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 9 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 10 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.

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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.

February 18, 2022 at 12:03am

#1026940

The Light

Back home, tired, but I have time to do a prompt from "Journalistic Intentions"

[18+]:

"If I learn one lesson,
count your blessings,
look to the rising sun,
and run run run."

And all I know from the prompt is that it's in the "Lyrics" category.

Is that first couplet supposed to rhyme? Because it doesn't. Not unless you're a Southerner and you force "blessings" to become "blessin" with maybe a secret "s" subvocalized at the end.

I get that songs don't always have meaningful lyrics. Even some songs I like. Hell, when Springsteen was starting out, he'd just sit on his bed with a guitar and a rhyming dictionary, then fit a bunch of other words into the rhythm. Well... mostly into the rhythm. He was much better at rhyme than rhythm.

Even so, he ended up with something resembling poetry:

Madman drummers bummers and Indians in the summer with a teenage diplomat
In the dumps with the mumps as the adolescent pumps his way into his hat
With a boulder on my shoulder feelin' kinda older I tripped the merry-go-round
With this very unpleasing sneezing and wheezing the calliope crashed to the ground

Internal rhymes abound in this, and there's even connections between the lines: "teenage" and "adolescent" in the first couplet; "merry-go-round" and "calliope" in the second.

Nonsense? Sure. But nonsense with some internal consistency.

Those lyrics are from "Blinded by the Light," which was the first song on his first released album, so it was what introduced Bruce to a weary world; the rest is... the opposite of silence. Even today, some people are under the misguided impression that the song was by Manfred Mann. Nothing against that version, but it was, indeed, a cover, and lacked Springsteen's frenetic energy. (Comments about the pronunciation of "cut loose like a deuce" therein will be summarily ignored.)

That doggerel in the prompt, though? Let's just say I hope that whatever song it's from has other redeeming qualities. A decent bass line, maybe, or at least a melody, one that doesn't all hover around one note like so many of today's songs do. Meaningful lyrics apart from those lines. Something.

The temptation to look up the lyrics is real; I might be ragging on a performer I actually like. Well, if so, so be it. Even Bruce had some dogs in the wolf pack. But I'm not going to do it. Not until I'm done here. And I'm not done.

"If I learn one lesson" -- Oh, surely some lyrical wisdom is to follow; let's keep listening...
"Count your blessings" -- Or maybe not; my mom used to use that line on Teen Me in a failed attempt to make him less angsty.
"Look to the rising sun" -- Mama always told me not to look into the sights of the sun.*
"And run, run, run" -- Um. Couldn't think of two other syllables besides those first two instances of "run?"

*Also a line from Blinded by the Light

Perhaps I simply still harbor resentment over having my really quite very real teen drama complaints (none of which I now remember or care about) minimized, so anything that includes the line "count your blessings" just triggers me. If so, oh well.

Also just to be clear, it doesn't matter to me how old or new a song is. Sure, a lot of new music sucks. A lot of old music sucked. You know why people don't often rag on the old music that sucked? Because people got tired of it, it went away, and they forgot all about it.

So. If I learn one lesson, use a chord progression; make the lyrics be fun, and don't stare at the sun.

...okay, so that's three lessons. Whatever.

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