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Saturday, Nov. 23, 2024

9:49pm EST

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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:

tj-turkey-jobble-jobble-hard-J

about 3 hours ago
In response to "Pursuit"

My grandmother used to say, "Pursuing happiness is like a dog chasing it's own tail."

2 Like

🌕 HuntersMoon

about 9 hours ago
In response to "Pursuit"

Happiness is reading your ramblings daily. True happiness is posting a snarky response.

3 Like

Robert Waltz

Nov 20, 2024 at 2:54pm
In response to "Powers of Three"

🌕 HuntersMoon

- that's pretty much what quantum computing is, from what I've read.

2 Like

Brandiwyn🎶

Nov 20, 2024 at 2:32pm
In response to "Losing Weight Is Easy"

Wordsmitty ✍️

1 Like

🌕 HuntersMoon

Nov 20, 2024 at 1:20pm
In response to "Powers of Three"

With binary, it's either on or off. Wouldn't ternary introduce Schodenger's Cat? On, off, or both?

2 Like

Turkey DrumStik

Nov 19, 2024 at 7:43pm
In response to "Paramount Purpose"

I suspect old web filter attempts

would have combusted when confronted with this article.

1 Like

tj-turkey-jobble-jobble-hard-J

Nov 19, 2024 at 4:04pm
In response to "Losing Weight Is Easy"

Some of them break out in hives if you ask them to add 10% to something.

Yesterday I purchased an item that cost $6.42 after tax was affixed by the register. I handed the cashier a ten dollar bill and watched as she placed it in the cash register, which like most now days, showed her how much change to return. She had to try three times to count out the proper amount, and as I was leaving commented to the bagger, "Why do people have to pay with cash, don't they know how hard it makes my job?"

I didn't say anything, but was trying desperately not to break out in hysterical laughter.

1 Like

🌕 HuntersMoon

Nov 19, 2024 at 9:28am
In response to "Paramount Purpose"

At your age, Bob, take heart in the fact that it can be restored...

2 Like

Wordsmitty ✍️

Nov 18, 2024 at 12:34pm
In response to "Losing Weight Is Easy"

This brings up the old question of whether your weight changes when you inhale a helium balloon to make your voice higher.

Maybe this has to do with the difference of a mug of beer vs a tankard of ale.

The universe has so many weighty/massive questions.

3 Like

Brandiwyn🎶

Nov 18, 2024 at 11:27am
In response to "That's the Spirit"

Maybe you’d go with “Thunder” because that one’s so terrible that only divine intervention can explain its success. Wow. Missed opportunity, here. How could you let this go without a shoutout to mighty Thor, son of the Allfather, wielder of Mjolnir, He with the Long Forehead? (Said in the voice of the Soup Nazi): "No Valhalla for YOU!"

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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