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Tuesday, Apr. 23, 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

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

Show Recent Comments

10 Most Recent Blog Comments:

GeminiGem of House Lannister

about 3 hours ago
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

Lazy Writer est 4/24/2008

about 6 hours ago
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

Aennaytte: Free & Wild in GoT

about 7 hours ago
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

Aennaytte: Free & Wild in GoT

about 7 hours ago
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

Aennaytte: Free & Wild in GoT

about 7 hours ago
In response to "Revisited: "Hack This""

In my last psychology class, we discussed the evolution of the brain. It is now settled science that we evolved to have not only larger, but also vastly more powerful brains than most other animals because we learned how to cook (process) our food.
Although we are only about 1/4 to 1/3 of a normal gorilla, we eat half of the calories in a day. Not only that, we prepare the food in a way that we get to use more of the calories that we ingest.
The result is that our brain takes up more space in our noggin than a gorilla brain takes up in his noggin. Not only that, each of our brain cells is smaller, so we can fit exponentially more of those into our head.

Now, using all that goodness ...

Aennaytte: Free & Wild in GoT

about 7 hours ago
In response to "Hope for Hops"

Wait! Why are beer and coffee affected by climate change? That wasn't supposed to happen. All we had to do was have a boat by the side of the house for when the sea levels rise and beg our property insurance to cover fire damage. How are we supposed to remember all of that if we don't get coffee to wake us up in the morning so that we can think and beer at night to dull our sense of doom?

Aennaytte: Free & Wild in GoT

about 7 hours ago
In response to "Going to the Dogs"

I have never been to New York, but it seems that my assessment years ago that Berlin, Germany and New York have some similarities. Both cities are known for being inhabited by rude people. And both places, as I now found out, are dog obsessed. And they have high rents.
Berliners though, have a habit of letting their dogs run off leash. Maybe New York is a little better than Berlin in that regard.

1 Like

Robert Waltz

Apr 21, 2024 at 1:48pm
In response to "Revisited: "Hack This""

Wordsmitty ✍️, far be it from me to speculate on the motivations of space aliens, but why in Xenu's name would they do such a thing to the universe?

1 Like

Wordsmitty ✍️

Apr 21, 2024 at 1:39pm
In response to "Revisited: "Hack This""

I thought humans got bigger brains because extraterrestrials affected our evolution, and we eat processed chicken and eggs.

1 Like

Robert Waltz

Apr 21, 2024 at 10:22am
In response to "Revisited: "Hack This""

🌕 HuntersMoon, ah, yes, my favorite words to hear (or read).

Also, that video is an excellent, if brief, explanation of the conundrum and how to resolve it.

1 Like

November 10, 2022 at 12:02am

#1040465

The Far Side

Well, it missed the full moon by a couple of days, but this one finally came up from my queue for me:

Why the Moon’s two faces are so different
The far side of the Moon is incredibly different from the Earth-facing side. 63 years later, we know why the Moon's faces are not alike.

Article has a helpful picture of both hemispheres of the moon up at the top, and lots of other pretty pictures scattered throughout.

The Moon, by far, is the brightest object and largest object that’s visible to human eyes in Earth’s night sky.

I was about to object to this until I realized that it means that it appears the largest, not that it's the largest object, in absolute terms, that we can see. That would probably be a star somewhere, depending on your definition of "object."

So to satisfy my urge to be pedantic, I'll point out that one can often see the moon in the daytime sky, as well, during certain phases when it doesn't appear too close to the sun.

With even an off-the-shelf pair of binoculars or the cheapest telescope you can find, there are two main features about the Moon that you can’t miss:

That it's made of cheese and it's round?

In addition, because the Moon’s orbit is elliptical, moving faster when it’s closest to Earth and slower when it’s farthest away, the face of the Moon that’s visible changes ever-so-slightly, a phenomenon known as lunar libration. Even though this means, over the course of many months, we could see up to a total of 59% of the Moon, it wasn’t until 63 years ago, when the Soviet spacecraft Luna 3 swung around to the far side of the Moon, that we got our first pictures of the far side of the Moon.

Because of this, most of the features of the far side are named in Russian.

Although it wasn’t very impressive in terms of image quality, it was remarkable for an unexpected reason: the near side of the Moon appears vastly different, in terms of both cratered features and maria features, from the far side that always faces away from us. This discovery came as quite a shock, and for decades, even as our imaging and understanding of this elusive side of our nearest planetary neighbor improved in quality, we lacked an explanation as to why this difference existed at all.

The rest of the article explains just that, and it's pretty cool, not only for the explanation, but for the observations, deductions, and science that got us there. Which is fascinating, but there's not a lot of point in rehashing it here.

Then:

No matter how wild or unusual your idea may be, if it has sufficiently strong explanatory power to account for what we observe, it just might be the necessary idea to solve whatever puzzle it is that you’re considering.

Until some better observations come along, of course, and change everything we think we know. But that, too, is part of the awesomeness of science.

2 Like • Comment • View 1 comment

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