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:

Write_Mikey_Write!

about 8 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 10 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

July 27, 2020 at 12:02am

#989226

The Not-Shining

Because I am a pessimist, I'm never unpleasantly surprised.

PROMPT July 27th

Write about a time you were caught off guard, surprised, or had the rug pulled out from under you. How did you recover?

At least, not that I can remember. As with embarrassing moments, my mind utterly blanks them out, hoarding them until I'm trying to sleep, at which point it dredges them up so that I can stay awake. I can only assume it's my stupid brain trying to avoid another episode of sleep paralysis by trying to avoid the "sleep" part in order to skip the "paralysis" part, and also the "dark figure menacing me" part.

But then I forget them all the next day. I'm left with the memory of a memory; the vague notion that something was keeping me awake, some self-disappointment from the past. But I can never remember what.

So I'd write about the time I literally had a rug pulled out from under me... except that never happened, so I'm left with the figurative meaning of that phrase, and I still can't remember anything of the sort. Well, my divorce, I suppose, but it's not like I didn't see that coming. Pessimist, remember?

I suppose there's the tortured tale of trails and travel, something I manage to remember probably because I wrote about it at the time. Naturally, I can't remember where. My offsite travel blog, maybe? Can't be arsed to find it right now. Link's to the left, there, if you want to look. Since so much time has passed, it's likely some of the details will be different. Memory does that.

About, oh, eight or nine years ago, I suppose, I decided to drive across the country, something that, at the time, I hadn't done before. So I got it in my head that if I'm going to drive across the country, I'm going to drive all the way across the country, from the easternmost point to the westernmost point in the continental US.

The easternmost point is easy: I parked near Quoddy Head Lighthouse, near Lubec, Maine, and scrambled across a few rocks to touch the Atlantic so I could honestly say I was as far east as I could go without getting too wet (apart from my fingers). Then I took the next several days to actually drive across the country, avoiding Canada and the interstates (nothing wrong with either, but I wanted to see the US, not Canada, and take my time doing so).

The westernmost point is a desolate spot of land in northwest Washington, near... well, it's not really near anything. There's a dot on the map called Ozette, which turns out to be a ranger station and a convenience store, with the convenience store being closed because it was December. From Ozette there's a trail through the rainforest that winds about three miles to the Pacific shore. Three miles is easy, especially since the terrain is relatively flat and like I said: trail. On level ground, such a walk normally takes about an hour; since the trail was a bit rough, it was, oh, maybe an hour and a half from the ranger station to my goal.

I got there, looked around a bit, saw absolutely nobody else, dipped the toe of my shoe (idiot me forgot to pack hiking boots) into the Pacific, and took one final look at the featureless western horizon... whereupon I noticed that the angle formed by the accursed daystar, my stupid self, and said horizon was really quite remarkably acute.

Like I said, it was December, close to the earliest sunsets of the year, and a far more northerly latitude that what I'm used to -- and while the trail wasn't too difficult, when it got dark, I'd be boned.

And it was about to get dark.

Oh, and did I mention that the PNW is crawling with brown bears? Well, the PNW is crawling with brown bears.

So I started booking ass back up the trail. Under the canopy, it got dark fast. Really, really fast. I estimated I was about 2 miles into the 3 mile return hike when it started getting difficult to see tree roots and rocks and such, though the packed dirt of the trail itself shone like a beacon between the lush, bear-concealing vegetation on either side.

I don't run. Well, I did, that evening, but normally I don't run. Tough on the knees and back. But I was motivated. Oh, and in addition to dark, it was starting to get cold, which I didn't much notice yet because I was running.

By the time I broke into the clearing at Ozette, the stars shone in the sky, but I didn't have time to appreciate them because that's when the cold hit me.

I ducked into the car, turned the heat up as far as it would go, and flipped the seat heater to High (yes, I have seat heaters, shut up).

None of it helped. I shivered all the way through to the nearest town (by "nearest" I mean "several hours away") with a motel.

The nightmare didn't end there, either. The motel was in Forks, and this was back when the Twilight movies were in full bloom. I couldn't spit without hitting a cardboard cutout of some hormonally-enhanced Hollywood vampire, werewolf, or insipid brat.

So that counts as being caught off guard, I suppose. Curse the blasted daystar and its utterly unpredictable rising and setting. But still. I'd do it again. Just in the summer. And with enough time to retreat afterward to somewhere -- anywhere -- that isn't Forks.

Preferably, a place with beer.

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