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Saturday, Apr. 27, 2024

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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

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Show Recent Comments

10 Most Recent Blog Comments:

Kåre Enga in Udon Thani

Apr 25, 2024 at 3:16am
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

Apr 25, 2024 at 3:09am
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

Apr 25, 2024 at 2:58am
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

Apr 24, 2024 at 4:26pm
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?

March 7, 2024 at 9:23am

#1065784

Zoned Out

It's about that time of year for most of us, so here's a timely article from Atlas Obscura (copied from The Conversation) to consider:

Why Daylight Saving Time Messes With Your Brain
To “spring forward” is more damaging to our health than to “fall back,” according to some surprising science.

While health shouldn't be the only measure of a thing's worth, if something's unhealthy, then it at least ought to be fun or provide some other benefit. And I'm not convinced switching clocks around twice a year is of any real benefit.

This bit is US-centric, but other countries use some variant of DST, so it may be relevant.

As people in the U.S. prepare to set their clocks ahead one hour on Sunday, March 10, 2024, I find myself bracing for the annual ritual of media stories about the disruptions to daily routines caused by switching from standard time to daylight saving time.

And now you've added to them. Oops... so have I. Damn.

But the effects go beyond simple inconvenience.

"Inconvenience" is a valid reason to do, or stop doing, something; let's not dismiss it.

Researchers are discovering that “springing ahead” each March is connected with serious negative health effects, including an uptick in heart attacks and teen sleep deprivation.

On the other hand, an unreplicated study or two is not a valid reason to do, or stop doing, something. Mind you, I'm not saying the findings are bullshit, or that I don't like them (I happen to be on the "no time switching" team, personally), just the same sort of thing I've been harping on all week.

I’ve studied the pros and cons of these twice-annual rituals for more than five years as a professor of neurology and pediatrics and the director of Vanderbilt University Medical Center’s sleep division.

Admittedly, those are some impressive credentials.

I do have one quibble, and it's not about neurology, medicine, or somnology:

However, the two time shifts—jolting as they may be—are not equal. Standard time most closely approximates natural light, with the sun directly overhead at or near noon.

This is misleading. Or, well, misleading in most locations. And I'll mansplain why.

First: On a world with 24 equal time zones whose boundaries exactly follow lines of longitude, set up so that the middle longitude of each time zone most closely matches the average zenith of the sun at noon, you still have edge cases: on each edge of the time zone, noon will be off from average solar zenith by 30 minutes. Imagine two people standing on either side of this boundary line of longitude, close enough to touch each other. For one of them, it'll be 11:30 am; for the other, 12:30 pm. And yet both would see the sun at its daily high point in the sky. Well, there would still be a day-to-day wobble, because of the complicated interplay of the Earth's elliptical orbit and its rotation, but hopefully, you get the idea. This offset changes as you cross the time zone going east or west, but only at the middle longitude is it near zero.

Second: For sociopolitical reasons, few areas follow the designated time zones. Some countries (even large ones like China) do away with time zones entirely. This leads to even greater divergence between noon and solar noon at many locations. Hell, some places split the difference (notably areas within Australia and India), and the time there is always something:30 when most of the world sees something:00.

Third: as I noted in "First," solar noon occurs at a different time (based on a standard 24-hour clock) every day, no matter where you're standing. The sun's location in the sky, throughout the year, at any given clock time, moves around not only north to south with the seasons, but also a bit east to west. This results in a very pleasing pattern known as an analemma

(that link, which will take you to Wikipedia, explains the whole thing better than I can in a relatively short blog entry).

I'm focusing on solar noon here because that is traditionally what was meant by "noon:" when a sundial's shadow is shortest during the day.

I recently found an interesting map

that displays the difference between solar time and clock time, by country and time zone. If you go to that link, you'll see a preponderance of red. This does not mean that the country or area in question voted Republican in the last elections; it means that politicians of all colors seem to favor solar time being behind standard time, rather than vice-versa. Note especially Argentina, which should be mostly in the same time zone as the far east of Canada but is not; and China, which, as I said above, only acknowledges one official time.

Other things I found interesting from that map: 1) if you look closely, you can see the 1/2 hour time zone areas; 2) England and France are in different time zones despite being mostly the same longitude (which is terribly on point for both countries); and 3) even London, which through accidents of history defines the basis of all time zones, isn't set for solar noon = clock noon.

Whew. That was a lengthy diversion, especially considering the huge number of people who insist that DST gives them an extra hour of daylight. It does not. It only moves the clock around. Hell, even the end of the article is worded like that.

So I won't quote further from the article; suffice it to say that the author lays out her case along with some fascinating history and statistics. She seems to be on the side of "just adopt standard time year-round," though to that I'd add "bring the time zone boundaries closer to where they ought to be based on UTC."

In closing, then, I'll just add this: it doesn't much matter to me one way or the other, as I'm retired and a night owl. So I don't care on a personal level; I just like things to be logical and consistent, which I know is asking too much of humanity. The most common objection I've heard to year-round DST is: "I don't want to wake up and go to work in the dark." That argument is a condemnation of capitalism and the Protestant work ethic, not a reason to keep switching clocks around.

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