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 an hour 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 3 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 1, 2023 at 12:51am

#1049063

Going South

A Monday update about my Sunday travels:

The weather didn't hold out for me as I drove south into North Carolina yesterday morning, and I discovered, or rather, confirmed what I already knew about certain modern car features: rain can confuse the hell out of them.

When I was looking for a new-to-me car last fall, I had a few requirements: I knew I wanted a Subaru Crosstrek with a decent sound system and a sunroof. Most everything else was negotiable. Well, it turned out that the one they had on the lot fit the bill; though the system wasn't as high-end as the one in my previous Subaru, it's good enough.

I wasn't looking for smart features specifically, but they came with the car. For instance, Android Auto for connecting my phone's Maps to the car's screen. But also things like lane sensing and adaptive cruise control. Well, that's what I call it; I don't know what the official name is. It's like, if you set the cruise control but the car in front of you insists on plodding along at 2 mph below your cruise speed, it senses this and slows down.

A lot of people don't like new things, so I've heard all the hand-wringing about it, but it works. The car doesn't force you to use it. What I wasn't sure of was how it would handle real rain. Yesterday, I found out. No, not the hard way; turns out that when the sensors are blocked, it warns you. Okay, cool. Fair enough. Something to consider if we manage to get real autonomous vehicles before civilization collapses: how it'll handle reduced visibility. Probably by using something other than visual sensors. Or maybe by giving up and waiting for the rain to slow down. I don't know; I'm not that kind of engineer.

The sunroof is mostly useless in the rain, too, but that's as expected. Hence the name. It's not a rainroof. I've heard it called a moonroof, too, and I always thought that the difference was one can be opened and the other is fixed like a windshield, but apparently not. Whatever. Or maybe one of them is for baring your butt at other drivers? That seems dangerous.

The weather cleared up around midday, and I ended up in a small town I'd never even heard of, let alone been to (Kinston, NC) which turns out to have a couple of good breweries, so I stayed. One of the breweries is affiliated with the motel downtown, which, really, I don't know why more people don't pursue that business model. "Can't drive? Stay safe in one of our rooms!" Maybe they're afraid of the cleaning that might be necessary.

Today (Monday) is my last full day on the road; I expect to return home on Tuesday, after which I'll return to my usual kinds of blog entries.

But so far, I'm calling the trip a success: I had some new experiences, mostly positive, though even the negatives, such as the annoying push-button hotel from a few days ago and the occasional lower-quality beer, were worth the inconvenience, as they gave me stories to tell.

That could, of course, turn around ("go south," as it were), and if I were a writer, I'd be able to come up with sixty different plot twists to ruin the trip. Oh, wait. I am a writer. Well, bad things can happen at any time, road trip or not, so I'll just keep enjoying the journey for as long as it lasts.

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