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:

Elycia Lee ☮

about 5 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 18 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.

1 Like

Annette in France

about 21 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.

1 Like

Elisa: Middle Aged Stik

about 23 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!

2 Like

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.

1 Like

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.

1 Like

Annette in France

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

I'm already out of Paris.

3 Like

March 27, 2020 at 12:11am

#979298

Lone

This is from three years ago, but it seems especially relevant now.

https://www.citylab.com/life/2017/04/how-to-break-the-dangerous-cycle-of-lonelin...

How to Break the Dangerous Cycle of Loneliness
Social isolation kills, and in the process it makes it harder to reach out to others. A psychologist explains how to stop the feedback loop.

I ventured out today, for the first time in a week and a half. Of course it was to buy beer. I encountered the brewmaster, who's a friend of sorts (always be friends with the brewmaster; that's your tip for today), and said that I've been pretending my self-isolation is a kind of virtue signaling instead of life as usual. I'll probably use that line, or something like it, in a Comedy newsletter.

Last year Surgeon General Vivek Murthy warned that Americans are “facing an epidemic of loneliness and social isolation.”

Funny, that word.

Loneliness is more dangerous than obesity, and it’s about as deadly as smoking. The threat is considered so serious that England has created an entire “Campaign to End Loneliness.”

So what if you're a fat smoker who is alone? I guess I'm fucked.

Also, if England wanted to end loneliness, maybe they shouldn't have self-isolated from the rest of Europe?

To learn more about this conundrum, and how to resolve it, I recently spoke with John Cacioppo, a psychologist at the University of Chicago who wrote a book on loneliness and has researched the phenomenon extensively.

Oh, hey, we haven't had a good book-shilling in here for hours.

First of all, let me qualify something. Living alone, being alone, and the size of your social network is only weakly related. Think about patients in hospitals: They aren’t alone, they have all the support they could ask for, but they tend to feel very lonely. There’s a difference between being alone and feeling alone.

I have a housemate, and we're friends, and I have other friends, including a bunch of people on this site. Physical distance isn't a barrier to friendship. Hard to drink with internet friends, but lately they've been doing workarounds for that. I'm kicking myself for not buying stock in Zoom before this whole mess started.

For instance, the internet has increased connectivity. But if you ever find yourself [looking at] your texts and emails at an event with your family, you may realize that those digital connections don’t mean that you feel more connected.

I grow weary of people ragging on internet-connected friendships. We've had this thing for more than a generation, now, and I was an early adopter specifically so that I could meet people - not get laid, but meet people - from other places. Family can be oppressive, and it's natural to want to escape their judgment in some cases.

And then the guy gets into evolutionary psychology, so I kinda skimmed that part. Pseudoscience.

Being with others doesn’t mean you’re going to feel connected, and being alone doesn’t mean you're going to feel lonely. It can, but usually we choose to be alone.

The majority of people are extroverts. I have to wonder if all this psych sorcery takes into account those of us who generally prefer to be alone.

Self-preservation depends more on your attention to your outcomes when you’re lonely than when you have lots of connections. [Sometimes] if you talk to a lonely person, they'll start talking to you and you can't get away. So, how do you share rather than just barrage? It’s about interactions, it’s about synergy, it's about mutuality.

Here's the thing about actual loneliness: it feeds on itself. I remember I was dating this chick back in college, and her roommates had put a thing on their bulletin board (we didn't have internet then) that went something like, "Wouldn't the world be a better place if depression and loneliness were attractive qualities?"

People sense loneliness and their instinct isn't to fix it, but to distance themselves from it, like it's some new virus coming around to infect them. This just isolates the lonely, depressed people more, and the cycle continues.

One of the biggest misunderstandings is what loneliness is. They equate it with being alone, and that leads to attempts to solve the problem that don’t solve the problem at all. And if you try enough times, you start to feel like, “Well, I’ll never be able to solve this, I’m just a worthless person.” And that's when you start getting social withdrawal.

Exactly. I need my alone time, but I also need some time with friends. Short periods, anyway.

Do volunteer service in something that you enjoy. I've developed the acronym EASE

And I stopped reading there (to be fair, it's near the end of the article). Whenever a shrink starts making up acronyms, they can be safely avoided. Oh, you're probably wondering what the One Weird Trick is to break the cycle. I'll save you the click: it's to go out and do stuff to help others. Thanks, that's helpful in these times. I'm going to crawl back into my shell now.

1 Like • Comment • View 2 comments

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