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!

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.

1 Like

Adherennium Plotting something

a day 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.

May 2, 2019 at 12:38am

#958035

The Lamp

Remember bringing something from home for “Show and Tell” at school when we were little? Today, I want you to do the same in your blog. Pick an object that means something to you and describe it. What does it remind you of? How did the object come into your life? Does it bring you comfort? What is the story behind the object?

I'm pretty sure we never had Show and Tell. I mean, maybe we did and I've just blocked that memory, as we do with many traumatic experiences - and getting up in front of a room full of, say, second-graders is, was, and always will be traumatic. Or maybe I can no more remember it than I can remember what I had for lunch on, say, June 23, 2002.

It was probably pizza, though. The lunch, that is, not the Show & Tell. I ate a lot of pizza in the noughties.

Still, I'm familiar with the concept from books and TV and such. There are myriad objects that I could use, since I'm a bit of a collector (which is, I suppose, a synonym for "hoarder," only I don't keep old pizza boxes around) and I have many dead older relatives. There's my comic book collection, for instance. Or my father's navigational sextant - he was a sailor back when a sextant was the way you figured out your position at sea. Or my mom's beaver fur coat that she never wore because it was a beaver fur coat; there's a story there, let me tell you.

But instead, I think I'll talk about an antique torchiere.

It wasn't always antique, of course. I think my grandmother bought it new. I'm not sure what era it's from, but it could be a hundred years old or more. Sitting on a black marble, oval base, bronze scrollwork supports a fluted cylinder that, at just under head height, flares into more bronze scrollwork that cradles a thick, molded, frosted glass bowl. Inside the bowl, of course, sits a lightbulb socket. But not just any lightbulb socket; this one is a discontinued size that is much larger in diameter than the sockets of today.

Originally, there were two of these lamps in my family. My parents had one, and my Aunt A in New York had the other. My other aunt, Aunt E, possessed a very different esthetic and was married to a minimalist style artist, so it wouldn't have fit with her other furniture.

Mind you, it doesn't fit with my furniture, either, but that's never stopped me before. At some point, you go from "nothing fits" to "what the hell, everything fits."

Now, Aunt A was what you'd know in literature as the "maiden aunt" archetype. She was a feminist before feminism was cool, so clearly, she was also an early hipster. She lived in a townhome in Queens with her brother, my uncle, who - well, that's a whole 'nother story; I just want you to imagine that it's the end of WWII, you're a Jewish soldier in the Army, you get sent to the European theater, and you've heard rumors about what the Germans have been doing in their "camps," but you don't want to believe it, because how could people do that to other people, and then you end up being among the troops that discover Dachau.

Suffice it to say that my uncle couldn't live on his own, so his sister took care of him.

So the one torchiere illuminated their house in Queens, while the other brightened the rural farmhouse where I spent my childhood in Virginia.

Now, a bit more background, if you can stomach it. My parents were absolutely committed to each other, but like the lamp, they were from another time. Their design seems out of place, now, but it was all I knew as a kid. What this came down to was that they often fought - and sometimes, these fights entered the physical realm. So it came to pass that during one of these altercations, the torchiere, which was never very stable to begin with, toppled to the floor and the ancient frosted glass bowl shattered into dozens of pieces.

My father, ever practical, kept using the lamp, but with the bare bulb it just wasn't the same anymore - a harsh, glaring light as opposed to the soft, indirect glow of the intact torchiere.

My uncle died in the early 90s; my mom, a few years later. Then my dad in the late noughties, and I have no idea what happened to the lamp that lived in my house. But when Aunt A died a couple years after my dad, Aunt E took me to the house in Queens for one last look at the townhome that shaped the "city" part of my childhood.

"Is there anything here that you want?" she asked.

I looked around at the odd mixture of ancient and modern, at the furniture that I knew so well from multiple visits. I had very little room, and most of the stuff, I figured, should go to my cousins, who have kids to pass anything important on to. But I knew that there was one thing - only one thing - that I needed, and that was the torchiere.

They insisted that I take a few other things, as well, such as the letters home that my uncle sent from the War, which included a record of his shock and dismay at finding out the truth about everything. One of these days I'm going to see if a Holocaust museum wants them, but first I'd have to actually look at them, and the one time I did that... well, I can't describe the feeling, really, except to say that it reminded me of the shattering of the glass bowl of my parents' torchiere.

So I schlepped that lamp back to Virginia in my car, taking care to protect the precious glass; and now it stands, incongruous, in my dining room. I've never plugged it in. I've never been able to be arsed to find the nonstandard bulb that would allow it to shine once more. But it's there, an eternal reminder of family.

Every time I think that people believe I'm somehow a lesser person for being single and never having had kids, I look at that lamp - that intact torchiere - and I remember that there are worse fates.

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