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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QPdoll

about 9 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 12 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

about 23 hours ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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November 10, 2023 at 10:51am

#1059254

Area 51

This one's from Atlas Obscura, and will surely put to rest the pervasive rumors about secret government alien conspiracies.

Area 51, Aliens, and the Truth (It’s Out There)
A national security historian explains what’s really going on at the infamous Nevada site.

Ah... but that's what they want you to think.

One of the reasons people can never be entirely sure about what is going on at Area 51 is that it is a highly classified secret military facility.

Someone is entirely sure about what's going on. They're not talking.

It was not until 2013 that the United States government even acknowledged the existence and name “Area 51.”

After which they heaved a great sigh and opened Area 52, because Area 51 wasn't secret enough anymore.

As a national security historian, I know there’s a long history of secrets at Area 51. I also know that none of those secrets have anything to do with space aliens.

Or is the author lying?

The base commonly referred to as Area 51 is located in a remote area of southern Nevada, roughly 100 miles (161 kilometers) from Las Vegas.

It's also immediately adjacent to where they did underground and open-air nuclear bomb tests. Perhaps the aliens require a higher level of ionizing radiation to survive.

Area 51 was selected in 1955 to test the U-2 in part because its remote location could help keep the plane secret.

Where the streets have no name...

Area 51 became the test site for other secret new aircraft. This included the A-12, which, like the U-2, was a fast-flying reconnaissance plane. The A-12 was first test flown at Homey Airport in 1962. It had a bulging disc-like center to carry additional fuel. Its shape and shiny titanium body could well have been responsible for some people’s reports about seeing spherical ships, also known as flying saucers.

Why? Were they made out of swamp gas?

When the government does not tell the public the full truth, no matter the reasons, secrets can lead to wild speculation. Secrecy can leave room for conspiracy theories to develop.

And when the government does tell the truth, conspiracy theorists assume it's lying.

In fact, many UFO sightings match almost exactly with dates and times of flights of then-classified experimental aircraft. We also know that prototype drones and more recent versions have been tested at the site.

"Many." So... even if you're telling the truth, others are still aliens?

In the end, there is no reason to think that anything other than earthly technologies have been behind the strange sights and sounds at Area 51.

Ah... but that's what they want you to think.

Okay, seriously (everything I wrote above should be taken with a grain of iodized salt), the government needs to test their super-secret spy technologies somewhere. The only place in the US more remote than the middle of freakin' Nevada would be in Alaska, and that presents logistical challenges, such as either sending stuff across Canada or over the ocean, not to mention it's cold. Though I'd wager there's secret shit going on in that state, too. There's secret shit going on in a lot of places, including a site not far from where I live.

One day, it'll all come out in the open... and then certain people still won't believe it.

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