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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 5 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 7 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

3 Like

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.

2 Like

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.

1 Like

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.

1 Like

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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

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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