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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 13 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 15 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.

May 1, 2020 at 12:24am

#982442

Summer Fun

And it's May. Specifically, it's May 1, the beginning of summer here in the only hemisphere that really matters. "Waltz, have you been drinking again? Summer starts in late June!"

Yes, and no. To both sentences. Confused? Read on! But first, 30DBC

PROMPT May 1st

Tell us something uplifting! We need good news now more than ever. What is something positive that happened or is about to happen in your life? What has made you smile recently?

Something positive... hmmm... where have I... oh yeah.

Long ago, in a faraway land (I think it was the early noughties and Massachusetts), there lived a guy named Randall K. Milholland.

Randy had, at the time, the typical aimlessness of young adulthood, and, according to that link and other sources, his friend suggested that he do "something positive" with his life.

So he started a webcomic called Something*Positive.

Here's a link

to the current comic.

I can't say I've been following that strip since the beginning, but after getting introduced to it in, oh, 2004 or so, I went back and read all of the archives. Sometimes autobiographical, often dark, usually funny, and almost always worth a read, this comic has been a mainstay of my internet existence right alongside WDC (I started here in 2004 also) and Fark.

I know that "dark humor is like food; not everyone gets it." What I tell people is if they click on the First Comic link and aren't immediately repulsed by its content, they'll probably enjoy the rest of the series.

I don't do "uplifting." I don't give a fuck about "good news." And I generally only smile when a) my cats do something amusing or b) something terrible has happened to someone who I think deserved it. Besides, no one can tell when I smile; I lack the genetic programming to show my teeth when I try. So it doesn't matter anyway. If you want "Something Positive," well, there it is.

But, hey, I'm not trying to subvert the prompt, here. I found some good news on the internet. This really isn't relevant to my life, except that I love SCIENCE! and I get excited when there's new discoveries in SCIENCE! Besides, it's on the Good News Network, so it's appropriate here:

https://www.goodnewsnetwork.org/researchers-make-plants-that-glow-sustainably/

For the First Time, Researchers Make Plants That Glow Sustainably—And They Could One Day Light Up Our Homes

Although glowing plants may seem like a work of science fiction, researchers have succeeded in creating plants that produce their own visible luminescence—and they say the possibilities for how we can use these plants are endless.

I mostly encounter references to glowing plants in fantasy, but okay, I'll make allowances for movies such as Avatar, which is technically science fiction. In fantasy, they're a good excuse to represent a "magical" feel, whereas in science fiction, it's more "otherworldly." At least, that's how I see it as a connoisseur of both genres.

This week in Nature Biotechnology, the scientists revealed that bioluminescence found in some mushrooms is metabolically similar to the natural processes common among plants.

Fun Fact For You: mushrooms, and all fungi really, are more closely related to animals than to plants. This seems counterintuitive, but genetic studies have established this to a high degree of confidence. Here's

a link to the current state of genetic descent research. In the image at that link, you can see, way over on the right, that plants split off from the path leading to animals and other... kingdoms? Phyla? Whatever they're calling it these days, earlier than fungi split off from animals.

But, in the end, all life apparently came from the same source, and horizontal gene transfer isn't unheard of in the natural world; sometimes, we just make it happen on purpose.

The plants described in this week’s discovery can also be used for practical and aesthetic purposes, most notably for creating glowing flowers and other ornamental plants—and while replacing street lights with glowing trees may prove fantastical, the plants produce a pleasant green aura that emanates from their living energy.

Or, and hear me out here, maybe "pleasant" is overselling the concept. Depends on your point of view, of course.

According to the authors, the plants can produce over a billion photons per minute.

Okay, this is misleading as hell. I mean, I'm not doubting it, but out of curiosity, I looked up how many photons a typical candle emits in a given time period.

The tl;dr is at the bottom of the page: 1.51*10²⁰ photons per second. Since a billion is 1*10⁹, a billion photons per minute just doesn't seem that impressive. Even if you make allowances for the analysis' simplifying assumptions used to calculate the photon emission of a candle, we're looking at a plant that emits several orders of magnitude less light than a simple candle.

But, as the saying goes, the marvel isn't in how well the bear dances; it's that it dances at all. I mean, come on: glowing plants.

Light Bio is a new company that plans to commercialize this novel technology in ornamental house plants in partnership with Planta.

Of course, if you've been following along, you'll know that house plants don't last long around me. No plant does, except poison ivy and ragweed. So, while this is cool, it will never do me any good because I guarantee you within a week I'll have a dead mutant plant to dispose of, and I have my doubts about the bioutility of dead glowing houseplants.

As for the teaser at the top of this entry...

First of all, I drank beer earlier, but I'm entirely over it now. That's the "yes and no" to the hypothetical question above.

As for the hypothetical statement, the thing is, we have two summers, and they overlap. Astronomical seasons begin on solstices and equinoxes. Equinotcta. Equinotctae? Equinocti? Whatever; my last Latin class was XL years ago. Point is, astronomical summer in the northern hemisphere runs from the June solstice to the September equinox.

But civil summer is roughly centered on the summer solstice. It begins about halfway between the spring equinox and the summer solstice, and ends between then and the autumnal equinox. Fall equinox. Whatever. In modern times, that's been defined with respect to the Gregorian calendar as May 1 through August 1 (again, for the northern hemisphere). May 1 is, or was, known in the Celtic regions of the British Isles as Beltane and they treated it like the beginning of summer, so there it is. Modern Pagans still celebrate the occasion as Beltane. June 21 (or whatever day close to that when the sun's highest in the sky) is called Midsummer because it was literally in the middle of the time period they considered "summer." This confused me for many years, until I figured out that people use the same word to mean different things.

The actual midpoint, the First Day of Summer, is usually closer to May 5, but no one cares except the most extreme pedants. Naturally, this means I care. But again, "whatever;" the temperature outside right now is roughly 55F, not a thermometer reading I care to associate with "summer."

To be fair, it is, as I write this, the witching hour. So, again in the spirit of the prompt, happy Beltane, and enjoy your summer! Even if you end up spending it indoors with your cats, like I plan on doing.

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