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.

February 21, 2022 at 12:03am

#1027122

Movie Knight

And now for something else from "Journalistic Intentions"

[18+] -- with a movie review at the bottom, coincidentally, because I finally made it back to the theater:

"Remember what the MPAA says; Horrific, Deplorable violence is okay, as long as people don't say any naughty woids! That's what this war is all about!"

Of all the quotes in this month's list, this is the only one that's the least bit familiar to me, but I can't for any price recall where it's from.

Doesn't matter.

People freak out over the weirdest shit. Like the word "shit," for example. And for a long time I haven't understood why it's perfectly okay to have violence in a movie, but not sex. Like, unless it's an R rated movie, you can depict someone sticking a sword into someone else, but not a penis. When, generally, the former is about violence and the latter is about love.

Of course, it isn't, always, I know. Sometimes it's about sexual assault. And that's still more okay to show in a movie than mutually pleasurable sex.

Which can only lead me to the conclusion that our society is fundamentally backwards. But we already knew that, didn't we?

As far as I can tell, other countries don't have this perversion. French movies, for example. All the ones I've ever heard of have involved someone trying to doink someone else and eventually succeeding. As far as I've been able to tell, that's the actual purpose of French cinema; everything else is extraneous. Whereas the purpose of American cinema is to have one side beat the hell out of the other side, usually the "good guys" and the "bad guys" respectively.

Anyone following along here already knows that one of my favorite shows is Star Trek. Has been for as long as I can remember, from way back when there was only one series and it only lasted three seasons. Since then, of course it's had its ups and downs, and certain gatekeepers love to bitch about some of the new series, going "That's not Trek!"

Well, whatever, they're entitled to their opinion of course, but not everything is going to follow the formula from the Original Series, and not everything can be as awesome as Deep Space Nine was. Point is, though, Star Trek has never been about violence. Sure, it's sometimes depicted, but the philosophy of the show has always put diplomacy and other means of conflict resolution first.

So when the series Discovery came out, I watched it with great interest, and for the most part I liked it. But one of the things that stuck out for me with Disco, as I like to call it, is that in (I think) the first season, one of the characters actually said "fuck" out loud.

I cheered.

Words are just words. If you're horrified by that word being uttered on a TV show, there's something wrong with you, not with the TV show. "But kids watch that stuff!" So what? Better they hear people cussing, or even see them boning, than watching them dismembering each other.

Let's get our priorities straight, okay?

One-Sentence Movie Review: Uncharted:

Spoiler: the good guys beat the hell out of the bad guys (as noted above); this is a fun movie, the cinematic equivalent of a frozen pizza: filling, but without much in the way of substance, and that's okay -- I never played the video games it was based on, so it's likely I missed a lot of in-jokes, but it's enough to watch Tom Holland be something other than Spider-Man and go haring off on a classic stunt-heavy, scenery-chewing treasure hunt quest.

Rating: 3/5

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