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:

Writer_Mike

23 minutes ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 17 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

about 20 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

July 5, 2023 at 11:14am

#1052191

Real and Complex

Well, that holiday's over. Hope everyone still has all their fingers. If not, I'm going to laugh, especially because people all around me were setting off illegal fireworks last night—some of them not quitting until 2am. No, it didn't keep me awake; I was awake anyway, listening for the siren sounds of screams and the screaming sounds of sirens. No luck either way, dammit. Well... one distant scream, but no follow-up siren.

Mean? Maybe. But at least I'm not faking positivity.

Unexpected Ways Toxic Positivity Sneaks Its Sunshiny Way Into Your Day

While this article is from almost exactly four years ago, I haven't seen much improvement in attitudes, so here it is.

Our culture's smiley insistence on feeling happy 24/7 now has its own name: "toxic positivity." And this happiness pressure has started to raise concern among mental health experts.

As I am not a mental health expert, I'm not concerned; I'm just annoyed. More annoyed than by the aforementioned fireworks last night.

I reached out to Dr. Allison Niebes-Davis, a licensed clinical psychologist, for her professional definition of the problem: "Toxic positivity is an oversimplified approach to difficult stuff. It insists on people only seeing the bright side, that they be happy, cheerful, and positive. Phrases like 'Just think positive,' 'It’ll get better,' and 'Everything happens for a reason' are frequent offenders."

Several sayings are guaranteed to light my fuse, and those are three of them. Mostly that last one. If I punch you in the mouth after saying it, there was a reason for that.

Actually, "It'll get better" doesn't bug me that much, but "What doesn't kill you makes you stronger" does (it's demonstrably untrue).

As for "just think positive," I'm positive that we're doomed, and this is just more evidence of that.

As Dr. Niebes-Davis explains to Romper, "Toxic positivity minimizes painful emotions and downplays difficult experiences. It sends a subtle but clear message that there’s no space for sadness or tough stuff. When we insist on only looking at the positive, we shame people for struggling, and as a result, they often feel isolated and alone."

I forgot to mention, in case you haven't clicked yet, that the site I linked above is unfortunately named Romper.

Below I point out the top three ways it sneaks its rainbow-patterned way into your day.

Because it's Romper, and not Cracked, the list actually counts up from 1.

1. Social Media

I avoid social media not just because of toxic positivity, but also its opposite. Occasionally, something good will happen there, but those rare instances usually sneak out into my attention sphere anyway.

Anyone who's spent any time on Insta is well aware of the endless #positivevibes posts, usually featuring someone in a stylish white fedora hanging off the back of a yacht somewhere off the coast of Croatia. (Guess what, Genius Fedora Woman? It's pretty easy to be positive in that sitch.)

See? Money can buy happiness.

Take stock of your emotional state post-scroll.

Or—and I know I'm fighting an uphill battle here—avoid social media entirely, or at least curate your feeds better. Or whatever it is you do these days. As of this writing, the former has gotten easier for Twatter, which to me is a positive thing.

2. Happiness Merch

I'm detecting a tendency on the author's part to use cutesy abbreviations, which is also setting me off like a Fourth of July firecracker.

There currently seems to be a surplus of products featuring not-so-subliminal messages of positivity. Happiness is clearly #onbrand.

Not to mention number signs. If Twutter goes away, will pound signs? Actually, the name of the symbol is "octothorpe," which I only mention because that's an awesome word.

Maybe you just lost your job. Or your dog. Or your mind. You certainly don't want to "Smile!" You probably want to go home and shatter your "Good Vibes Only" mug with ball-peen hammer. And if so, well that is a perfectly normal and healthy response! As Dr. Niebes-Davis wisely reminds us: "Some things just aren’t happy. Some situations just can’t be made shiny."

The phrase is, "you can't polish a turd." Or something to that effect.

3. Friends IRL

Get better friends.

How are they when you try to open up about your struggles? Are they able to just sit with you and your feelings? Or do they immediately try to talk you out of your emotions, and put a "positive" spin on your pain?

"I don't want to be around negative people, so either cheer up, or go away until you do."

While trying to "stay positive" can of course be a good and sometimes necessary strategy for getting up and plowing through yet another day as a mortal being scurrying over a dying rock floating through space...

I'm just including this bit because I love the phrasing, which makes up for the precious abbreviations earlier.

And that's what's it's all about, right? The positive and the negative. As writer Harlan Ellison so wisely mused: "For without pain, there can be no pleasure. Without sadness, there can be no happiness. Without misery there can be no beauty. And without these, life is endless, hopeless, doomed and damned."

All due respect to the great Harlan Ellison, I think the great philosopher duo of Beavis and Butt-Head put it more succinctly:

"If everything was cool, and nothing sucked, how would we know what was cool?"

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