Complex Numbers (Book)

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Complex Numbers

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/

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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

about 24 hours ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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This is a private area. Access has been restricted and you may not view it under its current settings. If you feel you should have access, please contact the area's owner and request access.

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

October 14, 2024 at 9:27am

#1078266

DeTECHtion

From Cracked, a countdown of the unseen:

5 Ways to Detect the Invisible Stuff Hiding From Us
Surround yourself with the right sensors, and they might just save us all

All around you are signals you have no way of perceiving on your own. For example, did you know that invisible waves travel through the air, transmitting words and images, decodable by the right device?

We only see a remarkably thin slice of the electromagnetic spectrum. There are good evolutionary reasons for this, but why interrupt a fact-comedy with pure fact?

Not everything imperceptible is on the EM spectrum, though.

5. Seismometers Spot Quakes and Save Trains

Not a single person has died in an accident aboard Japan’s bullet trains.

There was a movie a couple of years back called Bullet Train, set on a Japanese bullet train. Action/comedy, so the title is a pun. It's not much of a spoiler to say that bad things happen on the train and people die. But not in an accident caused by an earthquake. I had the opportunity to watch the movie again on a recent long flight, and it was even better than I thought it was the first time.

Even in 2011, when that 9.0 earthquake hit Japan, collapsing a bunch of bullet train stations and ripping up the track, seismometers on the ocean floor detected the first tremors and shut down the trains in time.

EM waves, including radio, travel faster than seismic waves, so these early detection systems aren't magic, but science.

4. The Mountain Pool of Ultrapure Water

Japan is also home to Super-Kamiokande, an observatory for detecting neutrinos. It’s a tank a hundred feet tall and hundred feet wide, located in a mine a thousand feet under a mountain.

Neutrinos are the second hardest things to detect, after dark matter. Well, maybe third, after dark matter and something so invisible we don't even know it exists, yet.

Also, I'm pretty sure the tank is measured in SI units, as is its depth.

Neutrinos travel at the speed of light and pass through the planet without affecting much of anything, but we still manage to detect them using Super-Kamiokande.

Far be it from me to quibble about something on a comedy site, but I'm going to do it anyway: Neutrinos have mass, and therefore don't quite travel at the speed of light. But, you know... close enough. (There was some sensationalist reporting a while back on neutrinos that appeared to exceed the speed of light, but that turned out to be a mistake.)

3. ShotSpotter, the Gunshot Detector

Sometimes, you hear a gunshot outside, and you shrug your shoulders, saying, “Eh, what can you do?”

Emigrate?

Starting in 1997, several cities in the U.S. set up a system of sensors that pick up the sounds of gunfire and relay the location of the source directly to police. In D.C., for example, the sensors picked up some 40,000 shots in a 20-square-mile area over the course of seven years.

That's the sound of freedom.

2. The Detectors that Detect Radar Detector Detectors

Naturally, such devices use waves of their own, which means they’re vulnerable to being spotted by radar detector detectors. Of course, that means you should equip your radar detector with a radar detector detector detector, which will shut your device down as soon as it realizes it’s being spotted.

And so the arms race continues.

Pretty sure my state is still the only one in the US where radar detectors are illegal, but I can't be arsed to find out for sure. Yes, I drive around a good bit. No, I don't have a radar detector.

1. The Parkinson’s Sniffer

This one may seem out of place, being a mutant superpower instead of a tech device. But if you read to the end, you see that they used the mutant's superpower to create the tech needed for a Parkinson's test.

No mention in the article of radiation and Geiger counters, but I suppose most people know about that one, anyway. In any case, all of these were known invisible things. It's the unknown invisible things you really have to worry about.

3 Like • Comment • View 1 comment

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