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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
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Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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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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