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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Turkey DrumStik

about 3 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

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Write_Mikey_Write!

a day 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.

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Adherennium Plotting something

Nov 14, 2024 at 12:10pm
In response to "Global Domination"

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

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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.

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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.

2 Like

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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.

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Turkey DrumStik

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

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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).

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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.

3 Like

January 2, 2019 at 1:02am

#948671

Mas Tequila!

I have mixed feelings about birthdays.

Mine is in February. As I've noted before, February is my most hated month. Even though the days are getting noticeably longer then, it's still cold, and it contains the most hated day of all: Valentine's Day.

Fortunately, my birthday is after that dreaded occasion, so there's less of that creepy blood-red decoration everywhere. Instead, stores start lining up for Easter. I'm no fan of pastels, either, but anything's better than crimson.

After celebrating as many birthdays as I have, they all start to blend together. I know most of my adult birthdays have involved tequila. That's kind of an unintended tradition for me.

It started on my 21st birthday. That is, for you foreigners, the first day one can legally purchase or drink tequila (or any other booze) in this benighted country. Now, at the time, I'd already been legally drinking beer and wine for two years. It was, as I recall, on my 19th birthday that my state passed a law raising the drinking age to 21 for everything (it didn't take effect until that June, and I was grandfathered in, anyway). Stupid fucking law - get 'em used to alcohol early, is what I say. It becomes less of a Thing that way, and you get less abuse later.

But, in my neverending quest for alcohol positivity, I digress. The tradition, as dictated by my college housemates at the time, was to quaff 21 shots of tequila on one's 21st birthday. Now that... that's not something I'd recommend to anyone who hasn't practiced, and legally, you haven't practiced by then. Legally. Ahem. The problem was, though the bottle was labeled "tequila," and it was, by the laws of both Mexico and the US, tequila, it wasn't tequila; it was Cuervo Gold.

See, to be tequila, it has to a) be at least 51% from agave and b) be manufactured in a particular region of Mexico. Anything with 51%+ agave made outside that area has to be called something else, usually mezcal (that's the one with the worm). Anything with less than 51% agave has to be called something else too, usually "disgusting rotgut that'll kill you." So Cuervo, attempting to make the cheapest shit possible that can still be called tequila, fills up the other 49% with Quetzalcoatl-knows-what, and adds caramel color to make it look somewhat like it's been aged in barrels the way the unpronounceable gods intended.

Dammit, I'm digressing again. What I mean to say is, while that incident put me understandably off tequila for several years, I love real tequila, the stuff made from 100% agave. It comes in four basic varieties: Blanco, which is unaged; Reposado, which is aged a little bit; Anejo, which is aged at least 1 year, and Extra Anejo, which is aged 3+ years.

DAMMIT.

Birthdays.

Right. I'm getting there.

Ever since I discovered real tequila, which was significantly after my 21st birthday, it's been an important part of my birthday celebration. In margaritas. In shots from a ski along with friends while I was on the island of St. Thomas. (You heard right. A snow ski. On a tropical island. In February. Using Mexican liquor.) A flight of really good, really expensive Anejo tequilas in a local bar. A flight of not so expensive tequilas in a bar in Vegas.

So, as you can imagine, all those birthdays kind of blend together.

Therefore, none of them can be considered my most memorable birthday. That would be the one that didn't involve tequila and, instead, involved an IV drip in a hospital bed while recovering from the heart attack and resulting surgery I enjoyed on the evening before my birthday.

Memorable, yes. But it sucked. And you can bet that as soon as I got out, there was only one thing I wanted, and fortunately, there's a tequila bar just a few blocks from the hospital.

I mean, why mess with tradition?

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