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🎶

about 22 hours 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.

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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.

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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

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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

July 4, 2024 at 9:38am

#1073524

He Said, Archly

Another random prompt from "Journalistic Intentions"

[18+] today.

St. Louis 1904

There are, in the world, very few cities with icons more iconic than the Gateway Arch. There's the Eiffel Tower, of course, arguably the most recognizable structure in the world. Across the Channel, there's the unmistakable steampunk intricacy of Elizabeth Tower, known worldwide but usually mistakenly labeled as Big Ben (which may be a case of metonomy, as Big Ben is the largest bell within the tower, but is more likely just mistakery). You have the sweeping, nested shells of the Sydney Opera House, the Washington Monument, maybe the Space Needle, CN Tower, and the Empire State Building. I'd be remiss if I didn't mention the Pyramids, but they're not technically a city symbol, located as they are outside the boundaries of Cairo.

Almost all of these establishment-shot-bait constructions have one thing in common: the "tower" bit (the Opera House being an exception). This inevitably leads to jokes about phalluses (Phalla? Phalli? Hell if I remember what Latin declension it belongs to), especially in the painfully obvious case of the Washington Monument (Father of his Country and all that).

Not so with the Gateway Arch. It's not a tower; it's... well... an arch. Two foundations, not one. Curvy, not straight (unless you view it from one side or the other, but who does that?) I'm not saying it's the equivalent expression of feminine potency, mind you; just that it's not an erection sticking straight up into the sky.

Thus, it might surprise younger folks and those not familiar with the US in general, the State of Misery, and St. Louis in particular, that the Arch is barely 60 years old.

Thus, when one considers the condition of St. Louis in 1904, one must utterly wipe any image of the Gateway Arch from one's internal visualization of that important Midwestern city.

That year, St. Louis, just over 100 years after the Louisiana Purchase, hosted a World's Fair, formally known as the Louisiana Purchase Exposition.

For other context, consider these historical milestones:

The Wright Brothers' first flight wasn't even a year in the past, having taken place in the middle of December 1903—but the Exposition hosted the first-ever airship competition (see above Wiki link). This led to St. Louis being just as associated with flight as Kitty Hawk, a legacy later immortalized in the name of Lindbergh's famous transatlantic airplane, the Spirit of St. Louis. (Okay, the plane's name had nothing to do

with StL's history regarding flight, and these days, you're taking your life, or at least your sanity, in your hands by flying Spirit across the Atlantic.)

Ford had yet to democratize the automobile, and the hot topic in cities around this time was streetcars. I have no desire (pun absolutely intended) to get into the arguments concerning these modes in this entry, and anyway, San Francisco became more associated with trolleys. This also led to more than a century of pain for philosophy students as they contemplated various permutations of the Trolley Problem.

And then, of course, there were the Olympics.

Apparently, the 1904 Summer Olympics

were meant to be held a bit further north, in Chicago, the first such event to be hosted outside Europe. In typical American fashion, the Expo organizers apparently stomped their feet and shouted, "NO!" So the games were moved to Misery, which, in those days, was kinda hard to get to (remember, airplanes were less than a year old; overland routes sucked, and navigating up the Mississippi takes time).

But, regardless, it turned world attention to the site that used to be a major Native American gathering place (see my April entry, "Things Go Better With Cahokia"

, where, apparently, a lot of people came from all over to participate in social and sporting events.

Some things, it seems, are universal.

At least the Gateway Arch is unique.

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