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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 13 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 29, 2021 at 12:03am

#1009331

Hammer Time

One of the stupidest things about this timeline is how much weight people put on what celebrities say.

Science and philosophy can, and should, coexist
MC Hammer's recent tweet has sparked a conversation about the complementary nature of these two fields

Another stupid thing about this timeline is Twitter.

Nevertheless, I can't find any real fault in what MC Hammer said here.

MC Hammer recently brought the hammer down on those who see science and philosophy as fundamentally opposed disciplines. He first tweeted a link to this paper, showing that STEM fields account for 21.3% of citations of philosophy of science journals. Some tweeters responded with unflattering, and inaccurate, characterizations of philosophy that put it at odds with science. In response, MC Hammer had some words of advice praised by scientists and philosophers alike: "It's not science vs Philosophy ... It's Science + Philosophy. Elevate your Thinking and Consciousness. When you measure include the measurer."

As both science and philosophy are interests of mine, I do see that sometimes they're at odds. And they have completely different ways of approaching ideas. I see this as a good thing.

MC Hammer's insistence on the complementary nature of science and philosophy is in line with this 2019 opinion paper, published in PNAS. The authors described a continuum of science and philosophy, as the two fields share "the tools of logic, conceptual analysis, and rigorous argumentation.”

The big difference, as I see it at least, is that philosophy doesn't have to cleave to objective truth. One can draw perfectly logical conclusions from false (or at least unverifiable) premises, leading to faulty conclusions even with rigorous logic, but that's not what science is supposed to be about.

Tweets by MC Hammer promoting these views will hopefully also help to break down harmful stereotypes of the disciplines that might prevent scientists and philosophers from working together for the good of society.

I still think entertainers, no matter how talented or intelligent, have entirely too much sway over popular opinion. It's like... I'm a Springsteen fan because I like his music, but anything he says about anything other than music carries no more weight with me than when anyone else says such things.

In any case, it's always been clear to me that, deliberately or not, philosophy guides science, and science can inform philosophy. Examples? Well, consider animal testing in the biological sciences. For a while, it was generally accepted, but as attitudes changed and we found out more (partly through animal testing), such things were curtailed and subject to ethical review. There's no purely scientific rationale for not doing tests on animals (including human animals), but there's plenty of philosophical reasons not to.

After all, sometimes I wonder if the phrase "curiosity killed the cat" is more about a human's curiosity than the poor cat's.

On a personal note, for anyone who's tired of seeing these things pop up in here, I've signed up for the 30DBC again for next month. It's not like I don't have dozens more articles in the queue; it's just time for something different again.

Speaking of something different, today (Thursday) is the day I officially reach full vaccine protection, two weeks after my second shot. While it's not a good idea to start running around licking doorknobs or bathing in people's sneezes, I intend to celebrate by going to *gasp* a movie theater.

Godzilla vs. Kong looks good. If by "good" I mean "thin plot, mediocre acting, but kaiju battles with budget-breaking CGI." Which in fact I do.

6 Like • Comment • View 3 comments

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