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 6 hours 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

about 8 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.

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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

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.

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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').

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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

Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

May 2, 2021 at 12:02am

#1009516

Making Sense of It All

PROMPT May 2nd

What smell or sounds brings back great memories of your childhood?

Come to think of it, I haven't heard a tyrannosaur's roar in quite some time, now...

But seriously, though. When I think "childhood," I can't help but remember school. Not that memories of school are all that great. I tend to remember the other kids being mean, teachers being nasty, and principals not understanding the entire concept of "humor." A typical conversation would be like:

Principal: "Why did you put a frog in Suzie's bag?"

Me: "...because it was funny."

(So, okay, it wasn't only the other kids who were mean. In my defense, Suzie was very cute and I was just trying to get her attention.)

Principal: "It was not funny."

Me: "Yes it was."

Principal: *sigh* *opens the Drawer and pulls out a paddle*

Yes, I may not remember dinosaurs, but I do remember, very vividly, that it was perfectly okay to whack someone's kid on the ass. I don't have any real opinion on the practice now, although I certainly wasn't a fan of it at the time.

My dad was fond of paddling. My mom was more progressive on the subject of discipline, opting instead for less physical punishments. I'm not sure that was any kind of improvement. Her favorite thing to do was to take away something I enjoyed. A paddling was something that was, generally, over with fairly quickly, and then I could go back to reading comic books. But if I got the comic books taken away, I was never sure when (or if) I'd get them back, so I'd have to, I don't know, do homework instead.

What I learned was: never show any interest in anything, because it could be taken away at the slightest infraction.

But I digress. Those are not great memories (even if they do make me chuckle nowadays). In between stints of playing practical jokes on my fellow students, I actually learned a few things in school as a kid. One of the things I learned, along with everyone else, was that we have five senses: sight, hearing, smell, touch, and taste.

As with many of the things you learn in school, this is, at best, an oversimplification; and, at worst, a falsehood.

We have other senses. The sensation of pain, for example, like when a paddle hits your ass. Sure, touch is involved, at first, for an instant, but the pain remains long after the paddling is over with. Or you could sprain your ankle or eat something gross on a dare, which would tie your guts into knots. And you can sense, say, the heat coming off of a stove without actually touching the stove. There's the sense of balance, and the sense of proprioception (which, for example, lets you know where your hand is even when it's behind your back). And don't forget the sense of humor, which is lacking in elementary school principals.

You could argue that all of those are extensions of the sense of touch, but if you want to go there, I can argue that the other four "classic" senses are also, at base, touch: sight is photons touching your retinas; hearing is pressure waves touching your eardrums; smell is molecules touching stuff in your nose; taste is similar molecules touching your tongue.

I don't know when or how we collectively decided that we only have five senses, but we did proclaim this as fact, same as we collectively decided there were only seven colors (that bit, I traced back to Newton, who decided on the number seven for mystical reasons).

What does appear to be true, though, is that of all of the senses (classical or otherwise), it's smell that is most powerful in evoking memories. And you know, science has pretty much figured out why. This article

discusses it, for anyone who's interested.

Smells have a stronger link to memory and emotion than any of the other senses, and neuroscience may know the reason why.

One of my favorite scents is the aroma of a fine single-malt scotch. That shit is aromatherapy for me. If I'm stressed, all I have to do is open a bottle and take a whiff. But, you know, since the bottle is open anyway, and the glass is right there in front of me, well, I might as well experience the taste as well. My sense of balance is a small price to pay for the delight of scent and flavor.

When you see, hear, touch, or taste something, that sensory information first heads to the thalamus, which acts as your brain's relay station. The thalamus then sends that information to the relevant brain areas, including the hippocampus, which is responsible for memory, and the amygdala, which does the emotional processing.

But with smells, it's different. Scents bypass the thalamus and go straight to the brain's smell center, known as the olfactory bulb. The olfactory bulb is directly connected to the amygdala and hippocampus, which might explain why the smell of something can so immediately trigger a detailed memory or even intense emotion.

Scotch, of course, wasn't a big part of my childhood. No, I don't come from a family of drunks; I got that way all on my own. There are certain odors that do trigger memories of being a kid. Like the smell of bullshit. No, really, literal bullshit; I grew upspent my childhood on a farm, and while we didn't have cattle, a neighboring farm did.

Still, I can't say that's a happy memory. One of the first things you learn as a kid on a farm is that if there is one single creature with even less of a sense of humor than a principal, it's a bull. Oh, sure, they look all placid and ruminatory, but if you even look at one of them sideways, they impart important lessons about pain.

But hay.

No, really, I mean hay, as in dried grass or clover. Slightly musty, a bit sharp, and somewhat sad because it's dead -- and yet promising, because it exists to keep horses alive. And bulls, but offering hay to a mad bull doesn't work the way you might think it would. Yes, the fine scent of hay will inevitably remind me of my childhood in fields of alfalfa. And those are mostly pleasant memories. Freshly mowed grass comes close, but it's not really the same thing; that odor is more suburban and rigid.

I can't say I miss being on a farm. Turns out I'm allergic to work, and farms are the Platonic ideal of "work." But I do have some few fond memories of it, which I generally recall if I'm driving through the countryside in the summer or fall with the window open.

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