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!

a day 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

a day 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').

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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.

July 17, 2019 at 12:06am

#962745

Hello Darkness...

Describe the earliest memory you have.

Anyone who's been following along already knows this one. And you're being inundated with it now. It's all over the internet. I can somewhat understand if you're burned out on it, even though it was not only the greatest achievement in history, but the greatest achievement humankind can ever make.

So I'm not going to talk about the time people first landed on another fucking world and most of the population of the original world, including me, watched.

I've said before that it's my earliest memory. It's definitely the earliest one I can hang a time on. But I have other early memories, ones that may or may not have taken place before July 20, 1969.

All of them are dreams.

I don't mean that I still dream about them; I don't. At least, I don't think so. But the images stayed with me. They might have been earlier, or later; I don't know. But I still think it's odd that I can remember a dream from fifty years ago (give or take) but not what I had for dinner last Tuesday.

The first one involved frogs on lily pads. It's dark, but in the way of dreams, I could see the flora and fauna clearly. The frogs told me something important. I have no idea what it was. Probably "Buy Apple stock in 1984." Stupid dreams.

Another one was of a five-pointed star, glowing in the darkness. Make of that what you will. I no longer see it, but the darkness is a recurring thing in my dreams. It never bothers me there. I move through it just as easily as I move through brightness in the waking world.

The third, but I think maybe the earliest, was of someplace underground - but bright. So bright. White stairs descending to a white hallway. At the bottom of the stairs, filling the hallway, gapes an abyss. The darkness again, black to the hallway's white. I stop at the edge of the pit. And that's all - I don't remember ever crossing it, falling in, or re-ascending.

I'm sure that has great portent, especially since, at three years of age (or thereabouts), I had little if any experience with stairs, and none with bright white hallways with vast yawning abysses in the floor. Much later in life, I made myself revisit the dream, but no matter how hard I tried, I could never make myself leap the abyss, or let myself fall into it. I keep thinking I will fall, one day, as will we all.

Whatever meaning these things have, if any, escapes me. At this point, I have to question even my memories of these dreams. But that's the way these things go, I suppose.

All that we see or seem
Is but a dream within a dream.

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