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.

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.

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

1 Like

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.

2 Like

October 10, 2022 at 12:01am

#1038966

S'more and more

Confession: I have never had an actual s'more.

A brief history of the s'more, America’s favorite campfire snack

The reason why should be obvious from the headline: an actual s'more requires a campfire, which would imply that I'm camping. Which in turn would imply that I'm *shudder* outside.

Well, okay, fine. I have gone camping. That's how I know I hate it. And sometimes, someone actually brought marshmallows. But never the particular combination of marshmallows, chocolate, and graham crackers with which to form s'mores.

But of course I've consumed that combination on other occasions, sometimes even heated.

This summer, millions of marshmallows will be toasted over fires across America. Many will be used as an ingredient in the quintessential summer snack: the s’more.

This article, of course, is from back when summer was fresh and new, full of promise and the possibility of redemption. That is to say, the beginning of summer in 2018, in the Before Time.

Eating gooey marshmallows and warm chocolate sandwiched between two graham crackers may feel like a primeval tradition.

Unless you actually think about it. Or read this article.

But every part of the process – including the coat hanger we unbend to use as a roasting spit – is a product of the Industrial Revolution.

How long before we make coat hangers illegal?

The oldest ingredient in the s'more’s holy trinity is the marshmallow, a sweet that gets its name from a plant called, appropriately enough, the marsh mallow.

Yeah, not really. As the article goes on to point out:

Today the marshmallow on your s’more contains no marsh mallow sap at all. It’s mostly corn syrup, cornstarch and gelatin.

Another vegetarian trap.

Chocolate is another ancient food. Mesoamericans have been eating or drinking it for 3,000 years.

Yes, from an entirely different continent than the marsh mallow. Which supports once again my claim that international food trade is one of the few actual benefits of endgame capitalism.

Also, what the ancients called "chocolate" (or xocolatl or whatever) bears about as much resemblance to a plastic Hershey's bar (from what I understand, that's the preferred ingredient in a s'more) as the marshmallow does to the marsh mallow.

The chocolate that the Mesoamericans ate was dark, grainy and tended to be somewhat bitter.

Much like me after my divorce.

Finally, the graham cracker was invented by the Presbyterian minister Sylvester Graham, who felt that a vegetarian diet would help suppress carnal urges, especially the scourge of “self-pollution” (read: masturbation).

I have it on good authority that it does no such thing, unless you simply clutch a vegetable or a graham cracker in each hand at all times.

The original graham cracker used unsifted whole-wheat flour. Graham felt that separating out the bran was against the wishes of God, who, according to Graham, must have had a reason for including bran.

Thus proving once more that you can be right for the wrong reasons.

As for how the graham cracker became a part of the s'more, the snack’s true origin remains unclear.

Probably some marketing guru was like, "Hey, here are three things that are approximately 97% sugar. Let's combine them!"

The first mention of this treat is in a 1927 edition of the Girl Scout manual “Tramping and Trailing with the Girl Scouts.” In a nod to the treat’s addictive qualities, it was dubbed “Some More.”

I don't think the GSA could get away with issuing a manual that includes the verb "tramping" these days.

The term s'more is first found the 1938 guide “Recreational Programs for Summer Camps,” by William Henry Gibson. Some think the s'more may be a homemade version of the Mallomar or the moon pie, two snacks introduced in the 1910s.

That would support my "marketing guru" theory.

Today, the s'more has become so popular that it’s inspired a range of spin-offs. You can eat a s'mores-flavored Pop Tart for breakfast...

Those are actually not bad, but my days of eating sugar bombs for breakfast every day are long behind me.

...munch on a s'mores candy bar for dessert...

One of these days, someone will explain to my satisfaction why it's socially acceptable to eat donuts for breakfast, but not cake. Or vice-versa.

...and even unwind after a long day at work with a s'mores martini.

Oh, hell to the power of no squared.

Okay, no, it's not that I wouldn't drink it. It's just that I'm an absolute purist when it comes to the word "martini." If it's not gin and vermouth with an olive, it's not a martini. I can accept vodka in place of gin, but then it's a vodka martini. I can even accept different garnishes (though some kind of garnish is essential). But in no sane universe does a sweet drink get to be called a "martini," regardless of the shape of the glass it's served in.

As I often tell my students, the health-conscious Sylvester Graham is probably rolling over in his grave after what became of his beloved cracker.

Only if they eat s'mores while masturbating.

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