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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

Show Recent Comments

10 Most Recent Blog Comments:

Aiva Raine

a day ago
In response to "Knead Some Dough?"

I grew up on mostly Wonder Bread despite our family having a large dry farm (dry farm being a farm that grows wheat and barley and safflower without irrigation- it relies solely upon mother nature to rain). I was never fond of it and looked forward to the times my mom would laboriously mill and grind the grain we brought back during harvest into flour and make home-made wheat bread. There was and still is nothing better than fresh whole wheat bread with a little butter and honey.

I think we did the Wonder Bread because my mom wasn't a fan of cooking and making bread is a time-consuming process.

I quit doing the white bread decades ago. There are so many kinds of healthy, no HFCS, etc whole grain breads out there that are delicious that I have no idea why any adult would choose the crappy Wonder Bread.

1 Like

Wordsmitty ✍️

May 13, 2024 at 1:18pm
In response to "The Breakfast and the Breakfurious"

And for those who rather not read the Smithsonian article, the History Channel episode that covers Pop Tarts (if you have a cable TV account):

https://www.history.com/shows/the-food-that-built-america/season-4/episode-1

Annette

May 10, 2024 at 12:05pm
In response to "Meditation Clause"

Everytime a stranger asks me to close my eyes, I keep my eyes open to see what they are up to.
Meditation is not for me.

But I think most people would have 45 minutes every day if they took that time away from social media or TV watching.

2 Like

Wordsmitty ✍️

May 9, 2024 at 1:15pm
In response to "Super Stew?"

But . . .

What's the difference between one being in the soup or in a stew? Is one thinner or thicker? How do writers use soup and stew?

1 Like

Elisa the Bunny Stik

May 9, 2024 at 10:11am
In response to "Super Stew?"

"Served in a pot? Maybe in your country."

Honestly, in the US I'd think soup would be served in a pot due to the quantity that ends up being consumed.

1 Like

Lazy Writer est 4/24/2008

May 8, 2024 at 3:02pm
In response to "Be Positive"

Mine always tell me to eat more fruits and veggies.

Annette

May 8, 2024 at 10:52am
In response to "Be Positive"

Asking doctors for advice has its own pitfalls. Because there is no money in recommending a sensible nutrition pattern, they will say things like: Look it up online.

2 Like

🌕 HuntersMoon

May 5, 2024 at 8:50am
In response to "Revisited: "Fail""

Shame on you for making fun of the Hoggs. Being a Virginian, Bob, you should know the reverence that the name Hog inspires recalling the glory days of the Washington Redskins (now the Commanders). The Hogs

1 Like

Charity Marie - <3

May 4, 2024 at 5:42pm
In response to "Overdraft Dodgers"

I have four different accounts and each one has a purpose. Savings, personal debit card, auto draft debit card, and a b8lls account. The bills account is the only one that can overdraft as it's for manual big bills like rent, cell, electric, etc. Stuff that mist be paid. The rest are blocked from overdraft and the savings account has no debit card attached. My bank accounts have been hacked too many times so I keep it this way to protect myself. It is very rare I ever pay an overdraft. I have a money market account for money greater than $10K so it gains interest and isn't touched except emergencies. That is a secure account too.

1 Like

Kåre Enga in Montana

May 3, 2024 at 4:23pm
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

April 8, 2023 at 10:40am

#1047803

Uncivil

I do like the occasional "well, actually..." piece, so here's one from Mental Floss:

7 Misconceptions About the Civil War

By "Civil War" they mean the American one in the 19th century, not the one we're in now, not another country's war, and not the Captain America sequel.

I should note that I didn't fact-check this, but I did go to some of the links and it looks like they did their due diligence. Just don't blame me for any continued inaccuracies on their part.

The American Civil War is a pivotal and ugly moment in American history, but it's more misunderstood than you would think.

I wouldn't think. Given the near-deific significance it's accorded by Americans, northern and southern alike, I'd expect people to get its history right. But as is often the case, I overestimate people—or, at least, ACW fanatics.

1. Misconception: Lincoln’s policies enjoyed widespread support in the North.

I find that, in hindsight, an event in history is usually more clear-cut than it was at the time. I wouldn't expect that everyone in the North supported Lincoln, or that everyone in the South wanted to secede. It's kind of like "the colonies wanted to break away from oppressive England" without considering how many colonists wanted to keep saying "pip-pip-toodleoo."

Take a look at the article for some well-sourced examples.

2. Misconception: Robert E. Lee and Jefferson Davis were staunch secessionists.

I've known this was wrong for a long time (again, details in the link), but pointing it out now seems to be a failure to read the room.

3. Misconception: The Emancipation Proclamation ended slavery

When President Abraham Lincoln delivered the Emancipation Proclamation on January 1, 1863, it declared: “[All] persons held as slaves within any State or designated part of a State, the people whereof shall then be in rebellion against the United States, shall be then, thenceforward, and forever free.”

Well, clearly not, as slavery still exists. Even after the 13th Amendment (which the article covers).

4. Misconception: All amputations were done without anesthesia.

Sure, but it makes for a more compelling movie to ignore that.

5. Misconception: Only men fought during the Civil War

Pretty sure that if you look at wars throughout history, there were almost always women finding a way to join in the fun.

6. Misconception: Abraham Lincoln was the keynote speaker on the day of the Gettysburg Address.

Honestly, this one's pretty interesting. Not surprising to me, but has details I wasn't aware of.

One thing that you might not know about the address is that Lincoln wasn’t pegged to be the main speaker on that day. That honor belonged to Edward Everett, a distinguished scholar and orator who took the stage before the president.

Everett’s speech would go on for around two hours, totaling upwards of 13,000 words...

After Everett finished his speech, the president shook his hand and told him, “I am more than gratified, I am grateful to you.” Then the Thunder-Stealer-in-Chief rang out with “Four score and seven years ago ...” and made Everett’s magnum opus a historical footnote in under 180 seconds.

The original Rap Battle.

7. Misconception: The war was fought entirely in the U.S.

Well, duh, because technically, the CSA wasn't part of the US; hence the war.

But no, this part's mostly about naval battles.

I'd also add misconception #8: Everyone knows the war is over.

4 Like • Comment • View 4 comments

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