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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:

Kåre Enga in Udon Thani

about 14 hours ago
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 Udon Thani

about 14 hours ago
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.

Kåre Enga in Udon Thani

about 14 hours ago
In response to "Reality Cubed"

My dreams may be different from your dreams but if we share them with others then some form of the dream may live on. Regardless, if we don't eat we die. "Joy gives us wings." "Reality bites."

Aiva Raine

a day ago
In response to "Tipped Out"

Agreed. I'll tip and well, for actual waitstaff at restaurants. I'll tip appropriately for bell-staff who assists with luggage at hotels, the more luggage, the bigger the tip. I'll give a tip to the valet as well. Beyond that- I don't tip. Certainly no tip for the fast food workers, coffee baristas, grocery store clerks, etc.

It annoys me to no end that we're being asked to tip for all kinds of jobs where they are being paid an hourly wage. I've worked as a waitress when I was young and my daughter currently does. In those jobs- she absolutely depends upon proper tipping to make a decent wage. And I hate the fact that she is dependent upon tips to do so.

It leaves women especially at the mercy of arsh**() on whether she wants to accept hostile, negative, sexist, harassing, demeaning, and negative behavior from her customers in order to make a living. It condones people treating their waitstaff like shite in order to get paid. While there is much improvement from the 30 years ago when I was working as a waitress in the skimpy outfits we were forced to wear- it's still an issue at non-chain, independent owned restaurants and diners.

Women are left to the mercies of jerk customers and their only recourse is the manager who they have to hope supports their waitstaff rather than the jerk customer. Tipping should be eliminated. All employees should be paid a livable wage and waitstaff especially should not be dependent upon tolerating crappy to obscene behavior from customers in order to make a living.

But, until that happens- I always tip at least 20% and often times more. Very rarely have I tipped lower and the service in those situations was abysmal and they still got 10%.

1 Like

GeminiGem of House Lannister

a day ago
In response to "Going to the Dogs"

I don't understand the intersection, either. I lived in a much smaller city and found that having dogs was a hassle (although it didn't stop me), what with the complainey neighbors, the extra licensing, etc. I live in a rural location now and it works great. I can't wrap my head around living in NYC with or without a dog, although I like the idea of a place that issues a canine and supplies upon moving there. Can that please be a thing?

1 Like

Lazy Writer est 4/24/2008

Apr 23, 2024 at 3:31pm
In response to "Going to the Dogs"

I love my little dog...but I live in a biggish house with a fenced backyard.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:14pm
In response to "That's Cole"

I knew and ate Kohlrabi before I ever heard of kale. Kohlrabi is so much better in every way.

That last quoted sentence is hilarious. I am glad that I wasn't bred to be a Brussels Sprout.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:11pm
In response to "Cash Stash"

The discussion of money is an interesting one. Remember the recent collapse of a few banks? I think it was about a year or so ago?
Yes, deposits were FDIC insured up to $250,000 per depositor. But some people and companies had more money in that bank than the FDIC was insuring.
I never followed up on what happened to those people. I don't run in circles of people who own that much cash.

My accounting professor took half a period to explain to us what happened and how the bank went from Silicone Valley darling to dead so fast.

He also told us that the value we give money is not backed by anything except our hope and faith that our economy will keep going.

We didn't discuss whether the value of money needed a lower income class of people.

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 3:04pm
In response to "Revisited: "Hack This""

In my last psychology class, we discussed the evolution of the brain. It is now settled science that we evolved to have not only larger, but also vastly more powerful brains than most other animals because we learned how to cook (process) our food.
Although we are only about 1/4 to 1/3 of a normal gorilla, we eat half of the calories in a day. Not only that, we prepare the food in a way that we get to use more of the calories that we ingest.
The result is that our brain takes up more space in our noggin than a gorilla brain takes up in his noggin. Not only that, each of our brain cells is smaller, so we can fit exponentially more of those into our head.

Now, using all that goodness ...

1 Like

Aennaytte: Free & Wild in GoT

Apr 23, 2024 at 2:58pm
In response to "Hope for Hops"

Wait! Why are beer and coffee affected by climate change? That wasn't supposed to happen. All we had to do was have a boat by the side of the house for when the sea levels rise and beg our property insurance to cover fire damage. How are we supposed to remember all of that if we don't get coffee to wake us up in the morning so that we can think and beer at night to dull our sense of doom?

August 3, 2020 at 12:05am

#989758

Dueling Scientists

Today's article was published in the Before Times, but its theme is very relevant today.

https://theconversation.com/when-science-gets-ugly-the-story-of-philipp-lenard-a...

