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

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Wordsmitty ✍️

about 16 hours ago
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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

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Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
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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
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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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September 6, 2023 at 8:29am

#1055247

The Big One

The RNG gives us another planetary article today. This one's about the big guy.

Jupiter revealed
The Juno spacecraft has been circling Jupiter since 2016. Here are four things we’ve learned so far about the biggest planet in the solar system.

In the four years since NASA’s Juno spacecraft went into orbit around Jupiter, it has slowly been coaxing the king of all the planets to reveal its deepest secrets — an astonishing catalog that includes daisy chains of continent-sized cyclones circling both of Jupiter’s poles; vast hailstorms of ammonia-laden “mushballs”; a bloated, fuzzy core at the planet’s center; and a convoluted magnetic field like nothing else in the solar system.

The article is from 2020, so add another three years to that, as well as possibly other new discoveries that I can't be arsed to search for right now. I did manage to check whether the ceiling fan (I mean, look at the probe

) is still active, which, apparently, it is.

Named after the goddess Juno, who was Jupiter’s wife in Roman mythology, the spacecraft was launched in August 2011 with the goal of understanding the giant planet’s origin and evolution.

"Jupiter's wife" glosses over a whole slew of godly shenanigans, including a bunch of extramarital affairs (not all of them consensual), and that Juno was also his sister.

Meanwhile, here are four of Juno’s greatest hits to date.

None of which are the hits she took out on Jupiter's illegitimate children. Or, well, they would be illegitimate, except that gods play by different rules.

Around the planet’s south pole, Juno spied five cyclones, each wider than the United States, parked around a central cyclone of the same size. Not to be outdone, the north pole revealed eight similar cyclones encircling their own polar vortex.

For context, those are pretty small for Jupiter. Compare the best-known storm on Jupiter's cloudy surface, the Great Red Spot (not to be confused with a teenager's acne), which at its peak was something like 3 times the diameter of our entire planet. (Later in the article, the GRS is described as being "a bit wider than Earth;" as with many of us, it's been subject to shrinkage over the last century or so.)

Because Jupiter is what astronomers call a gas giant planet, there is no point asking what conditions are like on its surface: It doesn’t have one. Instead, the hydrogen and helium gas that make up the bulk of Jupiter’s atmosphere simply get denser and denser the farther down you go, until the hydrogen becomes a liquid metal.

As with my last planetary entry, the whole "unclear boundary between phases" thing is mind-boggling. This adds another level of boggle with the metallic hydrogen thing, which has got to make astronomers irate, because to them, hydrogen and helium are nonmetals, while everything else is a metal. This is distinct from the chemistry definition of a metal, which has something to do with free, shared electrons. Hippie socialist electrons.

Metallic hydrogen, as I understand it, only occurs at unimaginable pressures, such as those found near the center of enormous planets.

As a digression, there's a weird thing about planets that I don't fully understand. Well, there are lots of those, but in particular, the center of any planet is subject to high pressures. That part makes sense, like how water pressure increases with depth. But because the core is surrounded by planet, the net force of gravity there is near zero (the mass pulls equally from all directions). I haven't reconciled this in my mind yet.

Although Jupiter doesn’t have a surface, researchers had a running argument before Juno’s arrival as to whether the planet had a core — a solid ball of heavier elements gathered at the planet’s center.

Need to settle a bar bet? Build a billion-dollar ceiling fan and send it to Jupiter.

I joke, but the probe really did shed some light on that subject, though there are, of course, still mysteries.

As for the planet's Olympic-class magnetic field:

It’s as if someone took a bar magnet, bent it almost in half, frayed one end, split the other end, and then stuck the whole thing in the planet at a cockeyed angle. In the north is the frayed end: Rather than emerging around one central spot, the magnetic field sprouts like weeds along a long high-latitude band. In the south is the split end: Some of the field plunges back into the planet around the south pole while some is concentrated in a spot just south of the equator.

Jupiter is, believe it or not, quite different from Earth, so it shouldn't be that surprising that its magnetic field is weird. Still, the article goes a bit into just how weird, and, well, its weird.

Anyway, apparently, some science got done and we need to do more science.

As always.

Just don't tell Juno we're sending another probe; she might get jealous.

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