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:

🌖 HuntersMoon

about 8 hours ago
In response to "Bon Voyage"

It's not exactly like having your credit cards hacked.

For those interested...

2 Like

Elisa: Snowman Stik

about 8 hours ago
In response to "Bon Voyage"

"People like a happy-ending success story, I hear."

Now to see if this one gets a good response here or if it needs to be further shared.

2 Like

🌖 HuntersMoon

a day ago
In response to "Up In Re-Volt"

Interesting... but the puns

@ Jayne: One of my all-time favorite scenes.

1 Like

tj wanderlust-words-in-motion

a day ago
In response to "Up In Re-Volt"

After reading this I realized that most of what I know about electricity is that if you get a shock, it Hertz!

4 Like

Jayne

a day ago
In response to "Up In Re-Volt"

If only there was some way to milk cows with without electricity...

I know what they're referring to, but I still think it sounds dumb lol

3 Like

🐕GeminiGem🎁

a day ago
In response to "Meat Me In the Lab"

I am a vegetarian for animal-rights reasons. I eat plant based meat, some are pretty good. The rest are an acquired taste. They can call cultivated meat "slaughter free" if they call the real thing "slaughter house delights" or something ridiculous like that.

2 Like

tj wanderlust-words-in-motion

Jan 16, 2025 at 4:18pm
In response to "Meat Me In the Lab"

In time, and once the bugs (possibly literal) are worked out, I think it's a great idea. Of course, twenty-five to fifty years down the road it's likely they'll find some kind of adverse health problems related to eating bio-meats, but if it's not that it would be something else. Life is like that and the answer isn't to not try, it's to push forward and overcome.

Will we (you and I) ever see the day every home has a food replicator in it instead of a microwave? Probably not, but it would not only solve a lot of food related issues, it would also solve a lot of problems of what to do with our waste products...

2 Like

🌖 HuntersMoon

Jan 16, 2025 at 10:11am
In response to "Meat Me In the Lab"

Congratulations, Bob! You managed to get through the whole subject without once mentioning Soylent Green.

Mmmmmm. Real meat!

2 Like

tj wanderlust-words-in-motion

Jan 15, 2025 at 10:18am
In response to "Walk the Prank"

If a practical joke backfires is it an impractical joke?

2 Like

tj wanderlust-words-in-motion

Jan 14, 2025 at 12:41pm
In response to "Failure Is Victory"

Just think, back in the mid seventeen hundreds, Ole Ben had himself an electric BBQ; he stunned the turkey, shocked himself and cooked the bird on a gadget called an electric jack. Personally, I think he should have used a pellet smoker.

2 Like

January 18, 2025 at 8:57am

#1082508

Bon Voyage

People like a happy-ending success story, I hear. This one's from Nautilus, and dated just last month:

Voyagers Ready to Go Dark
Earth’s most distant space probes prepare for their inevitable long night.

When the two Voyager probes launched into space in 1977, they were headed to uncharted territory. It was the first time humanity had sent robot spacecraft to study up close the four giant outer planets of our solar system: Jupiter, Saturn, Uranus, and Neptune.

I'll pause a moment for everyone to get the Uranus jokes out of their system.

...

Okay? Good.

Let me then emphasize that they did all this with 1970s technology. Personal computers hadn't even been invented yet.

Once the Voyagers’ tour of the four planets was complete in 1990, the world’s attention faded; but the probes continued to provide remarkable insights into the dynamics of the solar system, including ultraviolet sources among the stars and the boundary between the sun’s influence and interstellar space.

As I said above: success.

More than 45 years after they first launched, the Voyagers are now NASA’s longest-lived mission and the most distant human-made objects from the Earth—but they will one day soon go offline and drift silently into the final frontier, perhaps for eternity.

Well, maybe one of them will gain sentience and surprise future starship captains.

For McNutt, it’s a “pleasant surprise” that the Voyagers are still working after all these years: “I joke with people: If you go back and look at the original papers, the Voyagers were designed to work for four and a half years,” he says. “We’ve outlived the warranty by a factor of 10.”

It seems like a lot of NASA missions went on long past their expiration dates. I suppose that balances the few who fail early into the mission.

Even when the Voyagers can no longer communicate with Earth, it will not be the end of their mission. Both probes bear the famous 12-inch “golden record” of the sounds of Earth, greetings in more than 50 languages, music by Mozart and Chuck Berry, and a star map showing how to get here.

Kid Me thought that was a bad idea back then, and I still think it's a bad idea. It's like setting your home address in GPS: someone steals the GPS, knows where you live and that you're not home. Only in this case they'll know where we live and it doesn't matter if we're home.

But whatever. The chance of meeting other technological beings in this galaxy are minuscule at best, and if it happens, it'll be a long, long time from now, and that'll be the last remaining shred of human artifice.

There are no happy endings, you see. There are only writers who decide to stop the story early.

5 Like • Comment • View 2 comments

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