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Thursday, Nov. 21, 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

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:

Robert Waltz

a day ago
In response to "Powers of Three"

🌕 HuntersMoon

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

2 Like

Brandiwyn🎶

a day ago
In response to "Losing Weight Is Easy"

Wordsmitty ✍️

1 Like

🌕 HuntersMoon

a day ago
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

a day ago
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

a day ago
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!"

Brandiwyn🎶

Nov 18, 2024 at 11:19am
In response to "Losing Weight Is Easy"

Has anyone ever told you how annoyingly pedantic you are?

Damn your pedantic ways. I was hoping to justify a vacation to Venus (which I still argue would be a much better holiday destination than fr-fr-freezing Mars, Elon. Ugh.) Now I guess I'll have to settle for the Cancun plans we already made, since my mass (and waist diameter, and thigh diameter) will be exactly the same regardless.

3 Like

tj-turkey-jobble-jobble-hard-J

Nov 17, 2024 at 4:04pm
In response to "Revisited: "A Frank Discussion""

I became disgruntled with meat sold in local grocery stores; half the weight is water, the taste is bland, and it's "use by" date is usually one day away. I found a small town butcher/locker plant that sells locally raised meats and buy in bulk so it's actually fresher, cheaper, and better tasting. Recently, I decided to try my hand at making our own sausage as well. It's a bit of work, but I think it's worth the trouble.

1 Like

October 16, 2024 at 8:12am

#1078398

Can We?

From The Takeout, answers to questions you never asked.

Here's Why Canned Food Always Comes In A Ribbed Tin

No, it's not "for her pleasure."

You might think the ribbed design on canned food is just a quirky touch meant to make jellied cranberry sauce look more interesting, but it actually serves a crucial purpose.

I might be alone in this, but I like the jellied cranberry sauce way more than the other canned kind, and way, way more than homemade. The uniformity of it appeals to me. I like to punch out both ends of the can and use one of them as a plunger, pushing out precisely 1/4" of the stuff at a time, then slicing it off at the can top. It's a far superior experience and taste.

Not that I'd ever tell that to someone who sweated over a hot stove to prepare homemade.

This is because the corrugated ridges add strength to the tin by creating a series of tiny arches along the surface. These in turn more evenly distribute pressure, similar to how the arches of a bridge help it carry heavy loads.

Always amusing to see someone unfamiliar with a subject (in this case, structural engineering) attempt to describe it to others.

Wait... is that how I sound all the time?

But the ridges have another job that goes beyond just strength.

Thus once again demonstrating the concept that things don't have to have just one purpose.

In addition to making the can stronger, the ridges on a can help it withstand heat. During the canning process, the contents are sealed and then brought to high temperatures to kill off any bacteria, helping to ensure that the food inside stays safe to eat.

In theory.

These ridges also help the can remain intact after it leaves the factory, when it might have to endure extreme temperatures during transport and storage.

I have a severe addiction to Coke Zero, and I buy it by the case. Obvously, Coke cans aren't ribbed, though their bottoms and tops are reinforced (because of how cylinders perform with pressure inside). Sometimes, on rare occasions, the reinforcement fails, and the bottom goes from concave to convex. On other occasions, it's the top that blows out, making it a whole lot harder to open the damn can. Unlike with canned food, this doesn't indicate that the contents have gone bad.

I suppose it's possible that some cans fail along the cylinder, too, and those are just culled before sale. But it seems to me that the sides of the can are not the failure points.

In the early 19th century, food was stored in glass jars that were heated to high temperatures to kill bacteria and then sealed to keep the contents safe.

Home "canners" still use glass jars. Duh.

The next big leap came in 1810 when English inventor Peter Durand developed tin-plated iron cans... Before the invention of the can opener nearly 50 years later, people had to use tools like hammers and chisels to access the food inside.

Fifty years? Man, humanity really dropped the ball on that one. I guess they were too busy opening cans to develop a better way of opening cans.

5 Like • Comment • View 4 comments

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