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!

a day 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

a day 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.

October 26, 2019 at 12:01am

#968432

Born in the USA

I pick these links at random from a fairly long list of potential things to talk about. Today's link fortuitously coincides with the release of Western Stars, the Springsteen concert movie. But it's not about new music - it's about something that hit the record stores (for such things existed then) half of Bruce's lifetime ago.

https://www.npr.org/2019/03/26/706566556/bruce-springsteen-born-in-the-usa-ameri...

What Does 'Born In The U.S.A.' Really Mean?

In the summer of '84 I was working long, arduous hours, outdoors, as a surveyor's assistant. This is probably where I got my utter hatred of unenclosed spaces from; it certainly wasn't the farm I spent my childhood on. Getting covered with poison ivy and stung by angry yellow jackets can do that to a kid.

But that was the summer that Born in the USA was released, and I was a Springsteen fan even then. I thought about asking my boss for the day off so I could buy the album - in LP format, of course - but he wasn't the type who would understand. In one of the many lucky coincidences involving me and Springsteen-related things, though, it rained that day so I wasn't working after all.

So I went to the shopping mall (80s, remember), bought the album, and went home to listen to the thing. That's how we did things back then, pre-internet.

If you're listening closely, the lyrics of "Born in the U.S.A." make its subject pretty clear: The 1984 hit by Bruce Springsteen describes a Vietnam War veteran who returns home to desperate circumstances and few options. Listen only to its surging refrain, though, and you could mistake it for an uncomplicated celebration of patriotism. You wouldn't be the only one.

This was about the time I first started realizing that most people don't listen to music the way I do. The big "hit" from that album was "Dancing in the Dark," a decent enough song, but all anyone else could talk about was the music video. I felt it was the weakest song on the album, though - everything else sounded like it was done in two or three takes, but that one song suffered from typical 80s overproduction. I found out later that I was right - the rest of the album was basically live.

But that asshole Reagan - he talked about the title track, all right. And got it entirely wrong.

By playing on the hope, Reagan seemed to overlook the despair.

That's a... generous way to put it.

"After it came out, I read all over the place that nobody knew what it was about," [Springsteen] said before performing "Born in the U.S.A" to a crowd in 1995. "I'm sure that everybody here tonight understood it. If not — if there were any misunderstandings out there — my mother thanks you, my father thanks you and my children thank you, because I've learned that that's where the money is."

See? Still a capitalist.

Maybe the meaning of "Born in the U.S.A." is the distance between the grim verses and the joyous chorus. It's the space between frustrating facts and fierce pride — the demand to push American reality a bit closer to our ideals.

The title of this blog is "Complex Numbers," with the mathematical definition front and center, but it's actually a multileveled play on words - "numbers" also refers to pieces of music, and I'm fascinated by the emergence of complexity from simplicity.

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