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 2 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 4 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.

August 9, 2021 at 12:06am

#1015323

Captain's Log

The personal lives of celebrities are generally not ones I care much about, but sometimes I run across an article like this.

‘Take it easy, nothing matters in the end’: William Shatner at 90, on love, loss and Leonard Nimoy
His career has taken him from Shakespeare to Star Trek – and soon he will be swimming with sharks on TV. He discusses longevity, tragedy, friendship and success

The interview is from May, so I suppose the sharks thing has already happened -- I vaguely remember hearing something about that, but Shark Week interests me even less than the Olympics or the personal lives of celebrities.

I could have done without the breathless immediacy of the present tense in the article, but the content is interesting enough. It helps to read his words with the distinctive Shatner voice in your head. You know you have one. Everyone does.

His tone is often heavily ironical, as if he is daring you to accuse him of playing a joke on you. This has led to much discussion from fans about “the Shatner persona”, although Shatner scoffs at the phrase. “I don’t know what that even is,” he says.

I think they think you play up to their expectations, I say.

“What are their expectations? That I’m Captain Kirk? Well, I am Captain Kirk! I don’t know what people mean when they talk about my persona. I’m just myself. If you’re not yourself, who are you?”

Indeed. Even actors.

There is a website dedicated to Shatner’s toupee, but his youthful appearance goes much further than impressive hair. Has he had some serious work done?

“No. Have you?” he shoots back.

No, because I’m just a journalist and can’t afford it, I say.

“Ha! Well, I don’t have any secret potions. It must be genetic. I ride a lot of horses and I’m into the bewilderment of the world, so I open my heart and head into the curiosity of how things work,” he says. I’m not sure if “the bewilderment of the world” is an ingredient Olay can bottle, but it certainly works for Shatner. That, and horses.

I am not going to the Shatner Hell Toupee website. If you must, the link is in the article. The important thing is the way he put "I'm into the bewilderment of the world." I can relate, even if I know I'm not going to make it to 90.

As for the horses thing, I think I remember reading somewhere that the horseback scenes from Star Trek: Generations were filmed on Shatner's farm with Shatner's horses. Dude can ride, that's for sure.

“Yes. Are you married?” he asks, not very subtly changing the subject.

Well, I say, I’ve lived with someone for a decade.

“And how’s it going?” he asks.

Well, we have three kids so there’s not much time to think about that, I say.

“And you bore them all?”

I what?

“You. Bore. Them. All?”

I’m not sure if he means did I give birth to them or do they all find me incredibly tedious, but either way the answer is yes.

I have to think he was deliberately playing with words. Gotta admire a good punster.

Shatner has been thinking about other dawnings because over lockdown he has been working on another spoken-word album.

AAAAAAAAAAHHHHHH! Maybe I could watch Shark Week instead.

“I didn’t know what I wanted to do. I just drifted with the currents of happenstance,” he says.

I'm just leaving this here for anyone who still thinks that dedication, goal-setting, and hard work are the keys to success. They are not.

He is a good sport about it, but when I ask one too many Trek questions (ie two) he changes the subject and tells me he has recently done a project with a company called StoryFile, which will recreate him as a 3D talking hologram.

Life imitates science fiction.

It feels rude to ask a 90-year-old if he worries about death, so I ask instead what he wishes he had known at 20 that he knows at 90.

“Here’s an interesting answer!” he says perkily. “I’m glad I didn’t know because what you know at 90 is: take it easy, nothing matters in the end, what goes up must come down. If I’d known that at 20, I wouldn’t have done anything!”

And there it is, folks: the most important life philosophy one can adopt.

I only quoted a few things from the rather lengthy interview, but it's worth reading for the other bits, especially about him and Nimoy.

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