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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 8 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 10 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.

July 16, 2020 at 12:18am

#988287

Comet Sutra

Harbinger of Doom! The End is Nigh! Repent and follow the ways of SCIENCE!!!

PROMPT July 16th

NEOWISE comet: “Enjoy it while you can. The frozen ball of ice won’t return to the inner solar system for 6,800 years.” Talk about it! https://www.nytimes.com/article/neowise-comet.html

Officially designated C/2020 F3, Comet NEOWISE was discovered on March 27 and had until this week been visible only to committed comet viewers willing to wake up in the early pre-dawn hours.

You mean "willing to stay up until the late pre-dawn hours."

I'm actually in a crappy place to view the comet. Sight lines to the east and west are blocked by trees. Sure, I could drive somewhere, but I haven't driven my car for four months and I'm seeing how long a streak I can keep up. Damn thing probably won't start now, it's been so long. Fortunately, I have roadside assistance.

But I digress. Comet.

It shouldn't surprise anyone that I've been following along. It's all the Astronomy Picture of the Day could talk about for the last week or so. This one's my favorite:

https://apod.nasa.gov/apod/ap200714.html

You can follow the links there to more images.

Until SCIENCE!!! came along, as I'm sure you're aware, comets were considered harbingers of doom. There was good reason for this: Doom happened with astonishing frequency, so once you see a comet, some disaster would inevitably follow within days, months, years, or decades, and they could then blame it on the comet.

Okay, so that's not actually a good reason, but when did humans need a reason to be illogical?

Anyway, I could get into the science and the history, but the NYT article linked in the prompt covers it all pretty well. Except it doesn't go into the whole "harbinger of doom" thing, perhaps because it's obvious bull hockey. Still, the fact is, people believed that shit, and I think it's very important to know that the more we learn, the more we study, the further into the background superstition can recede.

I say "can," because there's a whole lot of superstition now. We call some of it "conspiracy theories," and no amount of debunking seems to nudge them off their ignorant perch.

Hopefully, I'll get a chance to see it sometime in the next few days. As the article notes, July 23 will be a high point for the comet in the sky, and I might be able to catch a glimpse from somewhere in my neighborhood.

I really wish Nerd Camp had happened this year (and in July instead of August). Then I would have been able to see it from high in the Rockies, through a professional telescope, hanging out with an actual astronomer. Specifically, this guy.

I got to meet him two years ago, and we had some great talks about science and science fiction.

He also makes even worse puns than I do.

Speaking of which, it's time to announce the winner of yesterday's Mini-Contest!

Several of you amused me. Decent pun, Fivesixer, good to see you branching out. I have to admit I didn't click on the YouTube link that bobturn provided; after I explicitly stated that I don't find pranks on my person to be amusing, I just assume any naked YouTube link is a rickroll. PuppyTales thinks she's funny, but she's not. And Elle - on hiatus pushed the boundaries of this blog's rating to near their shattering point. ForeverDreamer, your joke wasn't out of bounds. And, SandraLynn, I also heard someone stole all the toilets from the police station, and the cops have nothing to go on. I appreciated all of the other comments too, funny or not, but I gotta award the prize to the dark humor of:

Sumojo: The woman took her dog to the vet. He examined the dog and shook his head and sighed. “I’m going to have to put him down.” He said.
The woman was horrified, “ Why!” She asked
“He’s getting heavy.” The vet replied.

It would seem that other people agree with me, based on the number of "Likes" that one produced, but this isn't a democracy; that's just a coincidence. Sumojo, I'll send your Merit Badge in a little bit.

Thanks again for all the comments, and I'll do another one in a few days!

9 Like • Comment • View 6 comments

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