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

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

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Wordsmitty ✍️

about 18 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"

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WakeUpAndLive️~Happiness

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

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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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August 17, 2023 at 10:22am

#1054219

The Seventh Planet

So far, in my random meanderings around the solar system, I've hit Mercury, Earth, Pluto, and the mysterious, undiscovered-as-yet Planet Ix.

This... this is the one I've been dreading.

Uranus: Why we should visit the most unloved planet

I should point out that this article is from the BBC. The thing about the BBC is that they have this reputation, at least on my side of the pond, for being stodgy, straightforward, mostly reliable, and largely disinclined to the tabloid excesses of certain other UK outlets.

This reputation is only slightly deserved. They have a remarkable sense of humo(u)r sometimes, but often, you have to look for it. For example, their native video player? Some years ago, I noticed that the volume control slider goes to 11.

That cannot be an accident.

So with that out of the way, it's time to dive in...

The butt (snigger) of countless jokes, Uranus is almost certainly the most unloved planet in our solar system. It always seems to get overlooked when the mission invitations go out.

Okay, good, acknowledge the issue in the lede. People are going to make the jokes anyway; you might as well get ahead of them.

(To head off more of the tiresomely inevitable, yes, I've seen the Urectum clip from Futurama.)

Spacecraft have been sent to Mercury, Mars, Venus, Saturn and Jupiter. There is even one on its way to non-planet Pluto.

Dwarf planet. Also, this article is from 2014, before New Horizons zipped past Pluto. It's not like Uranus has changed much in 9 years, apart from progressing further in its orbit.

But Uranus (pronounced “yur-an-us” in polite astronomical circles) does not deserve its dull, or comic, reputation.

Pronouncing it Urine-us doesn't help matters. One could go back to its Greek origins and pronounce it "oor-an-ohs," but that's just weird. I usually call it "the seventh planet" to avoid the inevitable bad jokes, but that doesn't work either, and besides, I end up watching people mumbling "Mercury, Venus, Earth..." while counting on their fingers.

“Uranus really stands out,” says University of Oxford planetary scientist, Leigh Fletcher.

I'm just going to pause here while you get this out of your system.

With a volume 60 times that of Earth, Uranus is a compressed mass of toxic gases, including methane...

NOT HELPING.

“We don’t have a solid surface on any of these giant planets,” says Fletcher. “There’re no sharp boundaries, nothing to stand or sail on, but there’s a continuous progression from gas to liquid to some sort of solid.”

Okay, jokes aside, this is hard to wrap one's head around. We're used to pretty low pressures, and clear dividing lines between the three classical states of matter. Sure, you can always think of exceptions, like silly putty, but for the most part, gas, liquid, and solid (dammit, more pun fodder there) are well-defined. Even the extreme pressure at the bottom of the ocean isn't enough to push water into some sort of boundary state.

In a giant (Uranus is classed as an ice giant), the pressures get way higher than that. It's difficult to visualize, but it gets weird.

Circled by 26 small moons, a few faint rings and a weak magnetic field Uranus appears to be tipped over on its side. Every planet has a slight tilt when it spins – it gives us our seasons – but unlike every other planet in the solar system, Uranus rotates on an axis pointing almost directly at the Sun. Something that Fletcher describes as “really weird”.

This is a bit misleading, but only a bit. It's close enough in essence. For one thing, the axis doesn't point at the Sun (except for twice in an orbit); it would be more appropriate to say that the axis is roughly parallel to the plane of the solar system.

“Imagine a world where winter lasts 42 Earth years and you don’t see the Sun once during that time,” he says.

That's probably easier to imagine than the whole states-of-matter weirdness in the depths.

Fletcher is part of an international team that believes Uranus has been neglected for too long.

See? Subtle.

This group of space scientists and engineers from Europe, the United States and several other nations, including Japan, is working on a $600m mission proposal for the European Space Agency (ESA) with the aim of sending out a space probe, within the next 10 years, to discover why Uranus is so odd.

Go ahead, call it the Uranus probe. You know you want to.

From what I can tell, this mission never materialized. Bummer.

However, there is a good reason why, in the entire history of space exploration, only one mission has visited Uranus: it is extremely difficult.

And as the article notes, that one mission was a flyby.

There's also some issues with launch windows. Any mission out that far requires gravity assist from other planets, and they have to be in a good configuration to do that. So it's doubtful that we'll send a robot out there in my lifetime. That's okay. Once I'm dead, I won't be able to get mad at all the stupid ass-puns.

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