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Sunday, Nov. 17, 2024

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

tj-turkey-jobble-jobble-hard-J

about 6 hours ago
In response to "Revisited: "A Frank Discussion""

I became disgruntled with meat sold in local grocery stores; half the weight is water, the taste is bland, and it's "use by" date is usually one day away. I found a small town butcher/locker plant that sells locally raised meats and buy in bulk so it's actually fresher, cheaper, and better tasting. Recently, I decided to try my hand at making our own sausage as well. It's a bit of work, but I think it's worth the trouble.

1 Like

Turkey DrumStik

a day 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

November 28, 2023 at 12:58am

#1060236

Upward Spiral

I'll be on the road for some unknown amount of time tomorrow [edit: I meant today. I'm writing this before I go to sleep, so tomorrow is today and today is totally tomorrow, while yesterday is but a fading memory as usual], stopping who knows where, so I figured I'd get in this last entry before traveling.

Speaking of traveling, here are some places none of us will ever go (probably):

In the Milky Way’s Stars, a History of Violence
Our galaxy's stars keep a record of its past. By reading those stories, astronomers are learning more about how the Milky Way came to be — and about the galaxy we live in today.

The amazing thing is that we can figure any of this out at all. It's been barely a century since we learned that there's more to the universe than our familiar galaxy, the discovery of which leads the linked article.

Late in the evening of October 5, 1923, Edwin Hubble sat at the eyepiece of the Hooker telescope at the Mount Wilson Observatory, atop the mountains overlooking the Los Angeles basin.

Visiting that telescope was the highlight of my trip to L.A. last year.

One might think, "What use is a telescope in L.A.?" Well, one, it's on a pretty high mountain and, two, I don't think the area was quite as smoggy or light-polluted a hundred years ago.

With the cleaving of the cosmos into a home galaxy and a larger universe, the study of our finite home — and how it exists within that universe — could begin in earnest.

I mean, sure, it was a paradigm shift much like the heliocentric model of the solar system centuries earlier, and a Big Deal, but it's not like we hadn't been studying the stars.

The latest results, amassed over the past four years, are now painting a picture of our home as a unique place, at a unique time.

To recap, the history of our thinking about our place in the universe went something like:

1) We're special and everything revolves around us.
2) We're not that special and not the center of the universe.
3) The universe is unimaginably big, and we're a rounding error.
4) Wait, maybe we are special.

We have been lucky, it seems, to live near a particularly quiet star on the calm fringes of a middle-aged, oddly tilted, loosely spiraling galaxy that has been largely left alone for most of its existence.

As regular readers know, I'm not discounting the power of luck, but if those conditions hadn't been relatively stable, we wouldn't be here to congratulate ourselves on our luck.

From the Earth’s surface — if you are somewhere very dark — you can only see the bright stripe of the Milky Way’s galactic disk, edge-on. But the galaxy we live in is so much more complicated.

It's getting harder and harder to find a place where you can see that. Oh, incidentally, it's called the Milky Way because of its appearance to the eye: a pale streak, like a road ("way.") It's translated from Latin's "via lactea," which, like a lot of things in Rome, came from Greek: galaxías kýklos. Hence "galaxy." Point being that the word, now applied to collections of stars all across the universe, shares a root with "milk" words like lactose and lactate.

I mention this simply because I find it amusing, if horribly mammal-centric.

A supermassive black hole churns at its center, surrounded by the “bulge,” a knot of stars containing some of the galaxy’s oldest stellar denizens.

Good to know it's not just middle-aged people who develop a "bulge."

Another digression: black holes have a scary reputation, like they're going to eat everything around them. And they do, but only up to a certain distance. Past that distance, they work just like any other massive object; you could orbit a black hole for a very long time, in the same way the Earth has been orbiting the sun for a very long time without showing any signs of falling in.

Next comes the “thin disk” — the structure we can see — where most of the Milky Way’s stars, including the sun, are partitioned into gargantuan spiraling arms.

One thing I've never been really clear on is how the spiral arms formed or are maintained.

While the article doesn't shed much light (pun intended) on that, it does go into the latest observations and theories about our galaxy's structure and how it got here.

And it's all very fascinating, at least to me, but I don't have much to comment on the rest of the article except to note that, given all the paradigm shifts we've already seen in cosmology, I'd expect more. And I look forward to it.

2 Like • Comment • View 2 comments

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