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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Write_Mikey_Write!

about an hour 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.

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Adherennium Plotting something

about 3 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.

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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.

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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.

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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.

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Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

May 4, 2023 at 1:06pm

#1049209

Lichen Thropy

In an effort to tie today's random article into today's sacred observance of Star Wars Day, I'll just quote our first introduction to the Force:

"The Force is what gives a Jedi his power. It's an energy field created by all living things. It surrounds us and penetrates us; it binds the galaxy together." -Obi-Wan Kenobi

Lichens and the Meaning of Life

Yeah, I know, it's kind of a stretch. But it relates to the unity of life, at least on our planet.

“When we try to pick out anything by itself, we find it hitched to everything else in the universe,” the great naturalist John Muir wrote in the middle of the nineteenth century.

Like I said. The Force.

Because of this delicate interconnectedness of life across time, space, and being, any littlest fragment of the universe can become a lens on the miraculous whole. Sometimes, it is the humblest life-forms that best intimate the majesty of life itself.

Take, for instance, lichens.

Not midichlorians.

Lichens — which are not to be confused with mosses — are some of Earth’s oldest life-forms: emissaries of the ocean gone terrestrial. For epochs, their exact nature was a mystery — until an improbable revolutionary illuminated that they are, in fact, part algae.

Also not to be confused with liches, the powerful undead that make for difficult final bosses in a D&D campaign. Or lycanthropes, also common in role-playing games.

In the final stretch of the nineteenth century, Peter Rabbit creator Beatrix Potter punctuated her writing and her painting with a series of experiments with spores, demonstrating that lichens — which Linnaeus considered the “poor peasants of the plant world” — are in fact not plants but a hybrid of fungi and algae: living reminders that the supreme vital force of life is not competition but interdependence, that we survive and thrive not through combat but through collaboration.

I will take this opportunity to emphasize that the person discovering this wasn't known as a biologist, but as a writer.

Okay, I'll stop with the pop culture references for now, and take this opportunity to talk about cooperation.

There's a popular interpretation of the theory of evolution that is often described as "survival of the fittest." This was not, as I think most people believe, a term coined by Charles Darwin, or even by a different biologist, but by an economist, Herbert Spencer.

The problem with this phrase is mostly that it's a vast oversimplification (though reportedly, Darwin didn't object to it) of an extraordinarily complicated topic. Now, I don't claim to be an expert on evolution or economics (and anyone claiming to be an expert on economics is automatically suspect to me), but stopping there, at that phrase, seems to lead to the conclusion that only the strong survive, and that evolution is all about competing for resources.

Certainly, there are elements of competition. But the primary driving force of evolution is, in my admittedly limited view, cooperation.

As an analogy, consider an American football game. Or, really, any team sport. You have two teams. They play a game and, usually, there's a winner and a loser. Sure, the teams are competing against each other—but within each team, it's only by cooperation that victory is even possible. Moreover, even the battling teams cooperate to an extent: they agree on a set of rules, and when a rule is broken, they agree on the penalties for doing so.

Without that cooperation, both within and between the teams, you don't have a football game. It is cooperation, not competition, that allows organisms to survive long enough to compete, and it obviously takes some cooperation for most organisms to reproduce.

Okay, enough of that. Back to lichens.

From the article, quoting biologist Lynn Margulis:

Like a farmer tending her apple trees and her field of corn, a lichen is a melding of lives. Once individuality dissolves, the scorecard of victors and victims makes little sense. Is corn oppressed? Does the farmer’s dependence on corn make her a victim? These questions are premised on a separation that does not exist. The heartbeat of humans and the flowering of domesticated plants are one life. “Alone” is not an option… Lichens add physical intimacy to this interdependence, fusing their bodies and intertwining the membranes of their cells, like cornstalks fused with the farmer, bound by evolution’s hand.

There is, of course, a lot more at the link. And it's not very technical; there's not even any math involved.

I'll just add one more thing, though: we all probably learned long ago that lichens are symbiotic organisms, usually consisting of an alga and a fungus. I remember that from high school biology. Most of this article takes this point of view, because that was the accepted science for most of the history of that field. I did see an article recently, though I'm having trouble finding it again, which stated that many lichens contain a third organism, a yeast. There's some discussion of the concept here.

So the yeast binds the other organisms together.

You know. Like the Force.

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