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').

1 Like

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

1 Like

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

December 5, 2020 at 12:02am

#999667

XXI

Today is, of course, the most important commemoration of the year.

"30-Day Blogging Challenge ON HIATUS" [13+]:
5. Bonfire Night!
Tell us about an evening around a bonfire.

or

Prohibition Ends in the US, 1933
Write whatever you like about this.

"JAFBG" [XGC]
Some people think it's wrong to lie to children about the existence of holiday icons like Santa and the Easter Bunny. How do you think parents should handle this?

I visited a friend's house once, a woman with a bunch of stepchildren, all girls, who at the time were all preteens. I don't remember their exact ages but none of them rolled their eyes so they couldn't have been teens yet. So at this time, sometime in December several years ago, I was hanging out with her and her husband in the kitchen and the kids were right nearby, definitely within earshot, doing kid stuff. Obviously I don't remember too many details, but what I do remember is that the subject of Santa Claus came up. And me, not thinking, said something to the effect of "Of course, we all know Santa's not real."

Oops.

I haven't been invited back.

I'm not going to try to tell people how to raise their kids. They won't listen to me anyway, since I'm not in their sacred club of kid-raisers. Not that they listen to other kid-raisers, either.

But since you asked, I'm inclined to go with: Hell yeah. Lie to the little bastards. Lie ALL THE TIME (except, once in a while, throw in some truth just to keep them guessing). This way they will learn a valuable life lesson: no one is to be trusted. No one, not even your parents. Especially not your parents.

Maybe then you'll raise a generation of sprogs who end up growing up distrusting the bullshit they see on the internet, as opposed to our current generations, who seem to believe whatever suits them, all evidence to the contrary.

Or maybe not. I don't know. This is why I celebrate XXIst Amendment Day. The XXIst Amendment to the US Constitution is the one that repealed the XVIIIth Amendment, which was Prohibition. Clearly, in the XIII years or so between those two amendments, two others were ratified, the XIXth and the XXth. But I don't remember what those were about, nor can I be arsed to look it up because they cannot possibly be as important as the XXIst.

Incidentally, Roman numerals are dumb.

Point being, since I all the time gotta deal with other peoples' kids, whether they're still technically kids or have theoretically grown up, drinking helps. Prohibition itself was based a big lie, like Santa Claus or the Easter Bunny -- that banning booze would alleviate many social problems. These beliefs were well-meaning, but without evidence, and they didn't really think their cunning plan all the way through.

They got a big wake-up call from the Truth Fairy.

My personal theory is that Prohibition was the proximate cause of the Great Depression, and Repeal was the proximate cause of getting out of the Great Depression. That's my story and I'm sticking to it. But it's (probably) a lie.

I'd better wrap this up before I wax too philosophical about the nature of ideas and reality. Believing something doesn't make it so, but fictions have power, whether it be to convince a nation that they shouldn't be drinking, or convincing kids that a bearded white guy will give you nice things if you don't misbehave. Some people need to believe in something, I suppose. Me, I believe I'm going to have a commemorative drink.

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