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:

tj-Merry Mischief Maker

about 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

a day ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

November 29, 2024 at 9:43am

#1080625

Shroom with a View

Several years ago, I did an entry on the Stoned Ape Hypothesis, and expressed great skepticism: "Expanded Consciousness"

I promptly forgot all about it, until this Big Think article pinged my radar.

A new spin on the “Stoned Ape Hypothesis”
The controversial theory about magic mushrooms and human evolution gets a much-needed update.

One might wonder (fairly) why I even give this attention if I dismiss it so readily. After all, I'm not here repeatedly sharing flat-Earth links, right? Well, it's different because, for one thing, it's not completely falsified the way flat-Earth doctrine is; for another, talking about it might help normalize the use of psychedelics.

In the realm of human evolution, few theories have captured the public imagination quite like the “Stoned Ape Hypothesis.”

It also might increase understanding of evolution in general, even if this particular hypothesis turns out to be a truckload of manure.

Originally proposed by ethnobotanist Terence McKenna in his 1992 book Food of the Gods, this provocative idea has recently resurged in popular discourse, thanks in large part to its discussion on Joe Rogan’s widely followed podcast.

Well, now I'm even less inclined to believe it.

However, matching the enthusiasm for the theory is the skepticism that opposes it, and critics have branded it “pseudoscience,” successfully demoting it from a legitimate scientific hypothesis to fringe status.

I'm not going that far. But I still haven't seen any real evidence.

Since most academics approve of this characterization, I’ve long felt motivated to “steelman” McKenna’s theory, which I think will prove to be more right than wrong.

Okay, fair enough. Your opinion, man.

The article goes on to do just that, and it's easy enough to follow.

McKenna’s highly amusing and admittedly speculative answer to the puzzle was that psychedelic substances helped spark the rapid evolution in human cognition, consciousness, and culture. According to his story, our early hominid ancestors would have inevitably encountered psychedelic fungi while foraging for food in locations like the African savanna. The psilocybin in these mushrooms would have provided adaptive advantages to those who consumed them, including enhanced cognition, creativity, and elevated states of consciousness.

Okay, so, what happened to the other species who consumed them? Because I would find it even harder to believe that it was only our ancestors who ate magic mushrooms. Did it have an effect on the antelope? The zebra? The... whatever the hell other foraging species roamed Africa at the same time?

Or maybe it only works on primates? Well, plenty of primates lived in places with shrooms, and we don't see them doing rocket science or writing novels.

For evolutionary theorists, this sounded too close to Lamarckism, the idea that acquired traits could be passed down to offspring, a theory that fell out of fashion with the emergence of Darwin’s theory of natural selection.

Which is exactly what I said in my earlier entry, but of course, I'm not a biologist. Still, I understand there's some leeway for heritability of certain acquired traits. This is called epigenetics (or so I'm told).

McKenna, though, had more than a few answers to these criticisms, which makes the theory difficult to judge as flat-out right or wrong, since some of his explanations could be more or less correct.

You don't get to just push a theory out there and expect us to judge it as "right" or "wrong." Like, I hereby theorize that there's life on Pluto. You can't prove me wrong, so it's a legitimate theory, right? No. No, it is not.

One promising alternative explanation, which you could say represents the “status quo alternative” to McKenna’s theory, is that social and cultural factors played a unique role, such that increasing social complexity created a natural selection pressure that strongly favored intelligence over physical attributes.

Looking around, I find the idea that, in humans, intelligence can be favored over physical attributes, almost as unlikely as the magic mushroom hypothesis.

...why would psychedelics then mostly disappear from our diet, rather than being a regular part of our contemporary lives, the way a drug like caffeine is?

Now, that right there is cultural bias. Other cultures incorporate, or used to incorporate before missionaries came along, psychedelics into their sacred rituals. (In the author's defense, he does acknowledge some of these instances later in the article.)

I realize that my statement there works in favor of Stoned Ape. That's okay. Skepticism doesn't mean outright rejection.

According to the New Stoned Ape Theory, psychedelics likely served as a “chemical catalyst” for a special kind of “cognitive-cultural phase transition,” characterized by a shift in perspective at the individual level that propagates through culture (“goes viral”) and restructures the worldview of society, bringing about a transition at the societal level.

Which, looking back at my earlier entry, I acknowledged as a possibility that I could accept (given evidence). I quote Younger Me: "And maybe - just maybe - I can see psychedelics being an engine for social evolution."

The article is fairly long, as BT articles tend to be. I'm not going to critique each claim, though there's plenty to critique. I'll just point out one other quote, one that claims to sum things up:

To summarize the theory in a sentence: Psychedelics, as “worldview shifters,” can create a cognitive phase transition whose spread creates a social phase transition — a shift in culture. It’s that simple!

I'm a big fan of Occam's Razor, but when it comes to evolution and human cognition, I reflexively distrust anything that's "that simple."

Which, again, doesn't mean it's wrong.

The article kind of undercuts itself at the end by proposing something even weirder and more speculative, but I'm not going to weigh in on that except to say that it sounds like the ramblings of someone who just ate mushrooms.

Which is fine. There's plenty of actual evidence that hallucinogens can, under certain circumstances, be beneficial. I just think the whole thing needs more science.

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