Complex Numbers (Book)

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Complex Numbers

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/

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tj-Merry Mischief Maker

about 21 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

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🌖 HuntersMoon

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

Just wait until we team up with the cats...

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Jayngle Bells

Dec 23, 2024 at 10:03am
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

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Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

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

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tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
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.

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Annette

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

Comments work.

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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.

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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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