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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Brandiwyn🎶

a day ago
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').

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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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

October 18, 2023 at 10:23am

#1057617

Rock Music

Mostly, I just think this is cool, so I'm sharing it. (From BBC)

What did Stonehenge sound like?
New research into the prehistoric site's acoustical properties is revealing that the stone circle may have been used for exclusive ceremonies.

While fascinating, I'll note that there's still a lot of educated speculation in here.

Through the doors of a university building, down a concrete hallway and inside a foam-covered room stands a shin-high replica of one of the most mysterious monuments ever built: Stonehenge.

Shin-high replica of Stonehenge? Did they borrow it from This Is Spinal Tap?

"We know that the acoustics of places influence how you use them, so understanding the sound of a prehistoric site is an important part of the archaeology," said Trevor Cox, professor and acoustics researcher at the University of Salford in Manchester.

As long as the theory is sound.

(You're goddamn right pun intended, and I'm WAY more proud of that one than I have any right to be.)

Despite being the world's best-known and most architecturally sophisticated ancient stone circle, archaeologists still don't know who built Stonehenge or what it was used for.

We know who built it: People. That's right; not aliens. Probably.

Thanks to Cox's recent studies, however, we now know a fascinating detail about one of the world's most enigmatic sites: it once acted as a giant echo chamber, amplifying sounds made inside the circle to those standing within, but shielding noise from those standing outside the circle.

Which is cool and all, but did the ancient builders do that on purpose? If so, how, without a scientific theory of sound, did they know? Well, it's very likely that Stonehenge didn't spring suddenly out of nowhere; I'd bet money that earlier henges were made out of material slightly easier to obtain, transport, and build with, such as wood. So, I'd guess (but it's only a guess) trial and error.

Unless, of course, it was aliens.

Once the stones were painted grey and arranged in the correct distribution according to the computer model, the challenges of the testing process began.

Another thing I wonder is: why gray? (Look, they use British spelling and I use US spelling.) Does color somehow influence the acoustic properties? I'd have guessed it would be more about material and texture than color, which just goes to show that my guesses are just guesses.

Through mathematical processing, Cox was able to create a computer model that simulates the acoustic properties of Stonehenge and can distort voices or music to give a sense of what they would sound like within the circle. The results surprised him: although Stonehenge has no roof or floor, sound bounces between the gaps in the stones and lingers within the space. In acoustics, lingering sound is known as reverberation.

Which musicians use to great effect, mostly electronically these days.

These results showed that Stonehenge would have allowed people inside the circle to hear each other quite well, while those outside would have been excluded from any ceremonies taking place. Cox's research adds to a growing body of evidence that Stonehenge may have been used for rituals reserved for a select few, with one study even pointing to the possibility of a hedge grown to shield the view from those not participating.

While this is sensible knowing what humans can be like (sometimes elitist), I wonder if the people excluded, if this speculation is true, were also the people who did the hard manual labor of cutting, moving, and erecting the stones. That, too, would be typical human behavior. Thus, probably not aliens.

Cox acknowledges that unanswered questions about the real Stonehenge make it difficult for him to draw definitive conclusions from his work with the scale model.

Like I said above: educated speculation.

But, in short, apparently people living in England have been rocking out for a lot longer than we thought.

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