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Saturday, May. 4, 2024

7:29am EDT

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Complex Numbers

by Robert Waltz

Rated: 18+ · Book · Personal · #1196512

Not for the faint of art.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

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Show Recent Comments

10 Most Recent Blog Comments:

Kåre Enga in Montana

about 15 hours ago
In response to "It's a Crime"

Thailand versus Montana versus New York versus Costa Rica.

Theft? Not in Thailand! Not a huge concern in Missoula, perhaps in NYC, definitely an issue in Costa Rica.

Violence? Not a huge concern in Thailand. More so in MT, NY and CR (where it wasn't a concern years ago), definitely a problem in Saint Louis and the US in general. El Salvador is known for violence.

Death by vehicle. Not a huge concern in either NY or MT. A rising problem in Costa Rica. Thailand is one of the worst places in the world in this respect.

You may be shot and killed in parts of the US but you're more likely to be run-over in Thailand.

1 Like

Simply Me

about 20 hours ago
In response to "It's a Crime"

I get it. The fear spreading jerks wants us to have an "us vs. them" attitude. The division gives them leverage. (they think)

1 Like

🌕 HuntersMoon

a day ago
In response to "Butt Of Course"

My wife tried waterskiing once... called it the 100-yard douche.

Talk about retaining water...

2 Like

Elisa the Bunny Stik

May 1, 2024 at 10:53am
In response to "Mad as Hell"

"But if you want to really see how anger is useful, you need to consider how it's used as a tool for manipulation."

This is a line that must be regarded with care. After all, there are people that get angry about the way they are being treated on a large scale and harness that anger to change their circumstances. There is some level of manipulation in this angry but still legitimate flavor of activism. We do have to consider the possibility that this angel's usefulness can have net benefits (such as environmental cleanup and protection; see the Live Canal as an example).

GeminiGem survived GoT

Apr 29, 2024 at 1:53pm
In response to "Supplementary"

Working in health care for a zillion years, I've heard it all. The whole thing about something being natural, therefore it would be good for a person is very, very, very...very prevalent. An internist I worked with had a fantastic comeback to that line of thinking. She would say, "Arsenic is natural, are you going to start taking that?"

1 Like

Lazy Writer est 4/24/2008

Apr 28, 2024 at 2:14pm
In response to "Revisited: "Exercise""

I will be 75 in July. My stepdaughter gave me a bike because I wanted to ride it again. Well, I started off shaky and stayed shaky...then I fell off it...A passing motorist stopped to help me get up...I have NOT been back on that bike and will not ever again.

🌕 HuntersMoon

Apr 28, 2024 at 10:43am
In response to "Revisited: "Exercise""

So, you're saying that now you pedal your ass differently?

Elisa the Bunny Stik

Apr 27, 2024 at 10:33am
In response to "Pleasure and Pain"

it illustrates why it's important to have people from different backgrounds, with different worldviews, working on scientific research.

This right here is why I get rather irritated by people poo-pooing college these days. I feel like that attitude effectively shits on minority populations that may not have strong finances but do possess persistent curiosity and want to learn about various subjects in-depth. In the 90s, these groups were exclusively told to go to trade schools even if they had the aptitude for college education. Nowadays the blanket "Everyone should consider a trade" mantra employs a more subtle (but still abundant) form of discrimination. I'm so over it.

1 Like

Kåre Enga in Montana

Apr 25, 2024 at 3:16am
In response to "Going to the Dogs"

I have ants in Thailand. Will I miss them in Montana? No.

Would it be nice to have a cat and a dog, here or there? Yes, but I have struggles keeping my neglected plants alive.

Not every situation is suitable for everything or everybody.

Kåre Enga in Montana

Apr 25, 2024 at 3:09am
In response to "Tipped Out"

I don't tip. In my travels I rarely find tipping the norm.

I also don't haggle.

When you add the tip... the true cost of America becomes evident. But what would I know? I travel while others who have double my income can't pay rent.

