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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 17 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 19 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.

March 5, 2020 at 12:02am

#977166

Moonlighting

Today's entry is for:

Journalistic Intentions

(18+)
This is for the journal keeping types that come to PLAY! New round starts July 1!
#2213121 by Turkey DrumStik

What is a "reactive visualizer?"

That was my first question, as I haven't encountered these videos before. The YouTube description is mute (see what I did there) on the subject, and I long ago trained myself to never look at YouTube comments.

And even after Googling, I'm still not sure. It looks like a "Piano Hero" attachment that drops the notes that you're supposed to strike?

But that was after watching the video. During, I was too mesmerized to care much about the technology.

Speaking of technology...

It turns out I have a fondness for things that take things from the past and give them a new twist.

It could be argued that just putting Beethoven on the internet does that, or giving out mp3s of singing monks, or whatever. But the medium itself is irrelevant; if it's just some dude sitting at a piano and playing a sonata, well, that's been the same for a few centuries now.

I once saw a video - I'd link it, but I don't want to overdo the embeds, so you'll just have to look for it if you're interested - of a rendition of (speaking of ol' Ludwig) Ode to Joy. Now, of all of the overdone Beethoven pieces, that's the overdonest, but it's arguably the second most recognizable tune from that composer (the first being, of course, the triumphant FUCK YOU of the opening notes of the Fifth). But this one had a twist. No, a few twists. First of all, it was an all-Japanese orchestra. If memory serves, it was also all-female, but don't make me swear to that because I can't be arsed to find it again now. Second, it was an all-theremin

orchestra. And finally, they played a version that was in Ragtime style.

Bottom line: a Japanese orchestra playing the music of an Austrian composer using 20th-century Russian technology, in an American style.

Or - and this one's my favorite - a mashup of electric guitar covers of one of the earliest known pieces of sheet music, Pachelbel's Canon in D.

To stray from music and into theater, I recently saw a video of a stage production of Much Ado About Nothing with David Tennant and Catherine Tate. That's right - The DoctorDonna, on stage, doing Shakespeare. It's glorious. Go find it.

Point being that modernization isn't a bad thing -- provided you know where the shit comes from in the first place.

As far as I'm concerned, this is why the internet is great. Well, that and cat videos. And maybe porn. No... cat videos.

Anyway, back to the video up there at the top.

I'd also never heard of Rousseau. Don't look at me like that; I appreciate classical music, but my heart belongs to rock. Some say it's made of rock, and who am I to argue? So I looked him up, briefly, but all I saw is a bunch of other reactive visualizer videos. Which is fine; I'll probably see more of them at some point, especially now that YouTube's algorithm has surely decided to inundate me with suggestions for other Rousseau videos. That's fine. It's something different from my usual suggestions of math / science explanations, rock and roll, cat antics, and My Little P- I mean, er, cooking vids.

And now I see that one of my suggestions is a video entitled "Does Quantum Immortality Save Schrödinger's Cat?" I haven't clicked on it yet, but I'm hoping it has a cookin' soundtrack.

3 Like • Comment • View 2 comments

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