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Sunday, Dec. 22, 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:

Annette

a day 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

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
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

August 29, 2024 at 10:54am

#1075879

Moon Time

I have a couple of articles about this, from different perspectives. This one, from Ars Technica, gets a little, well... technical.

Researchers figure out how to keep clocks on the Earth, Moon in sync
A single standardized Earth/Moon time would aid communications, enable lunar GPS.

I'm pretty sure I've talked about this sort of thing in here before, but if so, that was before they'd figured this out. I found one example: "Luna Ticks"

Our communications and GPS networks all depend on keeping careful track of the precise timing of signals—including accounting for the effects of relativity.

From what I understand, there are two opposite effects of relativity with GPS. One is the relative speed of the satellites; the other is their elevation. While opposite, they inconveniently don't cancel each other out.

You also have to account for distance from receiver to each satellite, because of the speed of light. I mention this because while the lag time to satellites in low earth orbit is small, the moon is about 1.3 light-seconds away, and a second and a third (pun intended, of course) is a long damn time for some things.

It's relatively easy to account for that on the Earth, where we're dealing with a single set of adjustments that can be programmed into electronics that need to keep track of these things. But plans are in place to send a large array of hardware to the Moon, which has a considerably lower gravitational field (faster clocks!), which means that objects can stay in orbit despite moving more slowly (also faster clocks!).

No mention here of accounting for the variable gravitational field of the Moon, which I've seen described as "lumpy." I trust they've figured out how to deal with that, but if not, I'm certain we'll hear about it in an "oopsie!" article in the future.

It would be easy to set up an equivalent system to track time on the Moon...

I think this author has a different definition of "easy" than I do.

I might have chosen "straightforward," instead.

...but that would inevitably see the clocks run out of sync with those on Earth—a serious problem for things like scientific observations.

Relativity put a dagger in the back of the concept of "simultaneous;" two space-time events that seem to be simultaneous for one observer can be asynchronous for another. This is generally not an issue on Earth, and doesn't matter to our day-to-day lives (as long as we're not physicists), but I expect it makes a difference when dealing with separate worlds.

So, the International Astronomical Union has a resolution that calls for a "Lunar Celestial Reference System" and "Lunar Coordinate Time" to handle things there.

Yawn. Come on, you can do better than that. "Loonie time" is right there on top of my brain. Give me a few minutes (and a few drinks), and I might even come up with some forced acronyms, like "Lunar Uncoupled Normative, Asynchronous, Temporal Isochronic ClockS."

And that's without benefit of beer.

Anyway,

We're getting ready to explore the Moon.

Son, we've been exploring the Moon since before I was born. Just not very often by way of humans visiting the place.

We'll have an increasing set of hardware, and eventually facilities on the lunar surface.

Or under it, as I noted in another entry: "Cave People"

All that could potentially be handled by an independent lunar positioning system, if we're willing to accept it marching to its own temporal beat. But that will become a problem if we're ultimately going to do things like perform astronomy from the Moon, as the precise timing of events will be critical.

Thing about lunar astronomy is... well, it's obvious that not having an atmosphere helps a lot with observations, but we have space telescopes for shit like that. No, the thing is, to the best of my limited understanding, you can put a telescope on the Moon and one on Earth and one out in space, and what you get is effectively one big giant telescope, able to resolve distant objects to greater precision.

That is, if all your you get all your ticks and tocks lined up right.

As always, it's possible I got something wrong there, but the point is, I can see how synchronization would be important when doing astronomy between Earth and some other world.

What does this look like? Well, a lot of deriving equations. The paper's body has 55 of them, and there are another 67 in the appendices. So, a lot of the paper ends up looking like this.

If you want to get heart palpitations, go to the article to find out what "this" is.

Still, using their system, they're able to calculate that an object near the surface of the Moon will pick up an extra 56 microseconds every day, which is a problem in situations where we may be relying on measuring time with nanosecond precision.

This sort of thing is why I strongly object to airy "time is an illusion" proclamations. Nothing that can be calculated to that degree of accuracy deserves to be reduced to "illusion." It may not be fundamental, sure, and it's probably an emergent property, but so is temperature.

If time is an illusion, then it's time (pun intended) to revise the definition of "illusion."

And the researchers say that their approach, while focused on the Earth/Moon system, is still generalizable. Which means that it should be possible to modify it and create a frame of reference that would work on both Earth and anywhere else in the Solar System.

Mars or bust!

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