#1386. If You Can't Stand the Heat...
ID #995608 entered on October 11, 2020 at 5:55am

#1385. Fail
ID #995512 entered on October 10, 2020 at 12:02am

#1384. The Pleasure Principle
ID #995429 entered on October 9, 2020 at 12:09am

#1383. Universal
ID #995330 entered on October 8, 2020 at 12:03am

#1382. Urine Good Hands
ID #995230 entered on October 7, 2020 at 12:03am

#1381. Road Hazard
ID #995149 entered on October 6, 2020 at 12:03am

#1380. Zodiacal Light
ID #995069 entered on October 5, 2020 at 12:02am

#1379. My Land
ID #994992 entered on October 4, 2020 at 12:04am

#1378. That's Not Kosher!
ID #994891 entered on October 3, 2020 at 12:03am

#1377. Dark and Stormy
ID #994824 entered on October 2, 2020 at 12:02am

#1376. Wasted
ID #994739 entered on October 1, 2020 at 12:18am

#1375. When September Ends
ID #994640 entered on September 30, 2020 at 12:02am

#1374. En-listed
ID #994530 entered on September 29, 2020 at 12:01am

#1373. Baggage
ID #994431 entered on September 28, 2020 at 12:06am

#1372. Not So Prompt
ID #994334 entered on September 27, 2020 at 2:27am

#1371. O'zapft is!
ID #994223 entered on September 26, 2020 at 12:08am

#1370. Don't You Forget About Me
ID #994138 entered on September 25, 2020 at 12:53am

#1369. Ex-Spend-Able
ID #994053 entered on September 24, 2020 at 12:03am

#1368. The Equation of Spontaneity
ID #993982 entered on September 23, 2020 at 12:12am

#1367. Déjà Vu All Over Again
ID #993899 entered on September 22, 2020 at 12:02am

#1366. Out Of Control
ID #993819 entered on September 21, 2020 at 12:05am

#1365. Patterns
ID #993759 entered on September 20, 2020 at 12:02am

#1364. Avalanche
ID #993687 entered on September 19, 2020 at 12:03am

#1363. Is Time Ever Really Free? Are We?
ID #993619 entered on September 18, 2020 at 12:01am

#1362. Lessons
ID #993535 entered on September 17, 2020 at 12:02am

#1361. Pappy? Happy.
ID #993461 entered on September 16, 2020 at 12:20am

#1360. It's an Ill Wind
ID #993382 entered on September 15, 2020 at 12:20am

#1359. The Inner Fright
ID #993295 entered on September 14, 2020 at 12:03am

#1358. Infinite Diversity in Infinite Combinations
ID #993223 entered on September 13, 2020 at 12:50am

#1357. Drinking about Thinking
ID #993134 entered on September 12, 2020 at 12:13am

#1356. Per Ardua Ad Astra
ID #993024 entered on September 11, 2020 at 12:02am

#1355. The Sound of Silence
ID #992939 entered on September 10, 2020 at 12:26am

#1354. Future Imperfect
ID #992827 entered on September 9, 2020 at 12:02am

#1353. To Explore Strange New Worlds
ID #992731 entered on September 8, 2020 at 12:02am

#1352. You CAN Take It With You
ID #992611 entered on September 7, 2020 at 1:30am

#1351. I Drink and I Know Things
ID #992517 entered on September 6, 2020 at 12:31am

#1350. The Force Is Strong in This One
ID #992410 entered on September 5, 2020 at 12:07am

#1349. Once in a Blue Moon
ID #992321 entered on September 4, 2020 at 12:01am

#1348. Everyone's a Critic
ID #992214 entered on September 3, 2020 at 12:07am

#1347. Mission: Difficult
ID #992094 entered on September 2, 2020 at 12:01am

#1346. Cheese? What Kind of Cheese? I Want Brie
ID #992001 entered on September 1, 2020 at 12:08am

#1345. Another Month in the Past
ID #991928 entered on August 31, 2020 at 12:01am

#1344. Nothing Matters
ID #991852 entered on August 30, 2020 at 12:50am

#1343. Rare Earth
ID #991792 entered on August 29, 2020 at 12:03am

#1342. Good? Evil? A Jedi Cares Not For These Things
ID #991729 entered on August 28, 2020 at 12:07am

#1341. No, It's Not 36.
ID #991669 entered on August 27, 2020 at 12:12am

#1340. For the Rest of It
ID #991593 entered on August 26, 2020 at 12:02am

#1339. Wave Function Collapse
ID #991528 entered on August 25, 2020 at 12:06am

#1338. Orange You Glad This Wasn't You?
ID #991450 entered on August 24, 2020 at 12:04am

#1337. Fur the Fun of It
ID #991386 entered on August 23, 2020 at 12:02am

#1336. Just the Tip
ID #991324 entered on August 22, 2020 at 12:06am

#1335. Quantum Speculation
ID #991258 entered on August 21, 2020 at 12:05am

#1334. Good Willpower Hunting
ID #991199 entered on August 20, 2020 at 12:18am

#1333. Mind Map
ID #991137 entered on August 19, 2020 at 12:10am

#1332. As the Turn Worlds (or whatever)
ID #991071 entered on August 18, 2020 at 12:02am

