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 11, 2023 at 10:30am

#1053952

Earth

Getting back to my list of solar system articles, this one's about a planet most of you know.

When will the Earth try to kill us again?
Most mass extinctions began with vast convulsions of Earth’s interior—can we detect that?

Why would you want to detect that? Are you going to try to warn the rest of us, who won't listen, and end up sending your kid away in a rocket to a distant planet where they'll have superpowers?

But seriously, though, unless we can do something about an impending disaster (or would that be displanet), just don't tell us. We're still doomed, only without the riots and panic.

Our planet Earth has extinguished large portions of its inhabitants several times since the dawn of animals.

Think of it like shaking an Etch-a-Sketch.

And if science tells us anything, it will surely try to kill us all again.

Except now, it's got competition. Well, it always did; the asteroid impact wasn't the planet's fault, except maybe for having the audacity to orbit into the path of the asteroid. But now we're the competition, too. Though, philosophically, we're part of the planet, so maybe it spawned us for the purpose of wiping out the latest Etch-a-Sketch.

But it seems like, in terms of mass extinction events, scary meteorites are in the minority.

In the words of researchers David Bond and Stephen Grasby, who reviewed the evidence in 2016: “Despite much searching, there remains only one confirmed example of a bolide impact coinciding with an extinction event.”

Real quick:

Asteroid: rock in space (though as I've said before, this should be the name for hemorrhoids)
Meteor: the same rock, in the atmosphere, leaving an ionization trail
Meteorite: the same rock, having hit the ground.
Bolide: the rock explodes before it hits the ground.

There's more to it than all that, but I find it unlikely that the Chicxulub rock was technically a bolide. But whatever.

Volcanism, on the other hand, has coincided with most, if not all, mass extinctions—it looks suspiciously like a serial killer, if you like.

I don't like. No. It's not like we can arrest Hephaestus.

This isn’t your regular Vesuvius/St. Helens/Hawaii style volcanism. It’s not even super-volcanoes like Yellowstone or Tambora. I’m talking about something far, far bigger: a rare, epic volcanic phenomenon called a Large Igneous Province or “LIP.”

LIP? What the fuck, geologists?

Anyway, the article goes on to describe these doomsday events, in excruciating detail, with color graphics even.

So if our serial killer is the volcanism associated with an LIP eruption, when will it strike again?

And can we beat it to the punch?

Their images reveal fat mantle plumes, regions of hot rock as wide as France, rising like chimneys through the mantle.

C'est très chaud.

LIPs may be on a cycle. On average, there’s one every 15 million years, with the last occurring 16 million years ago (the Columbia River LIP in northwestern USA). By that rough reckoning, we are overdue for another.

That's not how averages work, and a science author oughta know that.

Still, it could happen at any time. Sooner or later, as Susan Ivanova said on Babylon 5, boom. It's inevitable.

And if Olson is right, the field reversal will give us more than 30 million years’ advance warning of the next LIP (for perspective, human ancestors separated from chimp ancestors about 7 million years ago).

So, no need to worry about the sun turning into a red giant in 5 billion years or so. We're doomed long before that. Get your kid into a rocket now!

The climate effects of volcanic gases are deadlier still. Stratospheric sulfur from Laki cooled the planet by 1.3 degrees Celsius for three years, triggering one of the most severe winters on record in Europe, North America, Russia, and Japan.

That seems perfect right about now. Unfortunately, as the article notes, it's short-lived and followed by a much longer warming period.

At the end of the Permian, the Siberian Traps LIP erupted staggering quantities of lava for 300,000 years with relatively little environmental effect, just like the Paraná–Etendeka and Columbia River LIPs. Precise rock dating shows that Earth’s most severe mass extinction only began when sheets of magma, called “sills,” began to inject underground through sediments rich in fossil fuels, igniting them and baking off gases, as Seth Burgess describes:

Whew, good thing we've gotten rid of most of those fossil fuels, then.

Human mining and burning of fossil fuels mimics the most deadly LIPs. Even if LIP greenhouse gas emissions were larger and lasted far longer, our emission rates are far faster, so they are just as capable of overwhelming Earth’s neutralizing mechanisms. This is compounded by a cacophony of other man-made environmental disturbances (pollution, acid rain, deforestation, and so on).

Or not.

One of the talking points I hear from climate change deniers is that the climate has changed before. While technically true, in those cases, no one was around who could have caused it, or could have done anything about it.

We did, and we can.

But we won't.

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