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 3 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 4 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').

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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.

April 20, 2021 at 12:03am

#1008815

Real Men of Genius

It's been said that with genius comes a certain level of insanity.

4 Famous Smart People With Little-Known Problems

Would that the reverse were true. Anyway, from Cracked...

Pop culture loves to portray geniuses as dragging us ungrateful schmucks through history, whether we want to come along or not. So it's always good to remember that brilliant people still have the same problems as the rest of us ... except when their problems get much, much stupider than ours ...

I don't know; some of us have some really stupid problems.

4. Vladimir Nabokov Hated Sleep, Conducted Weird Dream Experiments

Vladimir Nabokov, the famous author of Pale Fire and that novel about a pedophile that's awkward to read on the bus, is considered a titan of literature, especially by dudes with bad facial hair who keep cutting off other people in their MFA classes. He was also a lifelong insomniac, and the condition took him to some weird, weird places.

Probably because it's been shown to a high level of certainty that lack of sleep messes you right up.

For starters, Nabokov called sleep "mental torture," "the most moronic fraternity in the world," and "the nightly betrayal of reason, humanity, genius," among other things offensive to anyone who considers napping their greatest skill.

For me, it's between napping and drinking.

3. George S. Patton Thought He'd Have To Fight Forever

Admittedly, he also wrote poems with titles like "The Turds of the Scouts," but his belief in reincarnation was legit, and he spent both World Wars claiming to know his way around old European towns and battlefields thanks to his time as a 14th-century French knight. Unfortunately, Sir Patton was one of many Frenchmen to meet his end at the Battle of Crecy, a crushing victory for the English, but it would have been suspicious if he'd claimed nothing but a string of huge successes.

What's suspicious is that all of his purported past lives were European. Couldn't throw some Native American, African, or Asian warriors in the mix? Racist fuck.

Exactly how Patton reconciled this with his devout Christian beliefs remains unclear, but he also claimed that he was at the Siege of Tyre, served under Caesar and Marc Antony, fought with the English at Agincourt because apparently when you reincarnate during a lengthy conflict, you can ask to switch to what looks like the winning side, served the House of Stuart during the English Civil War, and was a bigshot in Napoleon's Grande Armee.

See? If his past lives were any whiter, he'd be a ghost. Oh, wait.

Anyway, this may all sound ridiculous, but reincarnationresearch.com thinks he's currently living as James Mattis, and surely if anyone would know, it's those guys.

I have to wonder if Mattis knows about this.

2. Kurt Godel Would Only Eat Food Prepared By His Wife ... And Then She Got Sick

Kurt Godel is one of those mathematicians who was so smart you need a good math background to appreciate his incredible influence. His incompleteness theorems, contributions to set theory and modal logic, and many other accomplishments are all way over our head.

Mathematicians tend toward weirdness anyway. That's why they become mathematicians.

And Cracked, what's with omitting the umlaut? It's Kurt Gödel. It's not that hard to do, and it hasn't been fashionable to piss off Germans since 1945. Also, dude was metal, and nothing says "metal" like an umlaut.

Godel became obsessed with the thought of being poisoned, either intentionally via food or accidentally through, for example, the venting of gas from his refrigerator. His wife kept his fears in check by preparing and tasting his food for him, and long talks with some guy named Einstein helped keep him mentally grounded, but he was never the most stable genius.

Hm, take the slight risk of possible poison, or the sure risk of dying from malnutrition? If only there were a logical construct for managing risk.

1. Edward Gibbon's Balls Were Literally Too Big

18th-century politician and historian Edward Gibbon's big claim to fame is The History of the Decline and Fall of the Roman Empire, an epic 1.5 million word landmark in historical writing. While its central argument -- that Rome's decline can be blamed on its adoption of Christianity -- is considered wrong today, it's still seen as a huge leap forward in research standards and an entertaining read to boot. Books, shows, and games have aped its title, it's been referenced in pop culture like Mad Men, and countless dads once got copies in book subscription programs that they displayed prominently but never got around to actually reading. Whatever Gibbon's flaws, he was an esteemed, serious scholar ... who was undone by an inflamed testicle.

Right now you might be going "But Waltz, that's not insanity." I have no personal experience with this, but I'm pretty sure an inflamed ball would make any man nuts. So it fits. Also, it gives me an opportunity to make a "nuts" pun.

The intent here is not to mock a great man of history, although dying from big balls is very, very funny (Gibbon himself joked about it with close friends as he neared the end). The procedure was simply the best available at the time, and the risk was considered worth it, given what the unchecked problem would mean for his life. You are simply invited to reflect on the fact that, while you probably won't go down in history as a genius, you at least don't live in a time when preposterously swollen genitalia can take you to your grave.

No, it just gets you fame on the Dark Web.

Or so I've heard.

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