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

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

November 8, 2022 at 12:01am

#1040395

The Incredible Shrinking Brain

Full moon tonight. Also a lunar eclipse, but one that's too late to stay up for and definitely too early to wake up for.

So instead, we get to read about the Incredible Shrinking Brain.

No, the human brain did not shrink 3,000 years ago: research

Given the state of the world right now, I would think "thirty years ago" rather than "three thousand years ago."

Did the 12th century B.C.E.—a time when humans were forging great empires and developing new forms of written text—coincide with an evolutionary reduction in brain size?

Well, it's not in the headline; it's the lede. The answer is still "no."

Think again, says a UNLV-led team of researchers who refute a hypothesis that's growing increasingly popular among the science community.

One thing I should note: brain size isn't strongly correlated with intelligence. If it were, elephants would be ruling the planet. While I'm convinced they're intelligent, it's not like they've invented the internet, sliced bread, or beer.

Last year, a group of scientists made headlines when they concluded that the human brain shrank during the transition to modern urban societies about 3,000 years ago because, they said, our ancestors' ability to store information externally in social groups decreased our need to maintain large brains.

That's... not really how evolution works, anyway.

Their hypothesis, which explored decades-old ideas on the evolutionary reduction of modern human brain size, was based on a comparison to evolutionary patterns seen in ant colonies.

"What is this, a study for ants?"

That by itself would raise a bunch of red flags for me.

"We re-examined the dataset from DeSilva et al. and found that human brain size has not changed in 30,000 years, and probably not in 300,000 years," Villmoare said. "In fact, based on this dataset, we can identify no reduction in brain size in modern humans over any time-period since the origins of our species."

The date for the appearance of what's called "anatomically modern" humans, from what I've read, is about 100,000 years ago.

The UNLV team says the rise of agriculture and complex societies occurred at different times around the globe—meaning there should be variation in timing of skull changes seen in different populations. However, DeSilva's dataset sampled only 23 crania from the timeframe critical to the brain shrinkage hypothesis and lumped together specimens from locations including England, China, Mali, and Algeria.

The only information we have on human brains from that long ago is cranial capacity, which puts an upper limit on brain size. But, again, the relationship between brain size and intelligence isn't absolute.

I'm linking this article because, while I didn't see reports on the original study (the one that claimed brain size reduction 3,000 years ago), apparently, lots of people did. And when a study like that comes out, it's natural for people to want to share it, especially if it fits in with their worldview. In this case, it would be the (demonstrably false) worldview that we're stupid and never should have invented agriculture or civilization. Or possibly that we shouldn't be relying on what I like to call "auxiliary memory" (my smartphone) or we'll become even stupider. (Incidentally, there's a picture making the rounds purporting to show a misshapen human with a cell-phone-holding claw and a crooked neck and calling it "the future evolution of humankind;" that's also utter bullshit.)

It's like when a nutrition scientist says "dark chocolate is good for you," and people crow about it while stuffing their faces with Godiva, and then never hear that the study was suspect because it was funded by Willy Wonka.

So I'm here setting the record straight to the best of my ability. Now you can be That Person at the party who, upon hearing someone confidently proclaim that our ancestors' brains shrank 3,000 years ago, can go, "Well, ackshually..."

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