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 14 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 16 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.

April 25, 2020 at 12:19am

#981982

Expanded Consciousness

I'm just going to leave this here for your amusement.

https://www.inverse.com/article/34186-stoned-ape-hypothesis

The 'Stoned Ape' Theory Might Explain Our Extraordinary Evolution
A scientist resurfaces a psychedelic retelling of human evolution.

"Might?" It "might" also be utter twaddle.

In 1992, ethnobotanist and psychedelics advocate Terence McKenna argued in the book Food of the Gods that what enabled Homo erectus to evolve into Homo sapiens was its encounter with magic mushrooms and psilocybin, the psychedelic compound within them, on that evolutionary journey.

Hell of a trip, man.

He called this the Stoned Ape Hypothesis.

Because of course he did.

As early humans, he said we “ate our way to higher consciousness” by consuming these mushrooms, which, he hypothesized, grew out of animal manure. Psilocybin, he said, brought us “out of the animal mind and into the world of articulated speech and imagination.”

Granted, I'm no expert on biology, but that sounds awfully Lamarckian to me - heritability of acquired traits, which has even less support than the Stoned Ape Hypothesis itself.

And, because psilocybin mushrooms commonly grow in cow droppings, “the human-mushroom interspecies codependency was enhanced and deepened. It was at this time that religious ritual, calendar making, and natural magic came into their own.”

Take enough drugs and you, too, can confuse correlation with causation.

“What is really important for you to understand is that there was a sudden doubling of the human brain 200,000 years ago. From an evolutionary point of view, that’s an extraordinary expansion. And there is no explanation for this sudden increase in the human brain.”

So we're going to make one up that supports our confirmation bias.

“I suggest to you that Dennis and Terence were right on,” Stamets announced while acknowledging that the hypothesis was perhaps still unprovable. “I want you or anyone listening, or seeing this, to suspend your disbelief … I think this is a very, very plausible hypothesis for the sudden evolution of Homo sapiens from our primate relatives.”

And hey, science is full of hypotheses that started out ridiculous and ended up being... well, mostly true. Really, this one's no weirder than quantum theory. The difference is, quantum theory has experimental support, and makes predictions that can be, and have been, tested. Find a way to do that, and I'll have a nice steaming dish of crow.

McKenna’s theory chalks up the entirety of this complicated phenomenon to a single spark; to him, psilocybin mushrooms were the “evolutionary catalyst” that sparked consciousness by prompting early humans to engage in experiences like sex, community bonding, and spirituality.

Sex has been around since at least the beginnings of eukaryotic life. Community bonding is practiced in many different species. Our nearest cousins, the bonobos, are really, really good at these two things. As for spirituality, there are other explanations.

And yet, they’re equally stumped when asked to answer the question at the root of the debate over the Stoned Ape hypothesis and consciousness research in general: How did consciousness evolve? If it wasn’t psychedelic mushrooms that started the process, then what did?

That's a fair question, and it's good to have competing hypotheses for an answer. But, again... evidence.

If the science behind McKenna’s hypothesis is unstable, what worth does it have in the search for the origin of human consciousness?

I think they're asking the wrong question here. Look, I'm not coming down against the use of mind-altering drugs; that would be hypocritical (even though the ones discussed here are not my cup of mushroom soup). There's plenty of evidence that psychedelics, used carefully, can be beneficial. Maybe they can even expand an individual's consciousness, like the hype promotes. Let's not confuse personal growth with species evolution, though.

And maybe - just maybe - I can see psychedelics being an engine for social evolution. That is, some stoner decides something like, "Hey, we're all ONE, man," and passes that along to his or her fellow apes, who nod in agreement and then proceed to beat each other up over the last roll of toilet paper. But at least they feel sorry about that afterwards.

But in terms of species evolution? It's like the idea that if both of a child's parents had each lost their left pinky fingers in an accident, the kid would pop out with nine fingers. Lamarckian.

However, McKenna deserves credit for sparking an idea in the 1990s that scientists have only recently been able to prove: Psilocybin does alter consciousness and can trigger physical changes in the brain.

Not arguing against that.

The Stoned Ape hypothesis may now be lost to the annals of fringe science, but some remnant of its legacy remains. Now that scientists better understand the way psilocybin physically affects the brain, they can seriously investigate its potential to treat disorders like substance abuse, anxiety, and depression. If that happens — and it looks like it will — psilocybin will become a part of mainstream culture as an agent of positive change. And isn’t that ultimately what McKenna was advocating for?

So the article ends up in the same place I did. This sounds like beneficial research, but it shouldn't be sullied by association with ideas that, at best, are unprovable, and at worst, are made of the same stuff fom which magic mushrooms grow.

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