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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

a day 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

a day 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

a day 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

Apr 23, 2024 at 6:51pm
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?

February 21, 2021 at 12:03am

#1004906

Philosophy of Science

This one's a little on the deep side, and I can't claim to understand all of it. But I'm putting it out here anyway in the hope that someone will get something out of it.

Questioning Truth, Reality and the Role of Science
In an era when untestable ideas such as the multiverse hold sway, Michela Massimi defends science from those who think it hopelessly unmoored from physical reality.

Incidentally, my browser's spellchecker doesn't recognize the legitimacy of "untestable." There's probably a metaphor in there somewhere, but I'm entirely too sober to tease it out right now.

It’s an interesting time to be making a case for philosophy in science. On the one hand, some scientists working on ideas such as string theory or the multiverse — ideas that reach far beyond our current means to test them — are forced to make a philosophical defense of research that can’t rely on traditional hypothesis testing. On the other hand, some physicists, such as Richard Feynman and Stephen Hawking, were notoriously dismissive of the value of the philosophy of science.

I expected better from Feynman, and Hawking always struck me as having something akin to a philosophical streak.

That value is asserted with gentle but firm assurance by Michela Massimi, the recent recipient of the Wilkins-Bernal-Medawar Medal, an award given annually by the U.K.’s Royal Society. Massimi’s prize speech, delivered earlier this week, defended both science and the philosophy of science from accusations of irrelevance. She argues that neither enterprise should be judged in purely utilitarian terms, and asserts that they should be allies in making the case for the social and intellectual value of the open-ended exploration of the physical world.

Personally, I think judging anything on "purely utilitarian terms" misses an important part of the human experience. Which is not to say I don't have a utilitarianist perspective on a lot of things; I just don't think that's the only viewpoint worthy of consideration.

The article goes on to interview Massimi, and while I won't quote most of it here, it's worth seeing both the questions and her answers. But this is one question that struck me as interesting -- not for the answer, appropriately enough, but for the question itself:

One criticism made is that science moves on, but philosophy stays with the same old questions. Has science motivated new philosophical questions?

Her answer is important from one perspective, but I'll offer another point of view: Yes, science motivates new philosophical questions. It does this all the time. These discussions even seep into popular culture. The very first true science fiction book, Shelley's Frankenstein, delves into the philosophical repercussions of applied science (albeit from a purely fictional perspective). The question isn't answered, of course -- we're still asking it, in stories, right up to the present day. And undoubtedly beyond.

Without science offering the possibility of creating something new in the world, there would be little reason to ask those questions on a serious level.

Later, a part of her answer to another question goes like:

Obviously it is not the job of philosophers to do science, or to give verdicts on one theory over another, or to tell scientists how they should go about their business. I suspect that some of the bad press against philosophers originates from the perception that they try to do these things. But I believe it is our job to contribute to public discourse on the value of science and to make sure that discussions about the role of evidence, the accuracy and reliability of scientific theories, and the effectiveness of methodological approaches are properly investigated.

Ethics is in the purview of philosophy. Perhaps not so much in physics, but other branches of science, particularly biology and related fields, absolutely has to consider ethics. Animal testing? Hell, human testing? The potential for cloning? The quote from the original Jurassic Park movie comes to mind: "Your scientists were so preoccupied with whether they could, they didn't stop to think if they should."

But this is, for me, the most important point she makes:

In this respect, I see philosophy of science as delivering on an important social function: making the general public more aware of the importance of science. I see philosophers of science as public intellectuals who speak up for science, and rectify common misconceptions or uninformed judgments that may feed into political lobbies, agendas and ultimately policy-making.

This article is from 2018, and since then I think at least one major thing has happened that demonstrates the value of science in the public sphere. While there is certainly research that is, at best, questionable -- nutrition science comes to mind -- I think one of the greatest issues facing us right now is general distrust of science and the dismissal of experts, in science or other fields, as no more useful than your uncle's deranged Failbook posts.

So yes, by all means, keep up the philosophy. I won't understand all of it. But I don't have to.

2 Like • Comment • View 1 comment

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