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

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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

June 28, 2024 at 9:33am

#1073277

It Ain't Wrong

Hey, look, an article about writing. Sort of. From thesaurus.com:

Why Is “Ain’t” Such A Controversial Word?

I still remember a rhyme from elementary school: Don't say ain't, or your mother will faint, and your father will fall in a bucket of paint.

The image was so compelling that I spent the next few days saying "ain't" every chance I got; sadly, the promised amusing scenario never presented itself.

Look, there are myriad reasons I never wanted kids, but high on the list was that they might be like me.

What’s all the fuss over ain’t about?

I always figured it was kind of like "y'all," which I did a whole entry on a while back: basically, an attempt to squelch Black vernacular.

Ain’t is a contraction that can mean am not, are not, and is not. It can also mean have not, has not, do not, does not, or did not. We ain’t joking: ain’t is incredibly versatile, a kind of one-stop-shop for saying something isn’t, didn’t, or wasn’t.

That ain't confusing at all.

Ain’t apparently begins as amn’t, a contraction for am not, which you can still hear in Ireland and Scotland today.

I also vaguely remember wondering why we didn't have "amn't" when we had contractions like aren't and weren't.

Ain’t is a perfectly valid word, but today, ain’t is considered nonstandard. At worst, it gets stigmatized for being “ignorant” or “low-class.” At best, it’s considered a no-no in formal writing.

Lots of things are frowned upon in formal writing that we can get away with in, say, texts or blog entries. But "ignorant" is how I'd describe elitist snobs who are out of touch with common ways of communication.

While a lot of people consider ain’t improper, it’s a very regular and legitimate part of many forms of English, including in Black English (AAVE). Be mindful that judging someone’s use of ain’t as “wrong” can be a very socially loaded act, to say the least.

So I was at least partially right. It's also more acceptable in the South here, though it's not as quintessentially Southern as "y'all."

Outside of regional and dialectical forms of English, ain’t also appears in a great many expressions (not to mention in important titles, from “Ain’t I a Woman?” to “Ain’t That a Shame”). In fact, it’s hard to imagine these sayings without ain’t. Try replacing them with their formal counterpoint and see if they still keep their charm.

At least it's more acceptable now than when I was a kid. I wouldn't expect to see it in business correspondence or scientific research papers (apart from those that study language, that is), but for everyday speech, there ain't nothin' wrong with it.

Maybe one of these days I'll do an entry on how a double negative is an intensifier, rather than mathematically turning a sentence a positive. But this ain't the time.

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