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:

tj-Merry Mischief Maker

about 11 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

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

July 24, 2024 at 12:00pm

#1074345

It's Out There

From aeon, an article on a subject I've discussed before:

Alien life is no joke
Not long ago the search for extraterrestrials was considered laughable nonsense. Today, it’s serious and scientific

Though I'm repeating myself here, I feel the need to note that "life" doesn't imply "technologically-capable life."

Suddenly, everyone is talking about aliens.

For those of us paying attention, people have been talking about it all my life, and even before.

The next big multibillion-dollar space telescope (the successor to the James Webb) will be tuned to search for signatures of alien life on alien planets and NASA has a robust, well-funded programme in astrobiology.

Good. I hope they find some. Not Klingons, of course. Even just evidence of microbes or their equivalent would be a Big Fucking Deal.

Meanwhile, from breathless newspaper articles about unexplained navy pilot sightings to United States congressional testimony with wild claims of government programmes hiding crashed saucers, UFOs and UAPs (unidentified anomalous phenomena) seem to be making their own journey from the fringes.

On that front, the science is far less secure.

For decades, scientists wanting to think seriously about life in the Universe faced what’s been called the ‘giggle factor’, which was directly related to UFOs and their culture.

I once tried to start a conspiracy theory that went like: We have been trained to mock UFO enthusiasts because the extraterrestrials among us wanted to ensure that anyone who called them out for what they were got scorn heaped upon them.

I suck at starting conspiracy theories.

But for me, as a researcher in the field of technosignatures (signs of advanced alien tech) – the new face of SETI – getting past the giggle factor poses an existential challenge.

Once again, there's an implied conflation between "life" and "technolife." I call it that to avoid the inevitable tired jokes about "intelligence." You humans can be considered "technolife," but there's nothing that we know of about evolution that requires its development. There's probably exolife in our near stellar vicinity (perhaps even in our solar system), but I'd be surprised if technolife was anything but rare.

It is, however, in our best interests to search for both. It's probably not in our best interests to keep beaming signals to Proxima Centauri with messages like "Visit scenic Earth!"

There follows a bit of history about seeing UFOs (more recently renamed to UAPs, likely in an attempt to avoid mockery). And also the history of people looking for alien life.

I have no doubt that people have seen weird shit in the sky. What I have an issue with is immediately jumping to the "must be aliens" conclusion. It's like seeing ghosts. Perhaps you did see something; that doesn't mean it's a dead person's disembodied spirit.

Remarkably, when it comes to exoplanets, we are now also able to see exactly which planets are in their star’s habitable zone, where liquid water (the key, we believe, for life) can exist. That means we know exactly where to look in our search for life (something Drake could only dream of).

The "habitable zone" thing is real, but again, some people seem to equate "habitable zone" with "definitely must harbor life." No, it's a probability thing. If you're looking for a bear, you look in the woods, not a desert. If you're looking for a fish, you look in the ocean, not a mountain. Yes, sometimes you get fish in mountain lakes or streams or whatever, but your best bet is ocean.

Also, keep in mind there's a real chance that Jupiter's moon Europa harbors (simple) life under its ice cover, and it's not in the Sun's "habitable zone."

We’ve also demonstrated that there is no reason to suppose that biosignatures will be more common than technosignatures.

Yeah, that sounds like wishful thinking to me.

Since the exact same techniques are required to search for both bio- and technosignatures, there’s every reason to carry out both kinds of search at the same time.

But please, keep looking.

But if my colleagues and I claimed we’d found life on another world, we’d be required to provide evidence that meets the highest scientific standards. While we should let future studies lead us where they may, there is simply no such evidence surrounding UFOs and UAPs that meets these standards today.

And that bit is really what I care about: evidence. Not this leap to conclusions that definitely wasn't instigated by the aliens hiding among us.

The next big space telescope NASA is planning will be called the Habitable Worlds Observatory.

I'm betting that's a working title and it'll end up being named after someone, like the others were. One hopes they choose someone not associated with bigotry, this time.

Just remember, when they find biosignatures, don't immediately leap to "little green men" or their pop-culture alien equivalent. We're trying to keep our existence a secret, dammit.

7 Like • Comment • View 3 comments

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