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:

tj-Merry Mischief Maker

about 7 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

about 12 hours ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

about 14 hours ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

a day 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?

3 Like

🌖 HuntersMoon

a day ago
In response to "Gray Matters"

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

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
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

Dec 24, 2024 at 5:16pm
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

April 26, 2024 at 3:55pm

#1069815

On Track

A friend of mine sent me this link a while back, and it's just interesting enough to comment on.

The U.S. is exploring a railroad for the moon. It has a good reason.
All aboard the lunar express.

I have a board game called Lunar Rails. The idea is that you need to create rail links between lunar mines and settlements to facilitate transport of resources and passengers. Occasionally, something bad will happen like a solar flare, moonquake, or meteor strike, and you have to fix your routes or work around the disaster. I haven't played it in a while, because a board game requires friends who can all commit to the same time to play in person, and we have enough problems getting people together to play RPGs over the internet.

My point, besides bitching about how hard it still is to get people together even though we're no longer quarantined, is that the idea is hardly new. What is new is, apparently, that people are starting to examine the feasibility of turning it from science fiction into reality.

The first U.S. transcontinental railroad, completed with a spike hammered into the track in 1869, transformed the nation. Perhaps the same will happen on the moon in the game.

The difference being that the US had major settlements on both coasts before the railroad was built. Last I checked, the number of permanent human settlements on the moon was still 0.

Also, that last railroad spike was, famously, made of gold. While there is gold on the moon,

it's unclear to me how much there is, and how accessible. Not a lot of point wasting fuel lofting a golden metaphor up there, so they'd need a local source. But any gold found on the moon would probably be more useful for more practical reasons.

The Defense Advanced Research Projects Agency, or DARPA — an ambitious federal innovations division — has begun collaborating with over a dozen companies on potential future lunar technologies, including a moon railroad. It's called the 10-Year Lunar Architecture Capability Study, or LunA-10, and its mission is to find technologies that will catalyze a self-perpetuating lunar economy.

Yeah, so the headline is (big surprise here) a bit misleading. They're looking at the whole picture, which you have to do because it's not like you can go up there and wing it.

Now as NASA, global space agencies, and companies return to the moon with robotic spacecraft, the future Nayak sees is one that must be able to progress. It might be mining helium-3 (an extremely rare resource on Earth that could be used in medical imaging, computing, and even energy), harvesting water ice to create rocket fuel for deep space missions, and beyond.

Amusingly, helium-3 (a really incredibly rare, stable isotope of helium with just one neutron instead of the usual two) plays a key role in the video game Starfield. Also amusingly, you can mine it from the moon in the game.

DARPA recently chose the aerospace and defense giant Northrup Grumman to create the concept for the railroad. "The envisioned lunar railroad network could transport humans, supplies, and resources for commercial ventures across the lunar surface — contributing to a space economy for the United States and international partners," the company wrote.

Why I find the railroad thing interesting, above and beyond all the other shit you'd need to establish colonies and/or industry on the moon, is not just because I have the Lunar Rails game, or even because I studied transportation engineering. It's that we sometimes think of railroads as outdated technology, so we might not think about them in a space setting. Since then, we've made trucks and planes and huge cargo ships... none of which would be possible on the moon. Well, maybe trucks, but there's a question of how to keep them powered. You might think "rockets," but then you run into using fuel again.

The engineering challenges are interesting, though. I mentioned power above, but with rail, you can power it from various solar arrays at different locations (it's not like the moon has fossil fuels or wind power) and send it along the track like they do in the Northeast Corridor . There's the extremes of temperature to deal with; how do you design rails that will withstand both extreme heat and extreme cold? Not to mention what kind of design they might need to prevent derailments in the much lower gravity.

Well. At least they don't have to worry about wind resistance or rain.

Now, some people might consider the whole thing a waste of money and Earthly resources. I'm not going to justify it in this entry. Maybe another time.

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