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 21 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 23 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.

January 27, 2020 at 12:06am

#974282

Round Here

PROMPT January 27

Invalid Photo #1056988

Discuss a time in your life when someone has tried to "fix" or "solve" a problem for you - but you didn't see it as a problem in the first place.

How do you generally handle unsolicited opinions/advice?

My ex-wife, shortly after she moved in: "You have too many books."

Me: How is that even possible?

I still miss the books she "helped" me dispose of. More than I miss her. To be fair, though, I once had a mole removed that I miss more than I miss her. Don't get me wrong, though; we're on speaking terms - we just don't have anything to speak about anymore.

Before I met her, I dated a woman I'll call B, because she doesn't have any Bs in her name and I'd rather keep people anonymous. We didn't date for very long, and after we broke up we stayed friends. Yes, it really does happen. We were still friends when my ex dumped me, so we started hanging out again - not dating, though. One time, B and I were in a store and for some reason I had the munchies [Narrator: It was weed] and wanted to buy a Rice Krispies Treat - you know, one of those horrid amalgamations of Rice Krispies cereal and melted marshmallows.

"Don't buy that crap," said B, or something very similar. "I'll get the ingredients and we'll make some."

Now, one of the reasons I got along with B [Narrator: apart from the fact that she always had good weed] was that there are few people in this world who are worse housekeepers than I am, and she was one of them, so after buying the ingredients, we went to my place to make it because her kitchen was unusable. She made the things and, in her defense, they really are better than the store-bought kind. But then, instead of a single Treat and a discarded wrapper, I ended up with an empty box of Rice Krispies, half a bag of evil puffy marshmallows, a lingering scent of cooked sugar in the house, two dirty cake pans, a bunch of dirty dishes, utensils, and the pot wherein the nasty white things were melted, and a refrigerator full of Rice Krispies Treats that it would take me a few weeks to eat before I decided the remainder had gone stale and dumped the leftovers.

I mean, seriously, I have to have a serious case of the munchies to eat those things, and that just doesn't happen very often.

Result: A lot more work, a lot more time, a lot more calories, a lot more money spent. Still, I remember that day fondly; we had a good time.

Remember a bunch of entries ago when I realized that I was being stupid about the Tilex or whatever? Well, I guess I'm not alone there. Regardless, I'll always listen to advice. Ever since the Tilex Incident, especially, I'm open to the possibility that I'm Just Doing It Wrong, whatever "It" is. Doesn't mean I'll always follow it, but at least I'll give it some thought.

Still, B ended up moving to the precise opposite end of the country from me [Narrator: weed is legal there] and I miss having her around. Almost as much as I miss my books. At least B and I text every now and then. Today's Complex Number always reminds me of her.

"This is..the first way Counting Crows ever sounded, it was me and Dave in bars and coffee houses playing open mics, doing this song this way. The song begins with a guy walking out the front door of his house, and leaving behind this woman . But the more he begins to leave people behind in his life, the more he feels like he's leaving himself behind as well. The less and less substantial he feels like he's becoming to himself. And that's sorta what the song's about because he feels that even as he disappears from the lives of people, he's disappearing more and more from his own life. The chorus is, he sorta keeps screaming out these idioms these lessons that your mother might say to you when you were a kid, sorta child lessons ya know, "round here we always stand up straight", "carving out our names". Things that you are told when you are a kid that you do these things that.. that when you're grown up it'll add up to something, you'll have a job, you'll have a life. I think for me and the character of the song they don't add up to anything it's just a bunch of crap kinda. Your life comes to you or doesn't come to you but those things don't really mean anything. By the end of the song he's so dismayed by this that he's kinda screaming out that he can stay up as long as he wants and that no one makes him wait...the sort of things that are important if you are a kid. You know that you don't have to go to bed, you don't have to do anything. The sorta of things that don't make any difference at all when you're an adult, they're nothing. And uh and uh this is a song about, about me."

- Adam Duritz; Counting Crows

Round here we're carving out our names
Round here we all look the same
Round here we talk just like lions, but we sacrifice like lambs
Round here she's slipping through my hands

Oh sleeping children better run like the wind,
Out of the lightning dream
Mama's little baby better get herself in out of the lightning

She says "it's only in my head."
She says "shh, I know it's only in my head."

But the girl on the car in the parking lot says
"man, you should try to take a
Shot. can't you see my walls are crumblin'"
Then she looks up at the building
And says she's thinkin' of jumping. she says
She's tired of life, she must be tired of something.

Round here she's always on my mind
Round here (hey man)i got lots of time
Round here we're never sent to bed early and nobody makes us wait
Round here we stay up very, very, very, very late...

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