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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Elycia Lee ☮

about 9 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

2 Like

Wordsmitty ✍️

about 22 hours ago
In response to "Two Castles and a Cave"

That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

1 Like

Annette in France

a day ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

2 Like

Kåre Enga in Montana

a day ago
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

2 Like

Annette in France

a day ago
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

2 Like

Brandiwyn🎶

a day ago
In response to "There Is A Such Thing As Coincidence"

Love it when stars align!

2 Like

Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

1 Like

Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

1 Like

Annette in France

Sep 24, 2024 at 9:55am
In response to "All Out of Louvre"

I'm already out of Paris.

3 Like

January 8, 2020 at 12:05am

#972922

The Thin Ice

PROMPT January 8th

Share a story from your childhood when you did something you weren’t supposed to. Did you get caught?

Who, me? Do something I wasn't supposed to? What are you implying?

...well, I suppose the statute of limitations has expired, so okay.

Truth is, I spent more time doing stuff I wasn't supposed to than I did doing stuff I was supposed to. There was the time I almost set the house on fire, all the times I secretly smoked, tractor races, my first beer (a Red White and Blue

, which also resulted in my first hangover as well as my first time puking off the side of a boat), hitchhiking into town, pranking my father into taking the entire tractor apart piece by piece, biking along winding country roads, walking to the store 3 miles away so I could buy cigarettes and play pinball...

Is it any wonder I didn't want to have kids? I certainly wouldn't want to deal with me.

But possibly the most memorable one was the Ice Incident.

First, however, I need to provide a bit of background. I didn't grow up in a nice suburban house with a lawn and a white picket fence and a dog. Well, a dog, yes. But no, I spent my childhood out in the sticks, in a rural area adjacent to an estuary of the Chesapeake. This is one of several estuaries - they're called "creeks," but they're more like miniature bays, situated at sea level and subject to tides - that form inlets off of the Potomac River in both Virginia and Maryland. It's navigable by small to medium sized boats, and, in fact, we lived next door to a busy marina.

Said marina was, however, not all that busy in the wintertime. Then, the winds would howl off the Potomac, rocking the sailboats and causing their rigging to produce a melodic sound, like titanic wind chimes. That is, until the creek froze over, which it did probably once every other winter. There was not enough room at the marina for drydocks, and so the ice would freeze the boats in place, silencing their music. At this time of year, we'd also hear, with preternatural clarity, whenever a train roared past on the tracks whose bridge crossed the creek about a mile inland, to the northwest.

One winter - I really don't remember when, but I was old enough to still have a sense of wonder but not old enough to know better - the creek froze solid for over a week. Normally, it would only freeze out to the shipping channel, but on this occasion, one could, if one were so inclined, simply walk across to the other side. Such activity was, of course, strictly forbidden by the inconvenient parental units.

After a few days of this solid expanse of frozen creek water, snow fell, and covered the ice to a depth of an inch or two. There wasn't enough wind, then, to blow the fallen snow into drifts, so it was like looking out over a vast, flat, white landscape.

This, of course, was like heroin for me; I felt an unquenchable thirst to go and put my stamp on the wide expanse of pristine and untouched snow, much as I had to carve my initials into every smooth-barked birch tree in the woods.

So I did.

Now, I wasn't completely and utterly stupid, even as a kid - just reckless, as kids are. I waited until my parents weren't paying attention, went outside to "play in the snow," and snuck down the hill to the water's edge. First, I stomped upon the ice; it budged not even the slightest amount. I stepped further out and stomped again, listening. Not even the barest hint of a cracking sound.

Emboldened, I walked out even further, marveling at the new perspectives the journey gave me, dragging my boots through the powdery flakes like Neil Armstrong kicking up lunar regolith. The ice held my childhood weight without deflecting so much as a millimeter, so I kept going - all the way to the train tracks.

I stood there, under the overpass, my heart racing as a train swooped by overhead, rumbling, shaking the bridge's pilings, shivering the ice. Looking back, then, I saw my footprints connected by scuff marks, exclamation points in the snow.

It was about then that I realized that even if I made it back without being seen, the evidence of my passing would tattle on me just as sure as Wayne Gordon did whenever I did something at school that I wasn't supposed to - which was, as I've noted, more often than not.

So, with the logic of a preteen or early-teens intrepid Arctic explorer, I guess I figured, "Well, I'm in for it now. I might as well make it worth the punishment." So on my way back, I paused to scuff a single word in the snow, a word that I was sure would be visible from someone standing on the edge of the bluff behind my house.

And then I went home and faced the second-worst punishment of my childhood (exceeded only by what happened to me when I nearly set the house on fire; my father had built the house himself, and he was not amused).

Oddly enough, I don't remember exactly what the punishment was. There were so many, and they all blended together. My father was old-school and believed in corporeal punishment; my mom had subscribed to the newfangled "take away something they care about" realm of childhood correction. Neither of these methods, it turns out, is especially good for developing well-adjusted adults. The former leads to people who think they can use violence to get what they want (an attitude that I had to fight against for a long time), and the latter provoked me to never show love for anything, because it could be taken away if I did the slightest thing wrong. When I did something really bad, I got it from both sides. I'm sure that Mom, at the very least, revoked my TV privileges, which at the time was a harsh punishment, but I later thanked her for.

But Dad? His displeasure with me for endangering myself was exceeded only by his towering anger over the message I'd scrawled in the snow.

Until this incident, I don't think he even realized that I knew the F-word.

Momma loves her baby
And daddy loves you too
And the sea may look warm to you babe
And the sky may look blue
Ooh baby
Ooh baby blue
Ooh babe

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