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:

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🎁

about 24 hours 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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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

December 12, 2024 at 9:23am

#1081071

Life Is

Another day, another book ad. But an interesting book ad, this one from Big Think. I promise something different tomorrow.

A bold challenge to the orthodox definition of life
In “Life As No One Knows It,” Sara Imari Walker explains why the key distinction between life and other kinds of “things” is how life uses information.

I'm not going to weigh in on whether she's right or wrong, or somewhere in between. That's above my pay grade (not that that's ever stopped me before). I do think it's an interesting approach that adds to the conversation of science, even if it's ultimately a categorization issue, like the planetary status of Pluto or the sea status of the Great Lakes.

Sara Imari Walker is not messing around. From the first lines of the physicist’s new book, Life As No One Knows It, she calls out some big-name public intellectuals for missing the boat on the ancient, fundamental question, “What Is Life?”

I'm not sure how ancient, or fundamental, that question really is. With regards to humans and other animals, our distant ancestors could pretty much figure out the difference between life and not-life. With plants, it may have been a bit trickier, as they tend to not move even when they're alive. But I think Jo Cavewoman would scoff at the question. Dog: life. Rock: not life. (Yes, I'm aware that belief in animism might counter what I just said, but I'm talking in generalities here.)

It probably took until we started looking through microscopes that we began to question the boundaries. Is a spermatozoon "life?" How about a virus?

Since then, it's my understanding that people have proposed several different definitions for life, all necessarily based on conditions on Earth, and scientists and philosophers have been arguing ever since, as scientists and philosophers love to do.

Subtitled The Physics of Life’s Emergence, one of the book’s major themes is a critique of the orthodox view in the physical sciences that life is an “epiphenomenon.”

"Epiphenomenon" is another word with a kind of slippery definition. I don't like to quote dictionaries as sources, because they're descriptive and not prescriptive, but the definition I found was "a secondary effect or byproduct that arises from but does not causally influence a process." Which, well, thanks? That doesn't help.

The Wikipedia article

on it is similarly confusing, at least to me, with the added bonus of also coming from a source people don't like to cite.

What's worse, in my view, is when people conflate "epiphenomenon" with "illusion:"

This is the argument, often heard in mainstream popular science, that life is a kind of illusion. It’s nothing special and fully explainable by way of atoms and their motions.

To address the latter assertion first: "nothing special" is a value judgement, and "fully explainable" is laughable hubris.

As for the "illusion" thing, well, I've banged on in here on several occasions against the "time is an illusion" declaration. But that can be generalized to anyone airily calling anything an "illusion." To me, an illusion is something that, upon further study, goes away: a stage magician's trick, or those seemingly moving lines in a popular optical illusion picture. But no matter how much we study, for instance, time, it doesn't go away. I think a better word description would be "emergent phenomenon," meaning that it's not fundamental, but rather a bulk property. Like temperature. One atom doesn't have a temperature; it only has a vibration or speed or... whatever. Get a bunch of atoms together, though, and the group has an average speed, which we read as temperature.

Or, to use everyone's favorite example, the chair you're sitting in. "It's an illusion," some philosophers claim (generally after taking a few bong hits). "It's not real." Well, look, any philosophy that doesn't start with "the chair is real" is a failure, in my view. Your ass isn't sinking through it; therefore, it's real. Sure, it's made of smaller pieces. On the macro scale, it's got a seat, probably a back (sometimes in one continuous piece), maybe arms, legs and/or casters, maybe a cushion for said ass. This doesn't make the chair any less real; it just means there's a deeper level to consider.

Similarly, the cushion, for example, is usually a fabric stretched over some stuffing. The fabric itself can be further broken down into individual fibers. The fibers, in turn, are made of molecules, some of which have a particular affinity for one another, giving the fiber some integrity. The molecules are made of atoms. The atoms contain electrons, protons, and neutrons. Those latter two, at least, can be further broken down until you're left with, basically, energy. And maybe there's something even more fundamental than that.

None of that makes the chair any less real. It just shows that our understanding can go deeper than surface reality. But surface reality is still reality.

And so it is with life. I know I'm alive, for now, and that's reality. I'm pretty sure my cats are, too, and the white deer I saw munching on leaves in my backyard yesterday. Not so sure about the leaves, it being December and all, but I am as certain as I can be of anything that they are a product of life.

Whew. Okay. Point is, I'd like to see these macro-level phenomena labeled something other than "illusion." It's misleading.

In the standard physics perspective on life, living systems are fully reducible to the atoms from which they are constructed.

Yeah, well, physics gonna physic. Just as with your chair, things can be studied at different scales. Biology is usually the science concerned with life. But biology is basically chemistry, and chemistry is basically physics. This doesn't make biology an illusion, either.

Still, they will argue, nothing fundamentally new is needed to explain life. If you had God’s computer you could, in principle, predict everything about life via those atoms and their laws.

I'm gonna deliberately misquote James T. Kirk here: "What does God need with a computer?"

Walker is not having any of this. For her, the key distinction between life and other kinds of “things” is the role of information.

Well, that's amusing. Not because it's not true—like I said, I'm not weighing in on that—but because from everything I've read, physics is moving toward the view that everything is, at base, information. Yes, that might be what energy can be broken down into. Or maybe not. I don't know. But "information theory" is a big deal in physics.

Whether there's something even more fundamental than information, I haven't heard.

Life needs information. It senses it, stores it, copies it, transmits it, and processes it. This insight is, for Walker, the way to understand those strange aspects of life like its ability to set its own goals and be a “self-creating and self-maintaining” agent.

Okay. Great. Let's see some science about it.

As usual, there's more at the link, if you're interested. Might want to sit down for it, though. You know. On that chair which is definitely real and hopefully not alive.

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