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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Show Recent Comments

10 Most Recent Blog Comments:

Wordsmitty ✍️

about an hour ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

1 Like

🌖 HuntersMoon

about 5 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

1 Like

tj-Merry Mischief Maker

about 24 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day 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

Dec 23, 2024 at 4:55pm
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.

4 Like

🌖 HuntersMoon

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

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

4 Like

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

4 Like

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.

1 Like

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.

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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