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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Elycia Lee ☮

about 3 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 16 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

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Annette in France

about 19 hours ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

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Elisa: Middle Aged Stik

about 21 hours ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

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Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

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Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

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Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

April 11, 2020 at 12:02am

#980742

All In A Dream

Many people dislike nightmares. I'm not a big fan myself, but they do sometimes give me story ideas.

https://www.theatlantic.com/magazine/archive/2019/04/dreams-what-do-they-mean/58...

Bad Dreams Are Good
Your night life prepares you for what’s to come.

I've come to be wary of anyone who states, conclusively, that they've finally figured out what purpose dreams have. But speculation? That's fine.

A handful of theories predominate. Sigmund Freud famously contended that they reveal hidden truths and wishes.

I'm pretty sure every single thing Freud said has been debunked, which is remarkable, given that even Donald Trump has been caught telling the truth at least once. You'd think Uncle Sig would have been right once or twice by accident, but no, I don't think so.

More recent research suggests that they may help us process intense emotions,

"process?"

or perhaps sort through and consolidate memories,

init defrag

or make sense of random neuron activity,

But what's doing the sense-making? Other neurons.

or rehearse responses to threatening situations.

Great, if I'm ever in a situation where I can fly away from a threat, I'll know exactly what I need to do.

Others argue that dreams have no evolutionary function, but simply dramatize personal concerns.

I'd be even more wary about "evolutionary function" arguments, pro or con.

Eight percent of dreams are about sex, a rate that holds for both women and men—though women are twice as likely as men to have sexual dreams about a public figure, while men are twice as likely to dream about multiple partners.

Sex? These days, some of us are wistfully dreaming about even casually touching another human being.

Anxiety is also rife: A study of Canadian university students found the most common dream topics, apart from sex, to be school, falling, being chased, and arriving too late for something.

So, Canadians are hyper-worried about being late. Interesting.

Which reminds me of a recurring dream I have. I suddenly remember, most of the way through the semester, that I signed up for a course I never showed up for, and now I have to take the exam.

I still have those dreams even though I haven't been in school for centuries.

A 1958 study determined that compared with Japanese people, Americans dreamed more about being locked up, losing a loved one, finding money, being inappropriately dressed or nude, or encountering an insane person. Japanese people were more likely to dream about school, trying repeatedly to do something, being paralyzed with fear, or “wild, violent beasts.”

Kaiju!

Speaking of trying repeatedly to do something in a dream, does anyone else have the thing where you keep trying to enter a number into your cell phone (or wherever), but you keep messing it up and have to start over? Or is that just me? It's like, in any dream, I find it impossible to ever type in the right numbers or spell anything correctly.

So the next time you dream about an education-related sexual experience in which you are both falling and being chased, don’t worry: It’s probably totally meaningless. Then again, your brain might be practicing so you’ll be ready if such an event ever comes to pass.

And like I said, if I'm ever in a situation where I can escape danger by zipping into the sky like Superman, or running away on all fours, my dreams will have prepared me well.

Here's a bit more on dream theory: https://www.sleep.org/articles/dream/

I probably would have gotten more out of the Atlantic article here if I'd bothered to follow the reference links. As it is, I remain unconvinced.

Perhaps I'll sleep on it.

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