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 11 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 13 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.

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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').

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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.

May 28, 2022 at 12:03am

#1032941

Trouble

There weren't a whole lot of interesting things that happened on this day in history, and I picked this one at random.

On This Day
28-May-1974
Collapse of the Sunningdale Agreement

Here's the thing, though. Well, two things.

1. What little I know about Irish issues can be stuffed into a thimble; and
2. The linked article is largely unsourced and suspect.

So I won't talk directly about the Sunningdale Agreement. You can go to that page and read it yourself if you want. I'm not saying it's not interesting, but anything I say I'd be talking out my ass.

Not that I've never done that, but I'm trying to go into recovery.

What I am aware of is that a big part of the problem has to do with religion. Catholics, Protestants, whatever. Blame Henry VIII, I guess. The whole thing is way above my pay grade.

A friend of mine, a Wiccan, took a trip to Ireland once. Someone asked her something like "So, are you Catholic or Protestant?" (This is apparently less volatile a question than which football club you support).

"Actually, I'm Pagan," she said.

"We're all Pagan, dear. Are you a Catholic Pagan or a Protestant Pagan?"

I have no real reason to believe that actually happened, but it's how she told it. I've seen it elsewhere, so it might have been a joke. Either way, it adheres to Waltz's First Rule of Comedy: Never let the facts get in the way of a good joke. Or a bad one. Especially a bad one.

Speaking of bad jokes, I'm also reminded of this old one:

An Englishman, a Scotsman, and an Irishman are all drinking in a pub, getting loud. Problem is, they're all stuttering terribly. The other patrons are starting to get annoyed. This gorgeous woman comes up to them and says, "Tell you what, boys. If any of you can tell me where you were born, without stuttering, I'll show you a night you will never forget."

The Englishman is the first to speak up. He braces himself, prepares, and says, "Liverp-p-p-p-pool. Shite."

The Scotsman is next. He goes, "Edinb-b-b-b-burgh. Ach, dammit."

The Irishman looks calmly at the woman and says, "London."

Surprised, she takes him by the hand and leads him upstairs.

Half hour later they come back down. Irish guy's got a big grin on his face, he sits down, quaffs the rest of his ale, and goes, "...d-d-d-d-derry."

I suppose that the most unbelievable part of the joke is where those three were drinking together amicably, but I suppose it can happen.

Point is, I guess, that wherever you are, someone's not going to like the government, and that's why shit like this is relevant. Because, as always, if you don't learn from history, you're doomed to repeat it. Well, not necessarily "you." Like I said, I'm not well-versed in this aspect of history, but then again, I'm not in charge of anything but a couple of cats and a bit of property.

And this is really why I'm doing these entries: to learn.

There's no perfect government, and some people will never be pleased with whatever government they end up with. Ideally, that's what democracy is for, but things rarely turn out ideally, so you get demonstrations, riots, insurrections, civil wars, not-so-civil wars. These only get worse when other nations get involved, too. Again, I can't speak to the Irish situation, but there's another country right now that recently annexed part of a different country, and still wants more. Not exactly parallel, I suppose, but another echo.

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