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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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.

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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.

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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 3, 2019 at 12:45am

#958116

Heritage

Tell us a fact about one of your ancestors. Where does your family come from? How far back can you trace your ancestry?

Real family or birth family?

It only matters in terms of what you're trying to prove for yourself, I suppose. Birth family contributes genetics, obviously. As I've noted before, though, I have no real interest in seeking them out. Last thing I need at my age is to find another mother. Yes, I know who she is. As of a couple of years ago, she was still alive. Still no interest in contacting her.

As for real family, I talked about them in my last entry. So here's a fact I don't think I've disclosed: One of my grandmothers died in 1918. Dad never told me how; I think the memory was always too painful for him, even though he would have been too young to remember her. From what I've been able to piece together, I suspect it was the flu epidemic that was rampant at the time.

And that's as far back as it goes. Like most people, I suppose, I got curious after a while, and when the internet made it possible for me to be lazy and still determine my heritage, I tried tracing it back. My legal name is extremely rare, so I figured it should be easy, right? Nope. Grandparents, end of story, as if they sprang from the ether to which my lineage will return when I'm gone.

Well, that's not entirely true. I know who my maternal great-grandfather was. From him descended not only my grandfather and mother - neither of whom was in any way famous - but a particular family who shall remain nameless for security reasons, but whose influence on the world is decidedly malign. Sometimes you learn things you'd rather not.

In the words of both Popeye and God, I am what I am. I don't need to trace lineage back to find myself; I'm right here. I might as well be a child of Earth; apart from visible features such as the pale skin and blue eyes that marks me as primarily of Northern European extraction, there's nothing to prove anything for or against a heritage from anywhere on the planet. I am, in short, a lineage of one, world without end, amen. Even that is merely an accident of genetics, which I've already argued against as a meaningful measure of anything.

There's a freedom in that, you know. I'm not boxed in (well, apart from having a fractured relationship with the accursed daystar, which keeps me from frolicking in the outdoors - pale skin, remember). There are enough factors keeping us apart from each other; why not just acknowledge our shared humanity and not worry about artificial divisions? Go back far enough and we're all descended from the same (relatively) hairless ape, anyway.

I mean, I get wanting to know where you come from. I just think it's more relevant to know where you're going.

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