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 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.

1 Like

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.

1 Like

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!

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

April 5, 2020 at 12:07am

#980273

Authentic

Related to my recent post about "be yourself:"

https://blogs.scientificamerican.com/observations/the-inconvenient-truth-about-y...

The Inconvenient Truth about Your “Authentic” Self
To actually feel authentic, you might have to betray your true nature

I can authentically say that I have never in my life worried about being "authentic." Likely, this is because I figured whatever I'm doing, it reflects a real aspect of my personality.

By Jennifer Beer on March 5, 2020

I solemnly swear that I didn't find this article doing a Google search for "beer."

Everyone wants to be authentic. You want to be true to yourself, not a slavish follower of social expectations.

"Everyone?" See above. And for some people, following social expectations is their purpose in life, their goal, their raison d'être.

Me, I have too hard a time figuring out what "social expectations" are to actually bother to adhere to them. For instance, all the complaining people are doing about being stuck at home, being bored. I seriously don't understand. There's so much to do! How can anyone be bored?

Although most people would define authenticity as acting in accordance with your idiosyncratic set of values and qualities, research has shown that people feel most authentic when they conform to a particular set of socially approved qualities, such as being extroverted, emotionally stable, conscientious, intellectual and agreeable.

This is going to be one of those articles that makes me feel like I'm an alien watching human society from the outside, isn't it? Yes. Yes, it is.

Authentic people have considerable self-knowledge and are motivated to learn more about themselves. They are equally interested in understanding their strengths and weaknesses, and they are willing to honestly reflect on feedback regardless of whether it is flattering or unflattering.

I never associated those qualities with "authenticity." To the extent I thought about them at all, I guess I consider them "maturity."

For example, introverted people are being authentic when they are quiet at a dinner party even if social convention dictates that guests should generate conversation.

I'm an introvert, and I'm far from quiet at dinner parties. Or, at least, I expect I would be -- I'm not the kind of person one invites to dinner parties, mostly because I had no idea that "social convention dictates that guests should generate conversation." I guess this is something you humans know that's been obscured to me?

Instead, research finds that people report feeling most authentic when their behavior confirms to a specific pattern of qualities: namely, when they are extroverted, emotionally stable, conscientious, intellectual and agreeable.

You... said... these things... up there.

Research has shown that we view people as less than fully human when they fail to conform to societal conventions.

Ah. Now I understand why I don't get invited to dinner parties.

So, when it comes time to actually make a judgment about our own authenticity, we may use criteria that are closer to how we judge the authenticity of an object such as food.

And you lost me there, too.

An aside about "authenticity" when it's applied to food: I find it irrelevant. The only thing I care about with food is: does it taste good? That's really my only criterion. Is it food? Then it's authentic, whether I like it or not. I've had people rag on a certain local restaurant because it's "not real Thai food." But it's good, so... so what? If you want real Thai food, go to Thailand. Even a hamburger in Thailand is Thai food.

The one exception is pizza. New York style pizza is pizza. That stuff they call "pizza" in Chicago is a casserole. I'm not saying it's bad. Just that it's not pizza.

And one time, in Washington State, I was on the road and hungry, and I passed by a strip mall with two restaurants that called themselves Mexican. One of them had bright neon light beer and tequila signs, with sportsball games visible through the windows on their huge screens. The other was sparse, with one TV set to a Spanish-speaking channel. I chose the latter. Not because I care about "authentic Mexican food," but because I hate sports and light beer.

So I guess that's my version of authenticity: doing what I like because I like it, not because someone else thinks I should. But sometimes, I also like to do what other people expect of me. I never claimed to be consistent. Or, hell, maybe I did, and now I'm contradicting myself.

I'm okay with that.

4 Like • Comment • View 2 comments

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