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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 22 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 24 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.

January 13, 2020 at 12:16am

#973273

Battle of Evermore

PROMPT January 13th

In your entry today, write about games! Board games, video games, card games, sports games, mind games, etc... Share a memorable story.

I'll skip the mind games, thanks very much. Had enough of those when I was married.

I gamble on occasion, but I've only played poker in a casino once, just to say I did it. I wasn't the first one out, and that's the best I can say about that. Mostly, I stick to blackjack; I understand the odds and strategies there. Roulette is too random, and dice games are, well, dicey. Slot machines are pretty much just vehicles to part people from money, but I sometimes play them anyway; they can be entertaining and, what the hell, sometimes I get some of the money back.

I don't like sports. Not watching, not participating. Last time I was at someone's house watching football, I was struck by just how little is actually going on, between all the penalty flags and shots of angry sideline coaches and idiotic commercial breaks. I know sports-watching is a big part of a lot of peoples' lives, though, and I'm not judging. You like what you like. The closest I get is sometimes I enjoy watching martial arts competitions.

In practice, the games I play most are single-player video games, preferably of the open-world variety. Examples are the Elder Scrolls titles and the Fallout series. I detest most multiplayer games, because there's a high percentage of assholes in the gaming "community."

But if I find a group of people who don't suck, I'll play the hell out of D&D (or other roleplaying games of its ilk). This kind of game emphasizes cooperation over competition; there are competitive board games that I play, but I have to avoid both smug winners and sore losers. As I've noted here before, I don't like winning or losing; I prefer the teamwork aspect. Also, at their core, RPGs are about storytelling, and, well... look where we are.

So, since I saw the prompt, I've been racking my gray matter trying to come up with a compelling gaming story. I know I have a lot of them - lots of memorable characters and groups over the years, and when I'm playing one of these games, often something will remind me of an epic battle or hilarious situation. Naturally, now, when I try to think of one, they disappear into the mist.

There was the time I was DMing and the party was tracking trolls. Said trolls had retreated to their lairs, which were in a series of caves in the side of a cliff. Trolls were better climbers than the party, and no one wanted to be the first one into a dark, smelly troll cave that was sure to be an ambush, so they hit on a solution: they tied the halfling to a rope and lowered him down from the top of the cliff, the idea being that if they saw a craggy troll hand swiping at the hapless thief, they knew there were trolls in that particular cave. But if they didn't, then that was probably the cave to enter so they could set up their own ambush. I mean, the halfling wasn't good for much else except picking the occasional lock and the party's pockets, so he was expendable.

Or there was the time when I was playing a halfling, myself (being sure to not be annoying to the other party members - well, not annoying enough that they'd use my character to fish for trolls). He ended up as werewolf chow, as I recall. Turns out that when you piss off a werewolf, you don't have to run faster than the werewolf; you just have to run faster than the slowest party member. Which is almost always the halfling.

And then there was the time I was playing a not-very-annoying human fighter-type with a big honkin' claymore. Our party managed to piss off not one, but two armies: the orcs caught us stealing from them, but we got away; and the human army we were supposed to be working for kicked us out because the party's bard managed to flub a Persuasion check (not easy to flub a persuasion check if you're a bard, but he managed somehow). So we spent the next two or three gaming sessions fighting off scouting parties from both sides until, having enough of that nonsense, we hit upon a plan:

First, we snuck into the orc camp at night and eliminated a few of them, being sure to "accidentally" drop a scroll of "orders" from the "human army" that read something like, "Your orders: go kill some of the orcs and make it look like the renegades [my adventuring party] did it. Signed, General Brod of [the human army]." The DM made the appropriate dice rolls, and the deception worked.

Next thing we know, we're watching from the forest as a hundred+ angry orcs descend upon the camp of our former employers. The orcs won, but only after taking heavy losses - heavy enough so that we were able to clean up the few remaining - and gave us the spoils of not one, but two, military camps.

Then, of course, the DM decided that was too easy and sicced a dragon on us, but that's another story.

The apples turn to brown and black, the tyrant's face is red.
Oh the war is common cry, pick up you swords and fly.
The sky is filled with good and bad
That mortals never know.
Oh, well, the night is long, the beads of time pass slow,
Tired eyes on the sunrise, waiting for the eastern glow.
The pain of war cannot exceed
The woe of aftermath,
The drums will shake the castle wall,
The ring wraiths ride in black, ride on.

5 Like • Comment • View 5 comments

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