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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

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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