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:

Turkey DrumStik

about 7 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

November 24, 2018 at 1:59am

#946152

Season

On every world, wherever people are, in the deepest part of the winter, at the exact mid-point, everybody stops and turns and hugs. As if to say, “Well done. Well done, everyone! We’re halfway out of the dark.” Back on Earth we call this Christmas. Or the Winter Solstice. On this world, the first settlers called it The Crystal Feast. You know what I call it? I call it expecting something for nothing!
-Doctor Who, A Christmas Carol (2010)

Well, Black Friday Eve and Black Friday are over.

This can only mean one thing: endless reminders that it's the holiday seasontime for rampant consumerism.

The best thing I can say about this time of year is that it's not February. The abundant food, holiday cheer, and festive atmosphere are largely lost on me. When I was a kid, my family observed Hanukkah, and that was nice: eight days of gifts and candle-lighting. I learned pretty early on, though, that Hanukkah isn't "Jewish Christmas," no matter what well-meaning Christians tried to make it out to be. Don't get me wrong; I appreciated the attempts at inclusion, once I got old enough to understand more about the world. Even acknowledging that I was different (but that's okay) is a big step up from the historically prevalent anti-Semitism. But it still set me apart.

By definition, most people don't know what it's like to live someplace where their observances aren't shared by the majority. It can set a person apart, keep him feeling like an outsider.

But I'm grateful for that. I think it's made me a better person, in general; I can empathize with the marginalized, the square pegs in the round holes, those who never quite felt like they fit in - even people who had a much worse time of it than I did. But I never really wanted to conform to the social norms.

I can fit in when I try. Both times I married, it was to women from Christian families. Though neither of my exes were what you'd call religious, they and their families took Christmas (and Thanksgiving) for granted, and were open-minded enough to include me in their celebrations. I'd hang out, drink eggnog (the good stuff), exchange presents, stuff my face with cookies and the occasional candy cane, and otherwise participate in their celebrations. Mostly, they didn't care that I was different; I never felt pressured into the more religious aspects of the observances. Not really surprising, since both my ex-wives were even more of the rebel type than I am.

The only thing about this time of year that ever really resonated with me was the astronomical aspect. Axial tilt: it's the reason for the season! Long before there was a Christmas, there were solstice observances. If you've been paying attention, you probably already know that Christmas is when it is because they co-opted earlier celebrations. To watch the days get shorter and shorter, until one day, on the solstice, the daylight is as short as it ever can be at a particular place in the northern hemisphere, and knowing that, from there, there's nowhere to go but up - that's cause enough to celebrate.

And the thing is, with all of our different religions and quasi-religions and other groups into which we like to divide ourselves, that - the solstice - is one event that, if we wanted to, we could all agree on. Well, almost all. It'd be different in the tropics, and completely reversed south of the equator, but (for now) we're all sharing one planet and we could, were we so inclined, all acknowledge the natural cycles of sun, moon, and planet. It could be a point of commonality, something that binds us together, not only as humans, but as part of the larger domain of life. Which, as far as we can know for certain, only exists here on Earth. (Okay, chances are it's more widespread in this vast universe than that, but we don't know for sure yet, not to "kitten" levels of certainty.)

But we can't have that, can we? No, we have to argue about whose holidays are more correct to celebrate, which foods we should and shouldn't eat, whose deity is the right one (if any), hell, whether or not it's rude to say "Happy Holidays" instead of "Merry Christmas." The best we can do, some of us anyway, is to appreciate our differences and our diversity, and be accepting of those who are different from us; but for the most part, we can barely even acknowledge that we're all human and all on this ball spinning in the darkness together, let alone agree on any one thing that unites us as a species.

And that's why I dislike this time of year. For some, it's a time of joy and wishes for peace, sure, and that's commendable. But in practice, it's more divisive than uniting, and that's what I, from my perspective as a thoughtful and cynical outsider, see in it.

So yeah, I'd like to think that we're "halfway out of the dark." But that's more of an ideal than the reality.

Still... I think I'll hold on to that ideal a little while longer.

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