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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:

🌕 HuntersMoon

about 14 hours ago
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

a day ago
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

Simply Me

May 3, 2024 at 11:58am
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

May 2, 2024 at 11:37am
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

April 22, 2022 at 12:07am

#1031161

Unkosher

Today's article might be a little fishy.

What Archaeology Tells Us About the Ancient History of Eating Kosher
A new study of fish remains deepens scholars’ understanding of how the dietary laws came to be

Just a few points to clarify here before we begin, for anyone who may not have a lot of background in the whole "kosher" thing:

1) Kashrut, the dietary laws, apply year-round
2) There are additional special dietary laws that apply only on Pesach (which ends today).
2a) Contrary to popular belief, God did not command us to eat gefilte fish, and I wouldn't do it even if he did.
3) The dietary requirements laid out in the Torah, which gets all the publicity, were later expanded and fenced in by far more stringent rules in the Talmud, a set of documents set down by religious leaders sometime around the first century C.E. This article seems to be about the pre-Talmudic commandments.
4) Not all practicing Jews today keep kosher, and some ignore the Talmud.

Now, on to the actual article, which I think is pretty fascinating as a historical investigation, regardless of one's religion or lack thereof.

After Adler spoke about his research on the mikveh, the Jewish ritual bath, Omri Lernau—senior research fellow at Haifa University and Israel’s top authority on all things fish—spoke about remains of aquatic creatures unearthed in ancient Judean settlements. He mentioned catfish, skate and shark.

Adler, who works at Israel’s Ariel University, was instantly intrigued. According to the Jewish laws of kashrut—the set of rules written in the Torah, the Hebrew Bible, that outline foods suitable for human consumption—these species are deemed non-kosher, and therefore unfit to eat.

There are at least two ways to approach this kind of thing. The first is to assume that every word in the Torah is literally true, laid down by you-know-who, and go from there. The second, which is my approach, is to ask why people wrote what they did in what would become the Bible, and one of the ways to answer that is through archaeology.

It's not my purpose here to ignite a religious argument, just to explain where I'm coming from in this entry.

According to the study, archaeologists have found the remains of three non-kosher species in the two ancients Judean settlements—the Kingdom of Israel in the region’s north and the Kingdom of Judah in the south. Judah residents in particular ate a lot of catfish.

I should become a copy editor.

Scholars don’t know exactly when these rules and practices were written down into the Torah, but in his upcoming book, Adler argues that evidence for its observance does not appear until the Hasmonean period that lasted from 140 B.C. to 37 B.C. And the point in history at which Judean citizens adopted the dietary rules prescribed in Torah into their lifestyles, essentially becoming kosher, is also not certain.

Another case for a copy editor. I could be wrong about this, but one does not "become kosher." One "keeps kosher," or "starts keeping kosher." Only the food is kosher or not.

The two scientists didn’t have to dig deep for the vestiges of aquatic life— Lernau had a collection of about 100,000 fish remains gathered from dozens of sites in Israel, which spans 10,000 years, from the Neolithic times to the present.

I'm imagining a bunch of those cartoon fish bones, like when an animated cat eats a fish and pulls the bones back out of its mouth. The reality is likely far less amusing.

The collection resides inside his home’s Fish Bone Cellar, which doubles as a bomb shelter during times of armed conflict.

"The Palestinians are shelling us again."

"Better go down to the Fish Bone Cellar."

Lidar Sapir-Hen, archaeozoologist at Tel Aviv University, who also studied the history of Judeans’ dietary restrictions but was not involved in this study, found similar evidence that Judeans weren’t following the laws of kashrut around similar dates that Adler examined. She had examined pig bones found in ancient Judean settlements. Pork is another type of non-kosher food and yet some digs yielded a number of pig remains.

Probably the most well-known proscription is the one against eating pork. My amateur opinion has long been that it's mostly a cultural thing, a desire to differentiate themselves from neighboring tribes. It's like if people in New Jersey looked at the people in Pennsylvania and said, "Those ungodly heathens eat cheesesteak, so we're going to pass a law that forbids cheesesteak." (Of course, cheesesteak isn't kosher either, as it blends a meat product with a dairy product, but that's not important right now.)

Other people have speculated that the anti-pork lobby was due to incomplete cooking practices and the possibility of trichinosis and other diseases that can lurk in improperly cooked pig. That explanation never really sat well with me, because it assumes that a) people before modern stoves couldn't cook for shit and b) other cultures were too stupid to notice that they were getting sick from eating pork. As a lot of people from other cultures survived munching on bacon, and even populated the rest of the world, neither assumption seems to be reasonable.

It's good to have actual archaeologists working on this, though. Perhaps they'll come up with a reason better than the above or "because God said so."

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