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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Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

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Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

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JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

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Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

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Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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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.

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