but bah humbug.

This is bad enough, though, I suppose. Kind of like the social media hounds who are always portraying themselves as shiny and happy all the time.

Call it FONO, or fear of a negative outlook. Also known as “dismissive positivity,” it’s expressed as an overbearing cheerfulness no matter how bad things are, a pep that denies emotional oxygen to anything but a rictus grin.

Ah, there's the definition. Nah... too close to FOMO for my taste. Used to be referred to as Pollyanna syndrome or something, after a character who was always, always cheerful no matter what happened.

You see it on Instagram, where the affective filter is always upbeat, usually followed by the hashtag #blessed.

No, I don't, because I avoid Instagram even more strenuously than I avoid Bookface and Twatter. And hashtags are a plague upon the land.

You hear it from the SoulCycle instructor exhorting every rider to swaggeringly sweat through the pain.

No, I don't, because... well, you know.

It’s available from the newly anointed chief creative officer for Vital Proteins, actress Jennifer Aniston, who claims that renewal isn’t only a result of its powders: Instead, “it’s within us.”

So what do we need of your patent nostrums?

You might even recognize it in the boss who insists that colleagues start every Zoom meeting by sharing a piece of good news to help keep moods buoyant amid the gloom.

Now, look, maybe this is going a bit too far. I don't think there's anything wrong with acknowledging that there's some good news during times of crisis. It's not just that every silver lining has a cloud, it's that there may be benefit, when you're bombarded by bad news all the time, to try to find one positive thing in your life. The problem comes when you focus only on the positive things, just as it would be a problem to focus only on the negative things.

In my own opinion, anyway.

For example: I hate February. I mean, I utterly despise this fucking month with a passion fit to burn stars. It's cold, it contains stupid observances like Groundhog Day, President's Day, and... well... that which shall not be mentioned but falls on today. It hosts a stupid sportsball game with stupid commercials. My birthday occurs in this hated month, which is a constant reminder that I have fewer of those in front of me than I have behind me. It's dreary and confining and depressing. On the bright side, most years it's only 28 days. As of today, it's half over. See? I can find a light in the darkness when I try.

Think of this mindset as one that responds to all human anxiety, or sadness, with uncompromising optimism. It can be found in sentences that start with those negating words “At least,” which are followed by a suggestion that however bad you’re feeling, at least you’ve got plenty else that should offset and outweigh it.

If I trust you enough to bitch about something, don't ever start a sentence with "At least." I will punch you, and then I'll feel bad about it, but at least I'll have had the satisfaction of punching you.

Ordinary Americans, casting around for inspiration and reassurance, became prime targets for these peddlers of perkiness.

"Peddlers of perkiness" would be an awesome name for a retro swing punk band.

Such magical thinking has paralleled the rise of professionals hired to be a personal cheerleader. Membership of the International Coaching Federation, a credentialing and training program accrediting body, has soared from almost 4,700 worldwide in 2001 to more than 41,000 today.

Wait. This exists? This is a thing? Forget what I said earlier. There is no positive side to this.

Successful people are the most likely to fall prey to this way of thinking, says Naomi Torres-Mackie.

I think I might be successful investing in tiny-violin futures.

For the current generation, the origins of this emotional cure-all lie in the 1990s, when then-president of the American Psychological Association, Martin Seligman, posited that pessimism is a learned behavior. Therefore it both could and should be avoided.

Okay, but I'm pretty sure faux positivity started long, long before this asshole. Let's see... oh, yeah, here.

"The modern positive thinking movement started in the late 1800s with a watchmaker named Phineas Quimby. Quimby became fascinated with the practice of mesmerism (a.k.a. hypnotism). He became the apprentice of a famous French mesmerist and traveled New England learning the trade. Once he could hypnotize on his own, he opened a practice and started having some success alleviating the symptoms of psychosomatic disorders. This lead him to believe that the body was a reflection of the mind and that all illness was caused by false beliefs."

Ah, yes, one of the oldest fallacies in the book. "An herb cured my illness, so ALL illness must be curable by herbs!"

My apologies to anyone who actually clicks on that second link. I started feeling nauseated just glancing at it. Consequently, I would say their assertion about Quimby is, at the very least, suspect. But my point is that people have been pushing bullshit positivity since LONG before the 1990s.

That observation snowballed into bestsellers such as The Secret, first published in 2006 by Australian TV executive-turned-author Rhonda Byrne. It was popularized after Oprah Winfrey championed its ethos. That breakout bunkum bible was essentially built on claims that the power of positive thinking would provide whatever you want, be it a baby or a Mercedes-Benz.

"Breakout bunkum bible?" I'm thinking bardcore / metal / steampunk fusion band.

But yes, "The Secret" is, with the possible exception of Twilight, the stupidest piece of trash published in this century. And yet, here I am, unpublished, so not so stupid for the author, is it?

So, How to Cope?

This is the central question behind about 90% of the shit I see online these days.

Ho, the neuropsychologist, has an unexpected suggestion to help calibrate a Pollyanna perspective: a session watching Disney-Pixar’s Inside Out, which animates and dramatizes human emotions.

This therapy session has been brought to you by Disney. Disney: We bring good things wherever we go! Be sure to subscribe to our streaming service, and remember, our theme parks are still open and waiting for your smiling (but masked and socially distant) faces! Remember Disney for all of your entertainment needs. Now featuring Star Wars, Marvel Comics, National Geographic, and the best lawyers in the business. Disney!

It’s no surprise that Byrne would also return now. Her sequel, The Greatest Secret, came out in November. Read it, the blurbs tout, and you can remove all negativity—as if doing so should be a central goal in life. (More than 80% of Amazon.com Inc.’s user reviews gave it five stars. It would be too negative to be negative, it seems.)

I so very badly want to give it a one-star review on general principles, but I have too much of a sense of honor to review something I haven't read, way too much self-respect to actually read it, and no intention of actually paying money for that dreck.

So there's my rant for the day, which I'll end by reiterating: I hate February.

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