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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 11 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 13 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.

June 30, 2020 at 12:23am

#986814

Ritual

Final entry for "Journalistic Intentions"

[18+]. I started June's blogging adventure with an entry, and I'm ending it with one. Also because 30DBC starts tomorrow, I'm going to wrap this one up.

11. You'll find love when you prepare the ritual correctly.
-Chibithulu (Alyssa)

Magic works.

I know, I know, "Waltz, you're full of shit talking about science and stuff and then proclaiming that magic works?"

Well, I'm also an atheist, but I assert that prayer works, too. It just doesn't matter who you pray to.

Obviously, I'm going to explain these assertions. Might want your beverage of choice for this one. And, at the end, you may still decide I'm full of shit. Or, you know, you can just skip to the starry bit below if you're only here to discover who won yesterday's Merit Badge mini-contest.

Most people, I think, associate the word "magic" with one of two things: stage illusions, or bending the universe to one's will.

The former is, as the name implies, illusory -- though I do enjoy a good magic performance -- and the latter is simply wrong. I'm not talking about stage magic here, though; just distinguishing it from the other kind.

In fantasy writing (oh hell, this would have made for a decent editorial, but I've already turned in this week's Fantasy newsletter), magic works something like this: someone casts a spell, the soundtrack blares out triumphant (if the character is a protagonist) or discordant (if they're an antagonist) music, the SFX team makes glowy, swirly lights, and (usually) whatever they're trying to do manifests itself in reality.

This is enjoyable for me, too, by the way. I do like a good fantasy story, especially if the writer keeps the magic system self-consistent. It's only later that I think, "But where did they get the energy to shoot that fireball / summon a demon from Hell / fly / talk to their dead ancestor / force someone to fall in love with them / whatever?"

But there are ancient (and modern) magical traditions here in the real world, though they're nowhere near as flashy. After all, we haven't yet figured out how to hack into the source code of the simulation we're living in, have we?

That's a joke. We're not living in a simulation, and even if we were, so what?

Thing is, no, you'll never be able to shoot fireballs, or do any of that other stuff. Not without technological assistance, and even then the demon-summoning thing is still questionable. There's nothing "magical" you can do to change the world around you, or the laws of physics.

But that doesn't mean you're powerless. We affect the world in other ways, mundane ways. We can affect how others see us by, for example, choosing to bathe instead of, you know... not bathe. Or put on make-up, or get a haircut, or dress a certain way. We can change our surroundings by rearranging furniture, cleaning our rooms (I really need to remember that bit someday), washing the dog.

Hell, some people have changed the world by merely writing.

So, no, there are no magical shortcuts to these things. We can't change things just by thinking about them; action is necessary. In that sense, magic is purely fantasy. Magic doesn't clean us; soap and water does. Magic doesn't transport us to faraway locales; feet, wheels, and wings do.

No, you can't change the world by just a thought, prayer, or ritual... but you can change yourself. Yoga, meditation, prayer, a ritual circle, whatever... these things can focus the mind, or reduce stress, or just give you a different perspective on things. The break from everyday routine alone could be enough to inspire you to finally write that novel, or psych you up to ask your boss for a raise, or whatever.

And to relate this back to the prompt, even if you could "make" someone else fall in love with you, well, that would be kinda evil, wouldn't it? Messing with someone's so-called free will is generally considered a Bad Guy thing (unless you're a Jedi, apparently). The stories about such things always end badly too: love potions, charm spells, whatever. There's never a Happily Ever After in such stories, and probably shouldn't be.

But just as you're more likely to find a date if you take a shower and comb your hair and maybe lose some weight and stop wearing socks with sandals, so too could modifying your own personality make it easier for someone to be attracted to you. And just as there is nothing wrong with grooming oneself, there is also nothing wrong with identifying the problem areas of your personality and working to change them. You're not messing with someone else's free will; you're only making yourself more attractive.

If doing so means drawing a hexagram and some runes on the floor, lighting some candles, and intoning some words, and that works for you, great. Or if it means talking to your invisible friend who happens to have created the universe, well, nothing wrong with it. Whatever works.

At least, in theory. None of it has worked for me yet. Maybe if I shaved occasionally?

Mini-Contest results!

Yesterday, I asked for words. Specifically, your favorite rare or obsolete words or expressions. As always, y'all didn't make it easy for me. But where would be the fun in that? I loved all the answers, and you'll all get another shot at this soon.

Elle - on hiatus

was the clear overachiever here, with five odd words in one comment (and a bonus sixth in another), so one Overachiever Merit Badge heading her way!

And yes, I'll find a way to do this even during 30DBC month. I've been trying to make Mondays Mini-Contest days, but we'll see where the prompts take us.

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