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!

about 8 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 10 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.

June 8, 2021 at 12:01am

#1011473

Out of the Valley

Sometimes you just have to issue a correction. Well, not me, obviously. But other people.

Neandertals don't deserve their bad, dim-witted reputation
Our hominin ancestor had bigger brains and probably went extinct with climate change. Who are we to judge?

Just kidding, of course. When I'm wrong about something, I'll admit it when I find out otherwise. But in this case, it's a matter of science doing what science does: being self-correcting.

Unfortunately, lots of people stick with what they first hear and then ignore any evidence to the contrary. Well, here's some contrary evidence.

On March 3, 2021, the governors of Texas and Mississippi announced that they were lifting their respective mask mandates, prompting criticism from President Biden, who called the move "Neanderthal thinking." Biden was implying that lifting the mandates was a primitive act — but this understanding of Neandertals is an outdated stereotype, unsupported by modern research.

That, by the way, is not an invitation to argue about politics or politicians.

Neandertals (Homo neanderthalensis) are an extinct species of hominin — the taxonomic name for a tribe that includes living and extinct humans and their ancestors. They lived throughout Eurasia up until about 40,000 years ago. One of the earliest Neandertal fossils was discovered in the Neander Valley in Germany, which is where the species gets its name. Interestingly, the contemporary word for “valley” in German is “tal,” so anthropologists tend to call the species “Neandertal,” not Neanderthal (dropping the 'h').

Completely irrelevant aside: Apparently, there were once very productive silver mines in another German-named valley, Joachimsthal (using the older "th" construction). Coins minted from there were called "Joachimsthalers" or "thalers." This got Anglicized as "dollars." Or, well, I may have some of the details wrong, but that's the gist of how "dollars" got their name.

This is one of the things I'll correct later if I'm wrong, but I can't be arsed to look it up again now. I'm planning on going to see Cruella on Thursday, and I'm busy delving into the source material (Dalmatians movies), so my time is somewhat limited. As far as I can tell, though, the "dal" in "dalmatian" has no linguistic connection to tal, thal, or dollars. Pounds, maybe, if you're Cruella.

If, as research suggests, Neandertals behaved similarly to humans, we either need to rethink our definition of human, or update our definition of Neandertal.

Given that they're all gone, except for traces of genetic material in most of us "sapiens," this may be a moot point.

Where did the brutish stereotype that plagues Neandertals originate? This association can be traced back to 1911, when French paleontologist Marcellin Boule described a newly discovered Neandertal fossil from the La Chapelle-aux-Saints site in France. Boule wrote that the fossil had primitive characteristics, including a hunched posture, big, divergent toes, and large brow ridges which, he argued, signaled a lack of intelligence.

I'm betting that was a conclusion born of phrenology. Science has had a few dead ends, and phrenology was one of them. Unfortunately, as with homeopathy, its effects linger.

Boule's original interpretation of Neandertals was based on an incorrect reconstruction of their skeletons.

But hey, you gotta start somewhere, right?

The wider build and shorter limbs common in Neandertal skeletons has also been seen as primitive compared to early humans, though researchers now argue these were adaptations to a colder climate.

"Primitive" is an unfortunate word. They were well-adapted to their environment, until they weren't. Calling them primitive is like calling a lion primitive compared to a tiger.

It's clear that incorrect early reports shaped assumptions about the behavioral and anatomical nature of Neandertals.

The brutish, sloped-forhead caveman stereotype, usually wearing skins and carrying a club, persists in cartoons to this day. Because it's a convenient shortcut for pre-agricultural humanity (or whatever), it's going to be hard to shake that image.

Dinosaurs once faced a similar issue: throughout the first half of the twentieth century, the general public believed dinosaurs were sluggish and cold-blooded. It wasn't until the 1960s that a small scientific revolution, sparked by new discoveries and led by paleontologist John Ostrom, suggested that dinosaurs were probably active, warm-blooded animals.

Like I said, science is self-correcting. Eventually.

Perhaps the Neandertals need their own revolution — a Neandertal rebirth, if you will — to help the general and scientific community understand that they were a sophisticated, cultured species who made art, had some similar anatomical features to humans, and who went extinct due to a changing climate, not inferiority.

Yeah, look, I get that the author is trying to make a point about climate change. Thing is, there weren't all that many Neandertals (comparatively speaking) in the first place, and obviously there was some interbreeding going on with Sapiens, who at this point number in the billions. Whatever your personal perception is of climate change, it's unlikely that we'll go extinct because of it. Which is not to say that things won't take a major turn for the worse.

In any case, I'm mostly presenting this in my long-standing crusade to counter outdated popular interpretations of science, not to argue about climate change or politics.

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