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.

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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
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 10, 2023 at 10:29am

#1047949

Hem

Today's entry for "Journalistic Intentions"

[18+] is about something that I never notice until it fails.

Hem

This word has a few definitions, but since the overall theme of this month's JI is clothing, I'm going to assume it's not referring to a makeshift fence or one of the sounds of indecision (along with "haw," which also pairs well with "hee.")

A while back, some bored scientists decided to program a computer to trace the evolution of language (specifically English, because these were UK scientists) backwards and forwards, both predicting where it might go from here, and delving deep into times from which few written records survive.

This, of course, was reported in various outlets as "Scientists discover oldest words in English!!!" Which is sensationalist and misleading as all fuck. There's an press release on it here,

but it only gets more sensationalized the further you get from the source.

If you can't be arsed to click, their candidates for Oldest Word in English are I, we, who, one, two, and three.

As these words supposedly predate English, and long predate Modern English, it's a matter of definition to call them the oldest words in the language. And a while back, I did a blog entry on the word "lox," which they think hasn't changed in 8000 years. Here's the blog entry; unfortunately, the link is already broken: "Lox Pie"

. I did, however, find an updated link

to the original article, or at least some version thereof.

So, what's all this hemming and hawing about? Well, in the course of researching this entry, I discovered that the word "hem," as used in fashion, dates at least from the 12th century C.E.,

and possibly from even earlier. Which makes it about 1/10th as old as "lox," but also puts it firmly into a time when some form of English was being used.

This wasn't, presumably, figured out by a computer that may or may not have been programmed correctly, but from what sparse written records survive. I can't be sure about that, but I would believe that "hem" is quite old, because, in general, shorter words live longer, and the idea of hemming a cloth garment is not something that's changed over time (though obviously, the technology to do it has).

One hems a garment to adjust its size, but the primary practical use is to keep a cut edge of cloth from fraying. Like a splice on the end of a rope, only in two dimensions.

So when it comes to discovering the origins of words, there's no need to be a-frayed.

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