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Wednesday, Jun. 26, 2024

10:06pm EDT

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Complex Numbers

by Robert Waltz

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 2 hours ago
In response to "The Oz Hypothesis"

The nekked wimmin look like something that I might have drawn in first grade.

1 Like

Aiva Raine

about 24 hours ago
In response to "Can't Spell Success without Sucs"

Ok- this made me truly LOL- "sipping a gin and tonic made with low-paid laborer tears"

Thank you. Totally agree with the rest of your article.

Money doesn't solve all problems but it sure as hell eliminates many and makes it easier to go places or buy things that can create some happiness. IE- a property somewhere quiet and peaceful and a trailer with which to chill on said property.

Rich people are typically clueless assholes.

1 Like

Anni Pon

a day ago
In response to "Can't Spell Success without Sucs"

I've never heard anyone say, "That guy is so successful, but he doesn't really make much money." Or anything similar. Sure, you can have a positive impact on the world without making lots of money, but people won't call it success.

2 Like

Annette

Jun 23, 2024 at 1:28pm
In response to "Revisited: "Experience""

This is how I coped:

1 Like

Elisa: Middle Aged Stik

Jun 23, 2024 at 9:51am
In response to "Revisited: "Experience""

Not gonna lie I kinda miss Zoom parties, at least for this community since people on WDC are so spread out. Meeting in person is pretty tricky and/or expensive. It's certainly more socializing than I've been doing regularly since, oh, 2022 or so.

As for server migration, well, I work in the IT world. I've been through that. And honestly, server migration is a bit nerve wracking, as some stuff can go missing. It's still more manageable than a complete redesign. Complete redesigns are awful. You can test in different environments that the public can't see till kingdom come, but things always go wrong when the public version is rolled out. A local grocery store recently redid their website, and the old site held thousands of recipes. When the new site was unveiled, none of those recipes made it over. I ended up reaching out on Messenger and was able to get a copy of a recipe I use somewhat often, but I'm kind of annoyed that the text recipes are currently MIA because somehow that data didn't transfer with the site rebuild. Absolute madness.

1 Like

Wordsmitty ✍️

Jun 22, 2024 at 5:56pm
In response to "Discomfort Zone"

Just so everyone is clear:

- There are credit cards and debit cards and even cash cards. They are treated differently by different places as some prefer debit and cash over credit (which usually chrages the merchant more).

- As Annette mentioned, using text and wi-fi is best and usually cheap or free included most US phone plans. However, 'data' outside N. America is not. If you want to wander streets and go hiking and check maps or GPS in Europe, make sure you have a data plan for Europe. That's where the thousands of dollar charges come in if you don't have that. Voice also is usually extra. So if you plan on a lot of talking, get a plan for that, too. (As I recall without an additional plan for Europe, standard charges were 25 or 50 cents for each text and and a dollar minute for voice.)

- On phones, you may have to turn on or unlock the ability to use European system as it communicates differently than the one used in the US. T-Mobile does have good info explanaing that and data plans once you find it on their site. I think there was a search to input your phone make & model to show you what will make it work outside N. America.

1 Like

Elisa: Middle Aged Stik

Jun 22, 2024 at 5:08pm
In response to "Discomfort Zone"

Funny thing is, everyone mentions day to day spending in cash, but I don't recall seeing a single person in the Netherlands use cash in the grocery store. I was in a couple different grocery stores and a convenience store outside normal tourist areas, and cards were preferred. Even in museums there were readers to get donations from cards instead of having boxes out to drop off cash donations. This may be less of a thing in smaller communities, but I do feel like some of the larger (non-tourist) communities are going cash free. The pandemic's legacy of contactless payments sure has been a weird one.

1 Like

Annette

Jun 22, 2024 at 4:51pm
In response to "Discomfort Zone"

I think that SIM card thing is over now. Check it out though. I didn't change my SIM card and I my bill was the exact same as always. I did only use texting and wi-fi though.

1 Like

Robert Waltz

Jun 22, 2024 at 4:46pm
In response to "Discomfort Zone"

Annette

, merci, tu as raison.

I have a pretty recent Samsung phone, and T-Mobile, but I seem to remember something about a different SIM card or activation or something. Easy enough to find out. I know there were horror stories for a while about people traveling and getting hit by surprise roaming fees in the thousands of dollars, so that would be something to avoid.

Thinking about going this autumn, because like you said, it's not summer. Also, hopefully fewer free-range tourist kids. One of the many great things about being me is that I have some flexibility, so I don't yet know exactly when.

