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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10 Most Recent Blog Comments:

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

November 9, 2021 at 12:02am

#1021211

Barber-ians at the Gate

It's hard for me not to think of a certain scene from Monty Python and the Holy Grail with this one.

PROMPT November 9^th

You are having a very realistic dream. In this dream, you are approaching a castle. What unusual reason do you have to be at this castle? What do you say to the two guards standing outside it to gain entrance?

And yes, I was referring to Castle Anthrax.

You are having a very realistic dream.

I have two kinds of realistic dreams: ones where I know it's a dream, and ones where I don't. I know people have wondered whether their actual waking life is, in fact, a dream, but in my experience, if I end up asking myself, then either it's reality, or I'm about to wake into reality. While I'm in the dream, I either know it or I don't.

In other words, no, I'm not a butterfly

dreaming I'm a man. That's just silly. Zhuangzi must have scored some of the really good opium.

Still, dreams are weird.

In this dream, you are approaching a castle.

I've seen mock castles here in the US, and seen (and toured) real ones in the UK. Obviously, the latter are more impressive. There was one somewhere in the North, but I don't recall the name, that was rumored to be haunted. I had a feeling I knew exactly who started these rumors, because inside this castle, there was a sign up: "See for yourself if the castle is haunted... if you dare! Spend the night, 100 quid." I'm paraphrasing, but that was the gist of it. Hey, British peers have to make money somehow, and telling everyone your castle is haunted is one of the better ideas I've seen.

Beats working for a living, I suppose. Haunted or not, 100 quid (at the time about $140) really isn't that steep for the experience of staying in an actual castle. (Of course it's not actually haunted; nothing is. But people can convince themselves of just about anything in the stony darkness.) I didn't do it, though; I was staying with friends.

And yet, my main experience with castles has been in the fantasy realm. D&D, mostly. I can't count the number of times a character of mine approached a castle. Usually using stealth, under cover of darkness.

That said, I can't recall ever actually dreaming about a castle. I suppose it would have some subconscious significance. Power, wealth, protection, something of that sort.

What unusual reason do you have to be at this castle?

The way my dreams usually work, it'd be something completely unrelated to what we think of as the normal uses of a castle. Let's go with "getting a haircut." If you've seen me recently, you'll recognize that this would be the kind of dream known as a "nightmare."

So in this hypothetical dream, there's a succubus bound to the castle who makes extra money by providing barber services. Hopefully, she can trim my goatee, too.

That is not a euphemism; this isn't that kind of dream.

What do you say to the two guards standing outside it to gain entrance?

So I'm approaching this dream castle to get a haircut and beard trim from the millennia-old succubus who resides therein. Again, since this is one of my dreams, I'd be going up to the guards and saying something completely unrelated to castles, haircuts, or succubi. Something like... I don't know... "The falcon cries at dawn."

Fortunately, that just happens to be the day's passphrase.

Unfortunately, the succubus ends up stealing all my vital energy. Which, in retrospect, I really should have seen coming. After all, who the hell gets a haircut from a succubus in a well-guarded British castle? I wake up just as the demon shoves my withered body out one of the castle's murder holes.

Dreams are weird. But then, so is my imagination.

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