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 2 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 4 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.

December 14, 2021 at 12:03am

#1023186

A Gift Beyond Price, Almost Free

Here's one from The New Yorker on why what you're doing is Bad and You Should Feel Bad.

The Hidden Costs of Streaming Music

Economics 101: There is no such thing as a free lunch.

Listening to music on the Internet feels clean, efficient, environmentally virtuous.

Alternatively, it feels like a lot of work and, while it has the advantage of breadth, sometimes finding a particular song can be a pain. And just try doing this while driving across country, through places with no service.

And I never thought of it as particularly environmentally virtuous.

Instead of accumulating heaps of vinyl or plastic, we unpocket our sleek devices and pluck tunes from the ether.

Kind of like radio, only without inane DJ chatter.

The ostensibly frictionless nature of online listening has other hidden or overlooked costs.

Not many things don't have such costs.

Exploitative regimes of labor enable the production of smartphone and computer components.

And everything else.

Conditions at Foxconn factories in China have long been notorious; recent reports suggest that the brutally abused Uighur minority has been pressed into the production of Apple devices.

And yet, no one cares enough to do anything about it.

Child laborers are involved in the mining of cobalt, which is used in iPhone batteries.

Gotta find some kind of use for kids.

Spotify, the dominant streaming service, needs huge quantities of energy to power its servers.

Huge compared to what? A house? A city? A Bitcoin mine?

No less problematic are the streaming services’ own exploitative practices, including their notoriously stingy royalty payments to working musicians.

And yet, again, we put up with it.

But Devine isn’t interested in inducing guilt; he simply wants us to become more aware of the materiality of music. He writes, “There is a highly intoxicating form of mystification at work in the ideology of musical culture more generally.” As a result, music is “seen as a special pursuit that somehow transcends the conditions of its production.” Devine’s critical history of recording formats throws a necessary wrench into that mythology of musical purity.

Yes, this guy's promoting his book. Fine. Anyway, I'm pretty sure Neal Peart said it better:

One likes to believe in the freedom of music
But glittering prizes and endless compromises
Shatter the illusion of integrity, yeah

And that song ("The Spirit of Radio") came out in 1980, when the height of music technology was the cassette player, you either bought LPs or tapes because CDs weren't a thing yet, and the internet was barely a spark in ARPA's eyes. Point being, it's not a new observation.

The linked article goes in to some of said history, and for once for this publication, it's succinct and worth reading.

“Musically, we may need to question our expectations of infinite access and infinite storage,” he writes. Our demand that all of musical history should be available at the touch of a finger has become gluttonous. It may seem a harmless form of consumer desire, but it leaves real scars on the face of the Earth.

Some sacrifices are going to be necessary, sure -- purposeful or otherwise. But no.

You can't have the Apocalypse without a soundtrack. It just isn't done.

And since I quoted the song above, here it is -- also to demonstrate that yeah, there is indeed "magic at your fingers," even today. For now.

Off on your way, hit the open road
There is magic at your fingers
For the Spirit ever lingers
Undemanding contact in your happy solitude

3 Like • Comment • View 2 comments

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