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Sunday, Nov. 17, 2024

5:34pm EST

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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 an hour 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

January 6, 2024 at 9:27am

#1061966

At Last

Last year, I featured several articles about planets. Over time, I hit every one... except this one, which has been languishing in obscurity due to the inherent randomness of life. Well, not life, but my system. Anyway, here it is, published way back in September of 2019:

A Love Letter to the Last Planet
Neptune really deserves more attention, if you ask me.

Oh, it deserves more attention, does it? Well, I can't disagree, but it's not like we can just stroll over there.

You see, I fell in love with Neptune as a kid, back when I was in the second grade.

Dude, I like astronomy too, but he's too old for you.

We all had to read about a planet and tell a few sentences about why we liked it. I got assigned Neptune.

What if you turned out not to like it? Huh? Did the teacher ever think of that? "Jupiter's pretty, but its radiation would kill anyone within a million miles." "Everyone's fascinated by Mars, but there's nothing there." Or, you know, phrased the way a second-grader would.

And what if the teacher made them draw the planet? I mean, most planets are pretty easy. Circle. Appropriate color. Lots of black background. But whoever got Saturn would be in trouble.

Anyway.

In August of 1989, when Voyager 2 started sending back the first clear pictures of Neptune and its moons, the mysteries began giving way to marvels. Neptune is a world of giant methane storms and strange rings.

Coincidentally, I read an article recently which noted that the Voyager 2 pictures were false-color, giving Neptune a much deeper blue than it would actually appear to have in the unlikely event that there was a human hanging out nearby. Ah, here it is.

Looks way more like that other planet, the one I won't name.

So, then the article goes into some of the things we know about Neptune, and they are indeed Amazing Science Facts (well, to the extent that anything we know can be).

Something else even more wondrous happens in those deep layers of superionic ice. Carbon atoms get squeezed out of the methane molecules mixed in with the water, creating clumps of crystallized carbon. You probably know crystallized carbon by its more common name, diamond. According to laboratory simulations, the diamonds inside Neptune could have grown to be a meter wide. They are denser than the surrounding ices, so they sink downward toward the planet’s core.

That’s right: Inside Neptune, it is raining meter-wide diamonds.

Somewhere out there, according to some astronomers, is at least one entire planet made of diamond. I don't know, maybe it was like Neptune, but got too close to a star and its lighter materials blew away. Point being, diamonds aren't as rare on Earth as they try to convince us they are; and Out There, they're probably even less rare.

A diamond deserves a ring, and Neptune has several of them — five of them, in fact. Unlike Saturn’s rings, these are thin, dark structures around the planet, too dim to be observed clearly from Earth.

I also recently saw a JWST image of... you know, that other ice giant, which also has rings. Spectacular image.

That telescope was meant to peer into the farthest reaches of the universe, and yet they pointed it at what's, comparatively speaking, right next door.

Look, I'm purposely avoiding the puns, here. Just pretend I've already made the one you're thinking of.

So yeah, my adult self agrees with my 8-year-old self that Neptune is a fascinating place. Voyager 2 gave us only a taste of what it’s really all about. And Neptune is just the prototype of a whole class of Neptune-size planets that appear to be common around other stars.

When Voyager 2 did its thing, we had no evidence for planets around other stars. I mean, sure, we figured they had to be there, but no direct evidence. Now, we've found thousands.

In general, the more we know, the more we understand. Sure, there are probably higher priorities than a mission to Neptune. But that doesn't mean we shouldn't do it.

4 Like • Comment • View 1 comment

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