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:

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').

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.

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

1 Like

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

September 18, 2023 at 10:32am

#1055909

Idol Speculation

Wood isn't known for its long-term durability. Smithsonian notes an exception.

This Wooden Sculpture Is Twice as Old as Stonehenge and the Pyramids
New findings about the 12,500-year-old Shigir Idol have major implications for the study of prehistory

Some environments preserve wood better than others, though:

Gold prospectors first discovered the so-called Shigir Idol at the bottom of a peat bog in Russia’s Ural mountain range in 1890.

For our purposes, I'm going to call this carved hunk of tree St. Peat.

The unique object—a nine-foot-tall totem pole composed of ten wooden fragments carved with expressive faces, eyes and limbs and decorated with geometric patterns—represents the oldest known surviving work of wooden ritual art in the world.

I'm wondering how they determined it was ritualistic and not just, you know, ars gratia artis. Was it like "Oh, our ancestors were all cowardly and superstitious, and whatever they did, they did to appease the gods and/or spirits?" Because they definitely weren't cowardly (couldn't be), and what religion they did have probably wasn't regarded as superstition any more than you regard your religion as superstition.

Based on extensive analysis, Terberger’s team now estimates that the object was likely crafted about 12,500 years ago, at the end of the Last Ice Age.

Technically, by some accounts, we're still in an Ice Age, just one that's in retreat.

“The landscape changed, and the art—figurative designs and naturalistic animals painted in caves and carved in rock—did, too, perhaps as a way to help people come to grips with the challenging environments they encountered.”

Or perhaps because art changes over time. Maybe not as quickly as today's deliberate art "movements," but then, as now, people get bored and/or inspired and make art. I've heard that hunter/gatherer societies had lots of free time, more than us civilized folks.

The debate has major implications for the study of prehistory, which tends to emphasize a Western-centric view of human development.

I imagine it's hard to make definitive conclusions about a society from one lone artifact, but that's no excuse to let one's preconceptions fill in the gaps.

Prevailing views over the past century, adds Terberger, regarded hunter-gatherers as “inferior to early agrarian communities emerging at that time in the Levant. At the same time, the archaeological evidence from the Urals and Siberia was underestimated and neglected.”

H-Gs aren't "inferior." Just different. Don't get me wrong; I like civilization. But it does have its downsides.

João Zilhão, a scholar at the University of Barcelona who was not involved in the study, tells the Times that the artifact’s remarkable survival reminds scientists of an important truth: that a lack of evidence of ancient art doesn’t mean it never existed. Rather, many ancient people created art objects out of perishable materials that could not withstand the test of time and were therefore left out of the archaeological record.

Much is made of cave art, and for good reason: it's a window into the thoughts of humans (and related species) of the past. But I find it difficult to believe that they confined their paintings to cave walls; that's just where the art would be best preserved. Artists today (using the term very broadly) would leave their mark everywhere, if they could; and some do. For all we know, every stone was covered in graffiti, every tree carved, every cliffside covered in murals.

We're not so different from the people who carved St. Peat, in other words: creative, curious, aware of our mortality, at least moderately intelligent, social, communicative. Sure, we likely have different priorities in life ("make money" instead of "look out for tigers," e.g.), and we, or at least most of us, have more knowledge about the world around us. But as I keep saying, don't conflate knowledge with intelligence.

And don't be so sure that a 12,500-year-old work of art was what you interpret it to be.

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