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 3 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 5 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.

September 11, 2020 at 12:02am

#993024

Per Ardua Ad Astra

Whew. I feared today's prompt would be about... well, something else.

Mini-Contest below! You could win a Merit Badge.

PROMPT September 11th

Write about your favorite childhood game or toy.

Teen years count as childhood, right? At least some of them do. I don't remember much about my pre-teen toys or games, though I do know my parents taught me important games like chess, checkers, and poker.

I think I was maybe 12 or 13 when I got my first rocket, a gift from an aunt. It was a cardboard and plastic thing with a Star Trek theme (I told you I've always been a fan), and, most importantly for Kid Me, you could put a solid-fuel engine into the thing and launch the sucker into the trees.

Sure, the tree thing was inadvertent. We had about a ten-acre field, plenty of room for launch and recovery, but somehow, something like 1 out of every 3 launches ended up with me climbing a goddamned tree. This is why I'm not Elon Musk.

The Star Trek-themed rocket was only the first of many. The basic design of most of them was similar: a long cardboard tube, nose cone, fins. Some were even multi-stage. Some had clear plastic payload chambers, and I should raise a memorial to all the beetles, worms, and mice that underwent several Gs of acceleration in the name of science. One even had a camera built into the nose cone -- a 110 mm film camera, as this was long before the days of digital photography. At perigee, the thing would automatically snap a blurry picture of the launch field and use it to determine which tree it would land in.

Most model rockets take some time and various degrees of skill to build. Glue (several types), sandpaper, balsa sealer for the fins, different color paints, decals, that sort of thing. Some were easier to build than others; hell, some of them were extraordinarily complex. Once built and dry, you could shove in an engine, install an electric igniter, hook it up to a battery, and launch it.

I didn't have much else to do in those days, what with living out in the sticks and it being a few years yet before home computers became a thing, so I built well over a hundred rockets.

Some of them are probably still in the trees.

In those several years, though, there was one model that was forever beyond my reach, both in terms of price and, I was sure, complexity of build, and that was the scaled-down Saturn V.

But I have never really lost my love for rocketry, even though, these days, I don't have a place to launch them. It's the building part that intrigues me. Every once in a while, I'll get it in my head to put a rocket together just for the craft of it.

And a few years ago, I finally got my hands on a Saturn V.

Unfortunately, the paint scheme foiled me. The thing's completely built, but for half the paint and the decals. I keep thinking one of these days I'll revisit it, touch up the parts that got fucked up because I'm not so great at masking, and put the decals on. Hell, I wanted to do it for the Apollo 11 50th anniversary, but never seemed to get around to it.

Still sitting there, though. Waiting. One day...

Merit Badge Mini-Contest!

Today is an important day. No, not just because of that, but because it happens to be my 16th anniversary here on Writing.com.

So no prompt for comments today -- just comment below and you might get a badge tomorrow. As usual, deadline is the end of the day today, September 11, WDC time. If you must have a prompt, you could talk about model rockets or how awesome I am, or how awesome my model rockets are even though I don't have any pictures of them up. But really, I just want to encourage comments, so anything goes, today. That's right - it's my birthday, but I'm the one giving a present. (Don't worry if I've given you a Merit Badge in the last couple of weeks; if applicable, I'll remind myself to send it out later for CR credit.)

6 Like • Comment • View 10 comments

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