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:

Write_Mikey_Write!

about 14 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 16 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

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.

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.

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