When science gets ugly – the story of Philipp Lenard and Albert Einstein

Never heard of Philipp Lenard? Neither had I until I read this article. His relative (pun intended) obscurity undermines the point that I think the article is trying to make. Let's take a look, shall we?

Scientists are not always as scientific as many suppose.

Well, duh. Scientists are humans. I know shows like to portray them as some sort of wizards, the Gandalfs of technology, but they're all people and thus subject to all of the biases, contradictions, prejudices, and just plain wrongness that plague the rest of us.

Recent well-publicized cases of scientific fraud prove that scientists can be as susceptible to the allures of wealth, power and fame as politicians...

Interesting comparison there. In science, theories that turn out to not fit observations are eventually discarded into the trash heap. In politics, well, for example, here in the US we have two houses of Congress and three branches of government, lots of different people thinking about stuff, so that eventually bad politics also gets discarded. Usually. Eventually.

Such breaches prove that scientists do not always base their work strictly on rigorous experimentation, data collection and analysis, and hypothesis testing.

"Prove" is kind of a dirty word here. I might have picked "demonstrate." That is, like I said, scientists are humans and thus fallible. However, that we know about these instances of bad science shows that science, as a process, is working as intended -- to smooth out the bumps in the road caused by individual bias and error.

In fact, scientists frequently disagree with one another, both as individuals and as representatives of competing schools of thought.

Feature. Not a bug.

The article goes on to provide a brief bio of both Lenard and Einstein. Lenard was "a German experimental physicist," while Einstein was "a Swiss theoretical physicist."

It's no secret that experimental and theoretical physicists are often at odds. They need each other, and collaborations have happened, but my impression is that, historically, members of each group consider the other somehow inferior. But, again, this is the intersection of humanity and science; eventually, the experimenters will either find results consistent with the theories, or they won't, in which case the theory has to be abandoned or modified.

Again, this is how science is supposed to work. It's not like scientists are supposed to be all one happy family, always agreeing with each other.

Lenard, meanwhile, was soon swept along in a wave of German nationalism that accompanied World War I. He became increasingly convinced of the existence of a distinctively German physics that needed to be defended against the plagiarized or frankly fabricated work emanating from other countries. Lenard also became more and more mired in anti-Semitism, accusing the “Jewish press” of, among other things, promoting Einstein’s dangerous work on relativity.

Ideally, science is universal. But scientists aren't always.

Lenard’s attacks on Einstein became increasingly vitriolic. He compared theoretical physicists to Cubist painters, who in his view were “unable to paint decently.” He lamented the fact that a “Jewish spirit” had come to rule over physics.

From everything I've heard, Einstein wasn't even a practicing Jew. So it wasn't even about religion; this was about ancestry.

Lenard’s conviction that science, “like everything else man produces,” was somehow grounded in bloodlines led him to become one of the early adherents of National Socialism.

And just to get this out of the way: Nazis were "socialists" the way North Korea is a "democratic republic." I can call myself a unicorn, but that doesn't make me a unicorn.

The story of Philipp Lenard reminds us that even scientists of the very highest caliber sometimes think, speak and act in utterly unscientific ways, swayed by prejudices that have no scientific basis.

And yet, as I said above, Einstein is a household word these days, while Philipp Lenard has been all but forgotten. Einstein's theories have been supported by evidence time and time again, from measurements of the precession of Mercury's orbit to the experiments in gravity wave detection. This is how it's supposed to work and, in general, it does.

And so science progresses. It's wrong to revere Einstein the man; I'm certain that he had his flaws just like everyone else. But again, his science has held up -- even though there seems to be growing evidence that they're going to have to tweak the equations to account for what they're calling dark matter and dark energy.

The problem comes in when people hear a pronouncement by someone with a degree or certification, take it as fact, and then ignore anything that contradicts it. So it's the last sentence of the article that concerns me:

They are human beings too, and members of the general public need to be careful to distinguish between a scientist whose arguments are based in evidence and one whose pronouncements stem from other, less reliable sources of conviction.

That's not always easy to do. Evidence, especially at the highest theoretical levels, isn't really accessible to "members of the general public." I mean, it is accessible, but not always comprehensible. But I know this much: if a so-called scientist, or doctor, or whatever, starts spouting off about demon sperm and lizard aliens, I think she can, and should, be summarily ignored -- at least until we have some hard evidence of lizard aliens.

4 Like • Comment

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