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 3 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.

1 Like

Adherennium Plotting something

about 5 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.

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.

September 8, 2020 at 12:02am

#992731

To Explore Strange New Worlds

Confidence is overrated.

PROMPT September 8th

What does confidence mean to you? Describe the traits of a confident person. Looking at the traits you listed, would you consider yourself a confident person?

It's interesting that this prompt is for September 8. Because September 8 is Star Trek Day -- the anniversary (54th this year) of the date of the airing of the first episode of Star Trek ("The Man Trap," an unforunately titled episode that was later serialized as "The Unreal McCoy.")

The reason it's interesting is that, as I have been a lifelong Trek fan, and having just binge-watched the entire Original Series, I can think of no better icon of "confidence" than Captain James T. Kirk.

Oh, sure, Picard. Fair enough. I'm not going to get into a debate over which one's a better captain of the Enterprise (besides, it's totally Christopher Pike, fight me). Why, I might as well debate who would win in a fair fight: The Hulk, or Superman (but it would depend on whether we're talking pre- or post-Crisis on Infinite Earths).

Ahem. Anyway. As I was saying, confidence is overrated.

See, in TV shows and movies, or books or plays or whatever, it's easy to write confidence, just like it's easier to write funny dialogue than it is to blast witty banter back and forth if the writer has days or weeks to plan out exactly what's going to be said. Similarly, if you're plotting a story, you know what's going to happen. You're writing it, after all. It's easy to portray your character as confident, or timid, or funny, or whatever attribute you choose for the character. Well. Not "easy," per se; most of us spend our entire lives trying to master the art of characterization.

But compared to being all confident and shit in real life? Cakewalk.

So, Kirk. Level-headed, first and foremost. Can contemplate imminent death without flinching. Calm. Competent - he knows how the Enterprise works, but, more importantly, he trusts and is trusted by his crew because he knows them at least as well as he knows his starship. Patient. Contemplative, but decisive once a decision point is reached.

All of those things are realistic enough, from what I've experienced. What sets Kirk apart from real-life people is that he almost always makes a decision that turns out to be the correct one -- and when he doesn't, it's an important plot point.

We schlubs toiling away in the Real World™ don't have that luxury.

There is, in life, a really remarkably blurry line between confidence and arrogance. I've heard it said that it usually doesn't matter what decision one makes, so long as one makes a decision. This is arrant nonsense. It's entirely possible to make an utterly disastrous decision with complete confidence. That doesn't stop the decision from being disastrous. But it does get you laid, at least, so there's that.

I mean, I'm sure that if you think hard enough, you can come up with a real-world example of a leader who is supremely confident... and almost always a disaster.

No, confidence is a trap. Just like everything else about movies and TV, the portrayal we see is idealized. You come to a decision point and you're like "What would Kirk do" or whatever, even if you don't have phasers, warp drive, or deflector shields. But you're working from incomplete information. You're always working from incomplete information, whereas a fictional character only has the appearance of working from incomplete information.

The important thing in a real-life situation isn't confidence. The important thing is being ready, able and willing to accept responsibility if things don't work out well after you've made the decision.

So... am I a confident person?

I sure as fuck hope not.

Anytime I'm supremely confident about something, that's when I find out I'm most misinformed. Hell, if you asked me "Do you think the sun will rise tomorrow," I'd have to answer, "Maybe. Probably. Might not. Small but nonzero chance it wont, in which case we're boned."

I don't trust confident people. Almost without exception, they're so full of themselves they don't have room for other opinions.

Speaking of Star Trek, in honor of Star Trek Day, I decided to confidently whip up a batch of Romulan Ale. This is the result:

In celebration of Star Trek Day, September 8, 2020

I had some guidance, but I mostly went my own way on it. Obviously, blue curacao figures prominently. The other ingredients will remain a strategic Empire secret, but suffice it to say there's a reason why this stuff is illegal in the Federation.

Maybe next year, I'll try to create some Saurian Brandy, or perhaps Klingon Bloodwine. Because if there's anything I have unearned confidence about, it's my mixological prowess.

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