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.

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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.

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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.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
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"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.

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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.

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Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
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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

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Aiva Raine

Nov 5, 2024 at 2:55pm
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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.

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August 5, 2024 at 10:28am

#1074799

Mach It So

There are complex questions with simple answers. I can't think of any right now, but I'm sure they exist. At the very least, the answer is "a large quantity of plastic explosive." But today's article, from Thrillist, is a simple question with a complex answer.

How Fast Can Airplanes Go, Exactly?
An aviation expert breaks down how fast the answer beyond "really, really fast."

Science and human ingenuity have given an innumerable number of gifts to society: indoor plumbing, medicine, electricity.

And most importantly, beer.

Sure, it's also given us drone warfare, AI, and those weird Amazon Go stores—but let's focus on the positive.

Those are positive. They generate positive cash flow for a small number of individuals, which is positive for them.

One of the most exciting advancements of the modern era is the commercial airplane, which has made traveling around our massive planet in less that 180 days a true possibility.

And you'll experience maximum discomfort while doing so.

Also, editing is dead. Not only did they use "that" instead of "than," but the famous Verne novel was 80 days, not 180. Now, ISS astronauts circle the globe (for it is, indeed, a globe) twice in a bit over 180 minutes.

Now that I've called that out, I'll probably commit a massive blunder in this entry, if I haven't already.

The first flyers traveled at max speeds of 31 miles per hour, back in 1903. Now, 104 years later, planes have gotten a little faster. But just how fast? It turns out the answer isn’t so simple—it's a lot more complicated than getting a basic miles per hour answer.

Even more complicated are the various units used to describe speed. Miles per hour, sure, but there's also km/s, knots, millimeters per day, etc.

Thrillist spoke with Luciano Stanzione, who leads AviationExperts.org, an aviation consulting agency, to better understand how fast the modern commercial airplane is capable of flying.

I'm of the considered opinion that everything on the internet now is either paywalled, or an ad. I didn't hit a paywall here, so I wonder what the ad is for. Certainly not an "aviation consulting agency," right? Possibly Thrillist itself, based on the blurb at the bottom of the page.

“The airspeed is measured in KIAS (indicated air speed in knots) and in Mach speed (reference value to the speed of sound, used in higher altitudes)," Stanzione tells Thrillist.

Oh, yeah, the Mach number, too. I imagine it'd be fractional for subsonic flights, but I don't recall any instance where it's like "this airplane has a top speed of Mach 0.8." At least, not until I read this article. But all these different units make for opaque comparisons, like if your car's efficiency is in miles per gallon and you're looking to compare it with one whose efficiency is listed in kilometers per liter. Or whatever.

Speaking of complicated, the speed of sound changes with air density, which as I understand it is generally correlated with altitude.

The speed that airplanes can reach in the air can’t be as easily determined by the same metrics in which we measure the speed of objects on the ground. So bear with Stanzione for this scientific breakdown for a second.

There follows said "scientific breakdown," which I think is easy enough to follow, but I won't quote it here. Essentially, there's airspeed and groundspeed (no mention of whether the plane's carrying coconuts, though).

How fast will airliners be able to go in the future?

"That is impossible to determine but manufacturers are working again on high-speed jets, above sonic speeds," Stanzione says. "For a few decades now, improvements on speed have not been major. Let's see if new supersonic developments change that."

All the science fiction I've read indicates that it won't matter, because we'll have suborbital ballistic flights, or teleportation. Given the constraints of physics and capitalism, I don't see either of those things happening anytime soon, if at all.

Whatever the actual future of air travel is, though, at whatever speeds, you can bet they'll continue to find ways to make the process more degrading, uncomfortable, humiliating, and dehumanizing.

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