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 11 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 14 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

a day ago
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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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October 28, 2023 at 10:00am

#1058195

Hunter's Moon

Full moon tonight. This article being chosen randomly from my rather lengthy queue is pure cosmic coincidence.

5 Ways the Moon Is Different From How You Picture It

Most sources on the internet insist that full moon names are linked to Gregorian calendar months. They are wrong. Giving names to full moons was a tradition of several different and disparate societies before that calendar was forced down their throats at gunpoint. Yes, this is related to my sporadic harping against the false definition of "Blue Moon."

Today's full moon is the Hunter's Moon, the second full moon after the northern hemisphere fall equinox. The corrupted sources say it's the full moon in the calendar month of October, but it doesn't always fall in October. Some cultures have different names for this instance of the full moon, but we're going with Hunter's Moon as it's certainly the best-known. So, shout-out here to our very own 🌕 HuntersMoon

!

The naming of full moons is a folkloric tradition (which is decidedly biased toward the northern hemisphere seasons), so has little to do with any scientific facts about the moon. But it's my self-imposed Quixotic duty to rail against attempts to shoehorn those names into a calendar that was never meant to track celestial events (though it's certainly useful for calculating solar returns).

Hence, to the article, which, like all good fact articles, is from Cracked:

If you haven’t been to the Moon, everything you think you know about it might be wrong. “The Moon could be made of green cheese,” people used to say, and we all know how wrong those people turned out to be.

I have another article in the queue that addresses the "green cheese" cliché. Maybe it'll turn up on the next full moon (which will be the Beaver Moon). But probably not.

Well, okay, no one ever really thought the Moon was made of green cheese. That was a figure of speech about people believing the absurd.

I wouldn't say "no one." Never underestimate the power of human ignorance.

However, people really were uncertain about the nature of the Moon until we reached there.

Well, yeah, that was one of the reasons we went there. The other being a dick-measuring contest against the USSR. Whatever; it got results.

I could smugly assert that I knew all these things already. And I'd be right. Except it's almost never right to be smug.

5. It’s Hot

Mind you, not all of the Moon is hot. Some of it’s very cold. But many people imagine the whole thing’s cold, because space is cold, and that isn’t true at all.

Neither is it true that space is cold. "Cold" and "hot" are imprecise terms for, respectively, things at a temperature below about 75F and things at a temperature above 75F. But in order to have a temperature, something has to be a... thing. A rock, a volume of water, even a gas above a certain density (our troposphere, e.g.). As a vacuum isn't a "thing" in this sense, it cannot have a temperature. The moon is a thing, so it does have a temperature, one which varies quite a lot depending on where you are on or in it.

The article goes into details, but also can't resist making a Uranus joke, so I won't give it the dignity of quoting from this section further.

4. It’s Dull

Hey, now, it may not be as smart as the Earth, but that's a low bar to clear.

Though the Moon is white, it doesn’t reflect light very well. That’s because it’s so rough. It’s not a polished cue ball, it’s rock and dust, which means it doesn’t have much specular reflectivity.

This, too, is misleading. While there's some color on the moon, it's mostly various shades of gray, ranging from nearly black to not quite as black. If it were white, even with a surface just as rough, it'd appear a hell of a lot brighter. It's kinda-sorta related to this optical illusion.

Again, folks, don't get your science facts from Uranus joke sites.

Still, apart from that, this section has some cool photos.

3. It’s Smooth

Well, yeah, in stories and poetry, it is often linked to romance.

A few seconds ago, we argued that the Moon is rough, not smooth. On the other hand, it’s also a lot more smooth than people imagine.

So's Earth. Sure, up close, we have mountains, valleys, and a nice beard of forest, but I've read (though never bothered to confirm) that those variations are proportionately less than those of a cue ball.

2. It Rings

Worth reading this section, but I'll just say this shouldn't be surprising. Sound propagates through solids.

1. It’s Far

Only by Earth standards. By cosmic standards, the distance to the moon is undetectable.

But, again, the article goes into way more detail, because it's hard for us humans to contemplate cosmic distances. And this time, I have no real quibbles.

This section's references to Starfield make me want to play that game. But I'm already hopelessly immersed in Baldur's Gate 3, so maybe later.

I'll just close by suggesting that, if it's clear tonight, venture (shudder) outdoors and take a look at the Hunter's Moon. Whatever the facts... it's just plain cool.

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