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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 4 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 4 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 5 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

about 15 hours 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

a day ago
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

a day ago
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

a day ago
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

a day ago
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

a day ago
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?

June 2, 2021 at 12:04am

#1011143

Survivor

This one's been hanging out in my queue for a long time, not being deleted. It's a miracle!

Why Earth’s History Appears So Miraculous
The strange, cosmic reason our evolutionary path will look ever luckier the longer we survive.

Because if the planet had been destroyed, we wouldn't be here to note it?

This is known as an observer selection effect, and the same sort of bias might apply not only to perforated planes, but to whole worlds as well.

"Might?" It's also a type of survivorship bias, and kin to the Anthropic Principle.

When I was a kid, lo these many aeons ago, it was common for us to ride around in the backs of cars or pickups without car seats, or even seat belts. I did it, and I'm still alive, so obviously from my point of view, that was a perfectly acceptable practice.

If, of course, you don't look at childhood death/injury statistics from that era.

It could be that we’ve been shielded from these existential threats by our very existence.

Rarely have I seen a more perfect reversal of cause and effect. It's remarkable, really.

As Sandberg and his co-authors Nick Bostrom and Milan Ćirković write, “The risks associated with catastrophes such as asteroidal/cometary impacts, supervolcanic episodes, and explosions of supernovas/gamma-ray bursts are based on their observed frequencies. As a result, the frequencies of catastrophes that destroy or are otherwise incompatible with the existence of observers are systematically underestimated.”

In other words -- as I interpret this, anyway -- we've been lucky so far, but might not be so lucky in the future. Based on the next few paragraphs in the article, I think I got it right.

If this is true, it might explain why our radio telescopes have reported only a stark silence from our cosmic neighborhood.

Or maybe it's because -- and yes, I know I've said this before -- what we're looking for is artificial signals, but evolution doesn't require the eventual emergence of technology-capable life. All it requires is fitness for survival, which takes many different forms; those qualities needed to build radios and rockets are irrelevant to species survival. Sure, one could argue that it allows for species survival through dispersal to other planets, but that's an effect, not a purpose.

One of the leading interpretations of this quantum weirdness is that all of the possible realities for the particle that were winnowed away in this act of observation actually are realized somewhere in branching-off parallel universes, by observers in parallel universes—parallel universes just as real as the one in which we happen to live. Though the universe may be infinite in distance it may also be infinitely divergent in this sort of ontological zoo.

Quibble: not actually infinite. But might as well be from our limited perspective, since the numbers involved are so freaking large.

This condition of eternal torment, where one might survive arbitrarily long by subjectively navigating the narrowing tributaries of the many-worlds time lines, staying alive through increasingly—and eventually astronomically—unlikely life paths, is known as quantum immortality, or quantum hell.

I comment on these things as I read them, not after (usually). I was just going to bring up the concept of quantum immortality. But I like "quantum hell" better.

“And eternal inflation implies an infinite universe. If there’s an infinite universe then yeah, if we wipe ourselves out there will still be plenty of life in the universe somewhere. And plenty of humans ... and plenty of Peters and plenty of Anthonys sitting in offices.”

This is a common misconception about infinity. The concept of infinity doesn't mean that there's definitely another "you" somewhere. Consider the set of positive integers, which is an infinite set. Never, not once in that entire set, does an number repeat. Sure, individual digits appear in different numbers, but there's only one 1, only one 42, only one 10¹⁰⁰. Infinity doesn't imply duplication.

Of course, the quote I quoted above was from someone far smarter than I am, so take my comment as you will.

To summarize, the article is worth contemplating, but I have to stress that it's all speculative. Philosophical, even. When I consider how many times I've come close to dying, but didn't, sometimes I wonder if the many-worlds hypothesis holds true, and the universe split into one containing alive-me and one containing dead-me.

But no, this is literally what Occam's Razor was invented for: "Never multiply entities unnecessarily." The many-worlds hypothesis may be compelling to physicists, but it's currently both unverifiable and unfalsifiable, and it literally, actually multiplies entities unnecessarily, when a more elegant answer is that we're only around to talk about this sort of thing because nothing has happened to prevent it.

5 Like • Comment • View 2 comments

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