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.

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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

April 13, 2023 at 9:44am

#1048137

Urine Trouble

I have a system (not a perfect one) in place to avoid duplicating links here. I also try not to repeat entry titles, but with 2300 entries, I know I've got some duplicates and maybe even triplicates.

What's worse, though, is when I thought I did an entry about something and now I can't find it. Such is the case with famed proto-astronomer Tycho Brahe. I mentioned him briefly in an entry a while back in response to someone's comment, but that seems to be the extent of things.

Until now.

The Father of Astronomy Was Killed by Piss

Unlike most Cracked articles, this one's not a numbered list.

To non-astronomers, there aren’t a huge amount of A-list astronomers. Most people could probably name Copernicus, Galileo and Hubble, but realistically, they learned those names from the dog in 1955 in Back to the Future, the lyrics to Bohemian Rhapsody and a big-ass telescope that occasionally features in “yo mamma’s so fat…” jokes.

Hubble's successor, the JWST, is not named after an astronomer but an administrator. Which I think is a step down, and that's not even getting into the controversy around Webb's policies.

Amazing telescope, though.

One guy whose name might ring only a faint bell is Tycho Brahe, a 16th-century Danish astronomer who made some remarkable innovations that helped to bring about the Scientific Revolution.

Speaking of telescopes, he did all his work without one because they hadn't been invented yet.

For exactly how influential this guy was, there's the article, and of course also the Wiki page

about him (which may or may not be less error-prone). I'm skipping it because the clickbait headline is about his personal life (and death), which was almost as interesting. I will note, however, that his contributions were significant enough for later astronomers (ones with telescopes) to name one of the largest and most recognizable lunar craters after him.

Speaking of cutting edges, Brahe had his nose sliced off at the age of 20. He got in a duel with his third cousin, who accidentally sliced a big chunk of his honker off...

And this is what he's probably most well-known for. No, I have no idea how he smelled.

The big-brained, metal-nosed polymath undone by humble urine.

No, he didn't discover Uranus. That would have to wait for telescopes.

The story goes that Brahe was at a banquet and needed a pee, but etiquette forbade him from leaving, so he held it in. Unfortunately, when he got home and could safely drain the lizard, nothing was forthcoming.

Sounds to me like he wasn't done in by piss, but by stringently following etiquette. The cautionary tale here is: fuck etiquette.

And then, in the words of Victor Thoren and John Christianson’s 1990 book The Lord of Uraniborg: A Biography of Tycho Brahe, “through five days and nights of sleepless agony, he pondered the agony of paying so great a price for, as he thought, having committed such a trivial offense.”

Uraniborg is unrelated to urine. It was, if you read the text I'm leaving out here, one of his pre-telescope observatories. Nor is it related to Uranus (other than maybe etymologically), or the Borg ("Borg," by my understanding, is or was a Danish word for "castle.")

He had several more days of pain, fever and delirium before dying. He was 54, and had spent 38 years documenting the stars.

Which is approximately 39 years longer than I can ever concentrate on one subject.

Just in case readers might be worried that this could happen to them, the article reassures us:

“The good news is that, while uncomfortable, occasionally holding back a pee shouldn’t harm us,” explains Ajay Deshpande, senior clinical lead at London Medical Laboratory.

Also, medical science has advanced somewhat in more than half a millennium. Hell, they can even reconstruct noses now. To your specifications. Which makes a plastic surgeon's office the only place where it's socially acceptable to pick your nose.

In other words, it isn’t out of the question that the father of astronomy sat on his keys, filled up with pee and died. All that time looking up, not enough looking down.

Perhaps this inspired Oscar Wilde's famous quotation: "We are all in the gutter, but some of us are looking at the stars."

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