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

June 10, 2023 at 9:08am

#1050891

sWords and Sorcery

Carpenters use saws, cooks use knives, surgeons use scalpels. Writers use words, which you would think wouldn't be in the same class of weapon, but at least according to Cracked, you'd be wrong.

5 Times Words Actually Killed

People like to pretend that the pen is mightier than the sword, but that’s mostly something nerds say right before they get packed into a school trashcan like a musket ball.

And we remember that shit when we grow up to make hiring decisions.

That doesn’t mean that words don’t have real power though — whether it’s just emotional damage or something that snowballs into a genuinely dangerous situation.

We know words have power, though not usually in the same way as, say, a machine gun.

5. A Deadly Joke

Wasn't that an Alan Moore Batman/Joker story? Oh, that was The Killing Joke.

One of the most famous documented deaths from laughing goes all the way back to Athens, with a philosopher known as Chrysippus of Soli.

"Known" may be overstating the case, as this is the first I remember hearing about him.

More interestingly, apparently the joke that ended Chrysippus’ life was his own, saving the Greeks from what would have been a highly confusing murder trial. Apparently, Chrysippus, during the Olympiad, saw a donkey eating figs and hit the ass with the stellar retort, “Now give him some pure wine to wash down those figs!”

Look, ancient comedy just doesn't hit the same way with modern audiences, unless they're stoned out of their gourds.

Regardless, he sent himself into such a disastrous laughing fit that he collapsed and started foaming at the mouth, later dying.

I can just imagine a comedy club audience looking on with a mixture of awe and disgust.

4. Thinking An Injured Man Was Gay

This one's not so funny.

Language is complicated, despite what that bastard Duolingo owl would like you to think.

Ah, yes, my personal nemesis.

Even a difference of a couple measly letters, run through one language to another, can completely muddle a very important sentence’s meaning.

Oh, it's worse than that. I did a whole blog entry a while back on why one probably shouldn't say "Je suis excité" to a French person, unless you're trying to bone each other.

He headed to the hospital and attempted to communicate in Norwegian that he was a hemofil, or hemophiliac. Unfortunately, the Danish doctor, with their ever-so-slightly different language, thought the man was simply coming out of the closet as a homofil, or homosexual. The doctor gave him what he must have thought was a very encouraging talk about how there was nothing wrong with that, and he needed no treatment.

At least the doctor wasn't bigoted?

3. A Suspected Wizard’s Final Words

When we do consider the potentially lethal power of words, we're usually thinking, like, curses or spells, things mostly relegated to folklore, fantasy, and role-playing games (I'm currently in an RPG playing a bard who can injure enemies by punning).

The “witch trials” of the old world have a pretty generous name, considering that they relied less on the rule of law and more on people’s flotation and fire resistance.

"Trial" had a somewhat different connotation then, closer to what we mean by "trial by fire." But of course, reading the sentence I just quoted, I instantly flashed back to Holy Grail: "And what do we burn, apart from witches?" "So, if she weighs the same as a duck..."

Corey, knowing he was fucked either way, but wanting his children to keep their inheritance, stayed quiet until the end, except for his extremely hardcore final words: “More weight.”

This doesn't quite fit the theme of the article, but it's still a badass story, true or not.

2. The Word That Launched Two Nukes

Technically, anytime someone on a battlefield shouts "Fire!" (and there are no flames in evidence), it generally results in death. But again, it's not the word that kills, but the bullets.

Japanese is an incredibly complicated language, with many words having highly nuanced or multiple meanings.

Which makes it even more suited to puns than English, from what little I understand of it. I really should learn more Japanese than "hentai" and "sushi."

If you were, for example, trying to negotiate a surrender during World War II, you’d think they’d want to make sure all the kanji was correctly crossed and dotted.

Or at least do your own English translation? The language wasn't exactly unknown in Japan.

The ultimatum demanded an immediate surrender or threatened “prompt and utter destruction.” Suzuki responded with the word mokusatsu. Correctly translated given the situation, he was basically using the old political chestnut of “no comment.” Instead, his response was translated as “silent contempt” or “not worthy of response.”

Again, though, it wasn't the word that killed a bunch of people. It was fission. And I can't imagine "no comment" would have been received with any greater diplomacy.

1. The Word That Almost Launched Even More Nuclear Weapons

Hey, now, I've been assured that "almost" only counts in horseshoes, hand grenades, and atomic b- oh.

The butchered blurb came from Nikita Khrushchev in 1956, from the end of a speech, which was translated as “we will bury you.”

Now forever immortalized in Sting lyrics.

In reality, though, he’d been saying something a little less directly murderous, and that was a more common Russian saying: Something effectively translated as “we will be there when you are buried,” or to translate into American euphemisms — “You’re digging your own grave.” Or: “It’s your funeral.”

I'm pretty sure that if you dig around (I can't be arsed), you can find many other examples of poor translations, especially of idiomatic language, that started or exacerbated wars. Just as many jokes get lost in translation, and most puns are utterly useless in other languages, threats, too, can be misinterpreted.

Or, you know, you can call yourself a jelly doughnut and have the entire world laugh at you. Unless, of course, "Ich bin ein Berliner" led directly to JFK's assassination.

I guess we'll never know

for sure.

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