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 5 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.

1 Like

🐕GeminiGem🎁

about 9 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

about 12 hours ago
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

about 12 hours 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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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

November 1, 2024 at 11:01am

#1079301

Cave In

Did my early voting today, so I can spend Election Day doing more important things, like drinking.

Unrelated to that or, well, anything else, really, is this Atlas Obscura article featuring an obscure city in an obscure state:

The Strange Heat Island Lurking Beneath Minneapolis
An urban explorer ventured deep below downtown in search of Schieks Cave. What he found changed science.

Greg Brick knew it was there, lurking beneath his city, hidden within the Minneapolis water and sewer system: an enticing geologic anomaly called Schieks Cave.

That sort of thing fascinates me, but I can't be arsed to do all the work or break the laws necessary to explore for myself.

When Brick arrived at the cave, he shined a flashlight into the thick darkness of the city’s underbelly. He found not just the natural void but also concrete walls that previous generations of civil engineers had built to support the natural structure.

If those civil engineers were anything like me, they wrote something like "put a concrete wall here" on a map, and some minimum-wage temps did the actual work.

The water was about 20 degrees hotter than it was supposed to be. It was more like the groundwater of Mississippi than that of Minneapolis. Something was warming the water beneath his city.

Well, clearly, that's because Minneapolis is actually a gateway to Hell.

[The cave's] discovery in 1904 by a city sewer engineer was initially kept secret lest the public fear Minneapolis had been built on unstable ground.

Apparently, it was, else generations of civil engineers wouldn't have specced out concrete support walls in the cave.

He finally had a way to access the cave—but there was another problem. Along the route, the raw sewage poured from shafts overhead, shooting bacteria into the air as it splashed down and creating what’s politely known as “coliform aerosols.” Unsurprisingly, Brick got sick.

Another reason for us civil engineers to stay at our desks.

In 2008, a separate team from the University of Minnesota had predicted that heat from Minneapolis’s urban surface was conducting itself deep underground, heating the groundwater there like a metropolitan microwave.

As the article notes, this turned out to be the correct explanation, not the "gateway to Hell" one. Much to my disappointment.

But it’s not all bad news: Canadian and European researchers recently suggested recycling underground heat and using it as a low-carbon way to heat homes, while also cooling the groundwater back down to normal temps.

I hope their "suggestion" included exactly how to do that.

Now, from the headline, I was expecting some discovery of dark matter or a violation of the Second Law of Thermodynamics or whatever. But no. Still, at least it's not boring.

3 Like • Comment • View 5 comments

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