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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Write_Mikey_Write!

about an hour ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 3 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

3 Like

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.

2 Like

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.

1 Like

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.

1 Like

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').

1 Like

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.

3 Like

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

November 17, 2023 at 6:42am

#1059626

The Way It Gose

After a bunch of wine celebrations, finally, a day about a beer.

International Happy Gose Day

Gose, pronounced GOZE-uh, is a tart wheat beer of German provenance.

It disappeared for a while, from what I've heard. Died out, ceased production, joined the choir invisible. Didn't get resurrected until people started drinking real beer again.

This German sour beer is customarily made with coriander, which gives it notes of flowers and citrus, and with salt, so it is usually quite salty.

"But what about the well-known German beer purity laws?" Gose was an exception, and besides, one needs to be very, very careful when promoting anything that has the words "German" and "purity" in it.

Lactic acid is often added, which makes it even sourer.

This is not nearly as disgusting as it sounds. Gose may not be my favorite style, by far, but it's not like it's fermented goat's milk or anything.

But the popularity of gose and International Happy Gose Day is not limited to Leipzig or even to Germany. The beer has been gaining in popularity in the United States in recent years.

Sour beers in general (there are other styles besides gose) have taken off. I suspect it's at least partly a backlash to years of bitter, overhopped IPAs, which themselves became popular in a backlash to mass-produced, flavorless, rice-adjunct lagers.

Americans, it seems, don't seem to believe in middle ground. Gotta take everything to an extreme.

I remember the first time I went to a brewery that specialized in sours. They offered a sampling of their entire tap list, 12 beers arranged in 3-ounce tasters on a platter, like a clock face.

When I was done, the tap lady asked me what I thought. I was just drunk enough to tell the truth: "Sorry, I like my beer to taste different going down than it does coming back up."

Harsh? Sure. But that's how I felt about sour beers at the time.

Since then, I've found some that I actually like, including a gose here and there. My favorite local brewery, for example, makes a good fruited one.

I don't know why they call today International Happy Gose Day. It's not even a pun. If it were Happy Gose Lucky Day, sure, I could get behind that. Or Day Gose By. Or Gose Fishing. Okay, I'll stop now.

And the article speaks of toasting with "Goseanna!" which I've never heard of before, and so I'm going to assume they're trolling until I get confirmation of that.

4 Like • Comment

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