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Complex Numbers

by Waltz Invictus

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:

Annette

about 17 hours ago
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

a day ago
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

a day ago
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

🌖 HuntersMoon

Dec 19, 2024 at 8:53am
In response to "Five Seconds"

2 Like

Aiva Raine

Dec 17, 2024 at 7:39pm
In response to "Need for Speed"

Even with the actual explanation- it still left me going- sooooo what's the difference between the special expensive "speed tape" they use for "temporary fixes" and duct tape? LOL- sure sounds like duct tape to me and they are using it for exactly the same reasons you would use duct tape. I'm guessing that the only difference between it and duct tape is the "speed tape" is slightly stronger and handles temperature differences better. It still comes down holding the plane together with tape which doesn't exactly fill me with confidence. Especially if it's a Boeing plane.

1 Like

tj-Merry Mischief Maker

Dec 17, 2024 at 6:14pm
In response to "Need for Speed"

I suppose they call it speed tape to keep people calm. If they called it what it is, temporary repair tape, they may lose passengers.

2 Like

Waltz Invictus

Dec 17, 2024 at 9:43am
In response to "Need for Speed"

Ichabod Crane-- Mince Pie

, I've heard that speed tape is extremely expensive. I can only assume the maintenance guys at Delta got a five-finger discount. But if you're going to steal from work, might as well go big and not just lift a box of paper clips.

1 Like

June 20, 2024 at 10:05am

#1072938

A Midsummer Daydream

Solstice today.

At least, it's today in my time zone. Australians have to be different, so it's tomorrow if you're in Oz, and winter instead of summer.

I mean, the event occurs at the same moment for everyone, but at different times depending on your location. It can all be very confusing, I know.

I know a guy who likes summer, but gets sad at the summer solstice, because it marks the start of shortening daylight hours. I can understand that. I always mark the winter solstice by noting that daylight will increase, but for some reason, I ignore the complementary effect six months later.

Probably an example of willful ignorance on my part. It happens.

Some people have a favorite season. I used to, I guess, but I came to the realization that you can't have a favorite season without having to experience the other seasons, too. I do have a most hated season (winter) and a most hated month (February), but that's about it.

Which is kind of odd, because, by some definitions, February is mostly in spring.

("Okay, Waltz, you need to get your head examined.")

No. Well, maybe, but not for that reason.

Astronomical seasons indeed go from solstice to equinox, or vice-versa. Summer (in the only hemisphere that matters) starts today and will end on September 22. It also features four, not the usual three, full moons this year; the one in August will thus be a Blue Moon.

Meteorological seasons go by calendar month: June 1 to August 31 for summer, September 1 to November 30 for autumn, etc. They do this because temperature and weather cycles tend to lag solar quarters.

And by folklore, in some cultures, the solstices and equinoxes mark the middle of their respective seasons, not the end points. Midsummer is today (again, being a hemisphere snob). Midwinter is at the winter solstice. I've never heard of midspring or midautumn, but they follow logically.

By the folkloric definition, spring starts around Groundhog Day, which is one reason why there's a Groundhog Day: will winter weather linger, or start acting like spring? (That rodent is wrong more than he's right, so it's wise to bet against him over the long term, Bill Murray movies notwithstanding.)

Similarly, Halloween (or Samhain) marks the beginning of winter by that definition. The actual date is closer to Guy Fawkes Day, but whatever.

Point being that as of today, per some definitions, summer's already half over.

Reason enough to be depressed, as if we needed another one what with... *gestures helplessly at everything* ...and all.

In short, the events—solstices and equinoxes—are objectively real transition points, but the meaning we put on them can vary.

Yes, I've written about this sort of thing before, notably at the last winter solstice, and made many of the same points. But I can't expect everyone to read every entry. Hell, I don't read every entry. Apologies to those who do, for the repetition.

But hey, this (astronomical) summer, we might get treated to a nova.

I wrote about it back in March: "Celestial Spheres"

So that's something for us astronerds to look forward to.

If, that is, we live in the northern hemisphere.

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