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Saturday, Dec. 21, 2024

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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:

tj-Merry Mischief Maker

about 13 hours 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

about 18 hours ago
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

about 18 hours ago
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

a day ago
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

a day ago
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

a day ago
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

Ichabod Crane-- Mince Pie

Dec 17, 2024 at 9:06am
In response to "Need for Speed"

One of the last jobs I had was with Delta. I did ask the maintanace guys about that tape once and they pointed out high speed will rip duct tape or any other tape right off. They used it personally for their boats sometimes because of its strength. But looking out a window at 400 miles an hour and 30 thousand feet up and see duct tape on the wing. Certainly made me nervous first time I saw it.

3 Like

September 9, 2024 at 9:39am

#1076518

Time Out

Back before the welcome Birthday Week diversion, I featured an article from Ars Technica on how they're coordinating time between Earth and Moon: "Moon Time"

Then, I wrote: "I have a couple of articles about this, from different perspectives." Well, today, the other article popped up. Being from Atlas Obscura, this one may be a little more approachable, but it also gives me more opportunity to be pedantic.

Moon Time Is a Thing Now—Here’s Why It Matters
Physicists recently created Coordinated Lunar Time, a time zone for our Moon.

On the Moon, there is no normal. There is no wind, no rain, no erosion.

Leaving aside that these things are "normal" for the moon, there is some wind (the solar wind) and some erosion, though that process works a good bit more slowly there.

Nothing flies overhead, and nothing green strains toward the sky.

...yet.

There are no daily rhythms like those we experience on Earth—no chirping of crickets, no sunset breeze.

I could live with the former, very easily. But there's also no air, so I wouldn't live very long.

A lunar day and a lunar night each last two Earth weeks.

Roughly. This corresponds with the phases of the moon, and is the result of the Moon always showing the same hemisphere to Earth, and its orbit around our planet. It's all very Newtonian.

What’s more, seconds tick by slightly faster than they do on our home world.

That effect is relativistic.

To understand why Moon time is so strange—and why scientists recently created a new and unique time zone just for the Moon—we have to spend a moment with Einstein.

People think of Einstein, they think of the famous equation. But time dilation is another thing entirely. He figured out a bunch of different shit; that's why they called him a genius.

Special relativity explains how different places in the universe perceive time and space differently; time seems to move more slowly in a larger gravitational field, relative to a lower gravity environment. And a clock moving through space ticks more slowly than a stationary one.

I have no quibbles here. It's just really hard to grok, because it's outside our everyday experience.

In the Hollywood interpretation of these theories, a character traveling in space for a long time, or under the gravitational influence of a black hole or other very large object, will age much more slowly than his loved ones on Earth.

As usual, Hollywood tends to exaggerate for the spectacle. You have to be traveling pretty damn close to light speed to experience that effect without precise measuring tools. And by "large," the author here means "massive." Black holes are generally pretty small, on a cosmic scale.

GPS satellites have to account for it in order to work. So every time you get directions on your smartphone, you experience the practical effects of special relativity.

I've noted this before, but it's relevant here.

In fact, all navigation is really about time. We invented longitude for this reason.

And I've talked about this, as well. Latitude is pretty easy to figure out. Longitude is a lot trickier.

First, we needed to invent a prime meridian, an imaginary line drawn from pole to pole along the planet’s surface. Then, for navigation, you need to figure out where you are relative to that fixed line. The simplest way to do this is to use a time scale that is the same at both locations. So longitude is really about clocks.

There's been a lot of talk lately, among some science video channels I occasionally watch, about how the Greenwich Meridian, the 0 degree longitude line, is in the "wrong" place. But ultimately, it doesn't matter where we put it, as long as everyone agrees on it. An earlier Prime Merdian went through Paris instead of a suburb of London, and I'm sure the French are just pleased as punch that everyone went with the English definition. So pleased that, even though France is mostly south of England, they use the next time zone to the east.

The only relevant thing here is that one needs to define a Prime Meridian for the Moon, which I'm pretty sure they've already done.

I say it should have gone through the center of Armstrong's first footprint, but no one listens to me.

Anyway, the article goes into how GPS accounts for relative time for Earth.

But we can’t just port this system over to the Moon. Clocks on the lunar surface actually tick faster than Earth clocks by almost 58 microseconds per day. It’s not much in a given day, but over the months it will add up, and it’s enough to disrupt the precise timing of GPS.

And thus, the calculation of latitude/longitude.

“It’s like having the entire Moon synchronized to one ‘time zone’ adjusted for the Moon’s gravity, rather than having clocks gradually drift out of sync with Earth’s time,” NIST physicist Bijunath Patla said in a statement earlier this month.

As good a summary as I've seen.

The physicists say their efforts to develop Coordinated Lunar Time could be applied to other places in the solar system, simply by adapting the clock system for any other world’s gravity.

Yeah, not really "any." The world would need to have a definable surface, which some of the larger planets don't. But we're not about to live in Jupiter's atmosphere anytime soon, so, fine.

It’s about time.

Thanks, now I can't use that pun in this entry's title.

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