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 21, 2020 at 12:11am

#998795

Aversion Therapy

This one could easily send me into dark places.

PROMPT November 21st

Write about a “cruicible moment” in your life. A challenging time that shaped you and altered your view on your life and/or the world.

So I racked my brains trying to think of a time that changed me without being too depressing. I mean, there was the 20 years I spent dealing with parents with dementia, one after the other, to the point where I was mourning them before they technically died... but 20 years is hardly a "moment." And no one would read it.

Instead, I'll go back to the time before their decline.

As I've noted before, I spent my childhood on a farm. It wasn't a large farm; we had a forest and a field and a garden and a changing cast of farm animals. It was work, sometimes, but I was never far from the shade of the house and a large body of water for cooling off in.

And so, when I went looking for a summer job in 1984, after my first year at university, I figured my childhood had prepared me sufficiently for the outdoor work of a surveyor's assistant, or "rodman." After all, hadn't I spent most summers toiling in the hot sun, and autumns trudging through the forest cutting and hauling firewood?

Summer in the Tidewater region of Virginia gets oppressive, even back then before the climate went to hell. You've head the expression, "It's not the heat; it's the humidity?" I'm 99% sure that was first uttered by a Jamestown settler. Oh, sure, that colony was famous for nearly getting wiped out in a brutal winter, but I guarantee you the summer wasn't any picnic, either.

So my boss, the surveyor, decided that the best way to handle the heat was to start work before the sun came up. I must reiterate that we're talking about summer here, the weeks surrounding the solstice, so morning arrived early. I didn't have an alarm clock, but I'd set a timer on my stereo so that rock music and the annoying voices of DJs would awaken me at like 3 am. Now, two things about that: One, like I said, I'd already spent a year at college, where 3am was almost bedtime. And two, my dad had been in the military, so he prided himself, like many men of his generation, on waking up at some ungodly cow-milking hour (for the record, we didn't have dairy cows).

Dad tried really, really hard not to be annoyed when, at 3 in the morning, Black Sabbath blasted suddenly from the enormous speakers in the room next to his.

Well? I wasn't about to spend hard-earned money on an alarm clock, and he could have bought me one at any time. It's his own damn fault.

Anyway, so then I'd grab a bowl of cereal or some such, then drive the ten miles to the surveyor's house, getting there usually just before the accursed daystar actually rose. We'd haul our stuff to whatever job site he was working on, and he usually tried to arrange things so we did open fields in the morning and then do our work in the woods at the height of the day. "Usually" and "tried" are the key words there; more often than not, I'd end up standing in the middle of a hot field, baking in the oven of the foul solar orb, waiting for him to take measurements -- and then haul the equipment on my back to the next spot.

That summer, I got serious muscles and a major tan.

I also got poison ivy, hornet stings, a sprained ankle, and back problems that will plague me until the day I die.

My boss didn't believe in eight-hour days. Now, don't get me wrong; I liked the guy, and he paid reasonably well for unskilled labor. And I wanted the job, not just for the money, but because I was studying civil engineering and this would turn out to be important for my career. A lot of civil engineers are also surveyors; I never got formally trained in it, but I picked up enough to be able to speak the jargon with the surveyors we worked with. Also, this guy was sweating right along with me, not just sitting back and watching me work. Anyway, as I was saying, he didn't believe in eight-hour days. We'd work until about noon, take a half-hour for lunch (usually, fortunately, in a nearby air-conditioned diner), and then slog through the afternoon. He'd generally use all available daylight, which -- again, summer -- got me back home around 9 pm.

Do the math. I'd eat a morose few bites of dinner, fall into bed, and wake up with Led Zeppelin blasting from my stereo in the darkness.

No time for anything else.

Oh, and on top of that, my boss didn't believe in weekends, either. We had Sundays off because his wife was religious (he wasn't, but you know how these things go), but apart from that and the rare occasion when it rained, we worked every day.

Now, I'm not going to say this didn't build character, or that I didn't, in a perverse way, enjoy it -- my dad was convinced that I'd end up being a lazy bum, and was endlessly surprised that I kept to this schedule. Mostly I did it just to show him I could. Sons are like that sometimes.

But what it ended up doing was turning me into a lazy bum.

See, once I got back to college, I returned to my college hours: study (or party) all night, arrange my class schedule to have the first class as late as possible in the day.

This was a far superior schedule for me. Also, I knew I had something to work toward: a lifetime of sitting on my ass in an office, with only the occasional foray into the not-so-great outdoors to look at job sites.

So yeah, that summer's experience changed me: it made me highly allergic to the outdoors, to waking up early, and to hard work. In my actual career, the day usually started at 8 am and proceeded, with a lunch break, until 5. Normal, ordinary hours, even if it didn't quite fit my natural nocturnal schedule (though the number of overtime hours I put in was frightening -- however, it was still office work).

I never did see that surveyor again after that summer. I hope he made enough money and got a chance to retire.

But I don't think he'd have enjoyed that very much.

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