1 Like

Annette

Jun 22, 2024 at 3:49pm
In response to "Discomfort Zone"

les bières en France sont mieux = meilleures (plural feminine beers)

The last time I went to France, I brought my Samsung Galaxy S8 and it worked perfectly for Google maps. I don't know what kind of phone you have, but if it's from 2018 or newer, you shouldn't have a problem with that.

I use T-Mobile and their plan includes sending text messages and using the internet (where available) without any upcharge. It's not a fancy plan, so I think it's a standard.

I don't know what kind of credit card you use, but Visa or MasterCard seem to be preferred to the others. I haven't had any problems using credit cards to rent cars or whatnot. I like to exchange some cash for small day to day purchases. It makes things easier and more fun too. Although I still think that the Euro looks like Monopoly money with its fake bridges on it.

When do you plan to go? I know, you said before your passport expires, but what does that mean?

Also, don't go in summer. It's too hot and they don't have air-conditioning. Anywhere!

1 Like

June 18, 2024 at 11:44am

#1072853

Luna Ticks

It's refreshing to find a science article that seems to get the facts straight (though necessarily simplified). Surprisingly, it's from CNN.

Why scientists say we need to send clocks to the moon — soon

Before I get into it, though: no, time is still not an illusion. It's just tricky. This is analogous (but not exactly so) to answering a question like "How long is this iron rod?" when the answer depends on temperature and other factors, and probably can't be answered past a certain degree of precision.

On the lunar surface, a single Earth day would be roughly 56 microseconds shorter than on our home planet — a tiny number that can lead to significant inconsistencies over time.

Sure, that doesn't sound like a lot. Over one (earth) year, it only adds up to about 2 hundredths of a second. But they're talking about long-term effects, and highly precise instruments.

Never mind that the length of a year (and a lunar month) change slowly over time, too.

Scientists aren’t just looking to create a new “time zone” on the moon, as some headlines have suggested, said Cheryl Gramling, the lunar position, navigation, and timing and standards lead at NASA’s Goddard Space Flight Center in Maryland. Rather, the space agency and its partners are looking to create an entirely new “time scale,” or system of measurement that accounts for that fact that seconds tick by faster on the moon, Gramling noted.

CNN calling out other sources' headlines for being misleading probably falls under "irony."

Astronauts on the moon, for example, are going to leave their habitats to explore the surface and carry out science investigations, she said. They’re also going to be communicating with one another or driving their moon buggies while on the lunar surface.

It should be obvious that they can't exactly use Google Maps to navigate. I have no idea if it would even be technically possible to set up an equivalent GPS array for the moon. Even if it is (our natural satellite's gravity field has been described as "lumpy"), I imagine the cost / benefit analysis might not work out.

GPS comes with its own time issues, too, having to do with relativistic effects, which the article notes later. And it requires a great deal of precision.

General relativity is complicated...

Ya THINK?

...but in broad terms, it’s a framework that explains how gravity affects space and time.

Eh, fair enough.

The article goes on to explain quite a bit more, fortunately mostly in plain language.

Then it gets into the bit I really wanted to address:

Accurate clockwork is one matter. But how future astronauts living and working on the lunar surface will experience time is a different question entirely.

On Earth, our sense of one day is governed by the fact that the planet completes one rotation every 24 hours, giving most locations a consistent cycle of daylight and darkened nights. On the moon, however, the equator receives roughly 14 days of sunlight followed by 14 days of darkness.

If we don't doom ourselves or give up entirely, this won't be the last time such adjustments have to be made. While there are robots on Mars now, there's always been talk about sending humans there. As I'm sure we all know, a Martian "day," (which they call a "sol" to distinguish it from our familiar 24-hour cycle) is longer than 24 hours, but not by much. As a reader (and sometimes writer) of science fiction, I've often wondered whether we'd keep our circadian rhythms there, or try to adjust them to the Martian day/night cycle.

Considering that any Mars colonies will be entirely indoors, and probably underground, I could see it going either way. In a way, Mars is harder than the moon to figure out this sort of thing, because it's much closer to the cycle we're used to.

Not to mention we'd have to answer these questions for any other planet we choose to visit. (Please note that this isn't an entry about whether we could or should. Mostly, I very much needed to make the pun in the title.)

I mean, sure, the article mentions this sort of thing, too, but without details. Guess I'll have to do what other writers do and just make stuff up.

4 Like • Comment • View 2 comments

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