#1331. The Real You
ID #990991 entered on August 17, 2020 at 12:08am

#1330. Changes in Attitude
ID #990910 entered on August 16, 2020 at 12:04am

#1329. Wait, Isn't Oxygen A GOOD Thing?
ID #990810 entered on August 15, 2020 at 12:08am

#1328. Go Home, Evolution. You're Drunk.
ID #990710 entered on August 14, 2020 at 12:01am

#1327. On Merit
ID #990638 entered on August 13, 2020 at 12:06am

#1326. I Got a Rock
ID #990565 entered on August 12, 2020 at 12:02am

#1325. Once More With Feeling
ID #990492 entered on August 11, 2020 at 12:20am

#1324. The The
ID #990401 entered on August 10, 2020 at 12:03am

#1323. Wine, Cheese, and Bread
ID #990304 entered on August 9, 2020 at 12:07am

#1322. The Great Gig in the Sky
ID #990216 entered on August 8, 2020 at 12:01am

#1321. Lox Pie
ID #990157 entered on August 7, 2020 at 12:02am

#1320. Panpsychoticism
ID #990052 entered on August 6, 2020 at 12:03am

#1319. Only the (not so) Lonely
ID #989926 entered on August 5, 2020 at 12:11am

#1318. Decisions, Decisions
ID #989830 entered on August 4, 2020 at 12:02am

#1317. Dueling Scientists
ID #989758 entered on August 3, 2020 at 12:05am

#1316. Eruv L'olam
ID #989699 entered on August 2, 2020 at 12:26am

#1315. Remember, Remember
ID #989626 entered on August 1, 2020 at 12:02am

#1314. Where Jokes Go to Die
ID #989563 entered on July 31, 2020 at 12:01am

#1313. Thoughts for Thirty
ID #989487 entered on July 30, 2020 at 12:21am

#1312. Humanity's Last Invention
ID #989410 entered on July 29, 2020 at 12:01am

#1311. Something is Rotten in Belgium
ID #989336 entered on July 28, 2020 at 12:21am

#1310. The Not-Shining
ID #989226 entered on July 27, 2020 at 12:02am

#1309. One True Pizza
ID #989146 entered on July 26, 2020 at 12:11am

#1308. Week
ID #989070 entered on July 25, 2020 at 12:03am

#1307. Freaky
ID #989000 entered on July 24, 2020 at 12:11am

#1306. Butcher
ID #988913 entered on July 23, 2020 at 12:02am

#1305. Vice-Virtue
ID #988828 entered on July 22, 2020 at 12:02am

#1304. Dear Me
ID #988737 entered on July 21, 2020 at 12:05am

#1303. Man on the Moon
ID #988635 entered on July 20, 2020 at 12:01am

#1302. Solitaire
ID #988560 entered on July 19, 2020 at 4:04am

#1301. Gnaw Bone
ID #988445 entered on July 18, 2020 at 12:06am

#1300. Driver's Seat
ID #988362 entered on July 17, 2020 at 12:01am

#1299. Comet Sutra
ID #988287 entered on July 16, 2020 at 12:18am

#1298. The Pun is Mightier than the Sword
ID #988201 entered on July 15, 2020 at 12:22am

#1297. It's a Trappist!
ID #988106 entered on July 14, 2020 at 12:17am

#1296. Summer Ale
ID #988003 entered on July 13, 2020 at 12:01am

#1295. Blog Standard
ID #987883 entered on July 12, 2020 at 12:01am

#1294. Sounding the Alarm
ID #987788 entered on July 11, 2020 at 12:01am

#1293. Catfall
ID #987703 entered on July 10, 2020 at 12:01am

#1292. That's Nice
ID #987614 entered on July 9, 2020 at 12:01am

#1291. All On That Day
ID #987524 entered on July 8, 2020 at 12:11am

#1290. You're My Inspiration
ID #987434 entered on July 7, 2020 at 12:12am

#1289. The Edge of the World
ID #987334 entered on July 6, 2020 at 11:42am

#1288. Inspiration and Aspiration
ID #987254 entered on July 5, 2020 at 12:23am

#1287. Forty
ID #987175 entered on July 4, 2020 at 12:14am

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