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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QPdoll

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I hope you're enjoying every minute in France! What an opportunity.

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With all that walking, I hope you stayed hydrated.

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Annette in France

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WakeUpAndLive️~Happiness

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Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
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WakeUpAndLive️~Happiness

a day ago
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WakeUpAndLive️~Happiness

a day ago
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Aiva Raine

a day ago
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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September 19, 2023 at 8:26am

#1055945

A Meaty Discussion

I wasn't aware there was still any doubt, here.

Animal magic: why intelligence isn’t just for humans
Meet the footballing bees, optimistic pigs and alien-like octopuses that are shaking up how we think about minds

Just a few gripes in the above before we get into the meaty bits: 1) Why muddy the waters by calling it "magic?" 2) As I'm sure we can all attest, intelligence isn't for all humans; 3) How do we know octopuses are alien-like, when we haven't met any aliens yet?

Hopefully the rest of the article, and the book it's pushing, isn't that sloppy.

Guardian link, so beware of British spellings. It's fairly long, so just a few highlights:

But what makes a pig optimistic? In 2010, researchers at Newcastle University showed that pigs reared in a pleasant, stimulating environment, with room to roam, plenty of straw, and “pig toys” to explore, show the optimistic response to the squeak significantly more often than pigs raised in a small, bleak, boring enclosure. In other words, if you want an optimistic pig, you must treat it not as pork but as a being with a mind, deserving the resources for a cognitively rich life.

I don't want an optimistic pig. I want tasty bacon. If treating it like Pig Royalty makes the bacon taste better, great.

We don’t, and probably never can, know what it feels like to be an optimistic pig. Objectively, there’s no reason to suppose that it feels like anything: that there is “something it is like” to be a pig, whether apparently happy or gloomy. Until rather recently, philosophers and scientists have been reluctant to grant a mind to any nonhuman entity.

Philosophically, we can't know what it feels like to be a different human, either. Oh, sure, we can take some guesses, and maybe even listen to them blather on (or read the blather) about this or that which makes them happy or angry or whatever—they seem to like that—but there's no way to really know what it's like to be them. Except that it probably sucks.

To René Descartes in the 17th century, and to behavioural psychologist BF Skinner in the 1950s, other animals were stimulus-response mechanisms that could be trained but lacked an inner life.

Perhaps we, too, are stimulus-response mechanisms, and our "inner life" is entirely illusory.

After all, as Charles Darwin pointed out, we all share an evolutionary heritage – and there is nothing in the evolutionary record to suggest that minds were a sudden innovation, let alone that such a thing occurred with the advent of humans.

This is true, but we didn't come by the "humans are special" philosophy by means of science, but through religion.

Consider the often maligned bird brain. Compared with bird neurodiversity, humans are a monoculture. Birds’ minds are scattered widely in mind-space, their differences and specialities tremendously varied. Some birds excel at navigation, others at learning complex songs or making elaborate nests.

Well, okay, but you can't compare {the set of all humans} with {the set of all birds}. Well, obviously, you can, but it would be like comparing the works of Beethoven with the entirety of EDM. Compare mammals with birds, or humans with ravens, or you have a categorization problem that confuses SAT-takers.

We award pride of place in the hierarchy of bird minds to tool-using species, especially corvids (crows, ravens, rooks). The most masterful of them is the New Caledonian crow of the south Pacific, which will design and store custom-made hooks for foraging, and even make tools with multiple parts. Among animals, great apes, dolphins, sea otters, elephants and octopuses are the only others known to use tools.

I have yet to hear of any nonhuman animal using a tool to make another tool. That, in my view, signals an ability to plan beyond the present and near future, and seems to be a key trait of humanity. Maybe it's happened and I just haven't heard of it.

Although animal communication can be subtle and complex, it’s generally thought that no animal besides a human uses symbolic communication, where one concept is represented by another, as it is in writing. None, that is, except perhaps the honeybee, which conveys information about a distant food source to its hive members by dancing.

Pretty sure symbolic communication is more widespread than that, but I'm no expert. Still, I doubt they grasp the concept of metaphor. Hell, half of humanity doesn't grasp the concept of metaphor.

The article goes on to describe octopus cognition, and it's fascinating too. But, lest all of this talk makes you want to become vegan (as for me, it just makes me want bacon):

Even this, however, might sound tame compared to the idea that plants have minds. Yet that proposition is no longer confined to the fringes of new-age belief; you can find it discussed (relatively) soberly in august scientific journals. There, it often goes by the name of “plant neurobiology” or, in a more extreme form, “biopsychism” – which supposes that every living being from bacteria up has sentience of a sort.

"Biopsychism" is at least more palatable to me than panpsychism, which I've ragged on in here before, at length. That's the unprovable and unfalsifiable philosophy that all matter has rudimentary consciousness.

We might not (and may never) agree about whether plants, fungi or bacteria have any kind of sentience, but they show enough attributes of cognition to warrant a place somewhere in this space. This perspective also promotes a calmer appraisal of artificial intelligence than the popular fevered fantasies about impending apocalypse at the hands of malevolent, soulless machines.

But those fantasies are fun.

Likewise, most of our fantasies about advanced alien intelligence suppose it to be like us but with better tech. That’s not just a sci-fi trope; the scientific search for extraterrestrial intelligence typically assumes that ET carves nature at the same joints as we do, recognising the same abstract laws of maths and physics. But the more we know about minds, the more we recognise that they conceptualise the world according to the possibilities they possess for sensing and intervening in it; nothing is inevitable.

There's good reason for this assumption: with it, we know what to look for. Again, though, I doubt the existence of technological aliens close enough for us to detect. We also have some idea how we might detect signs of any sort of life, including microbial, in the atmospheres of exoplanets... because we've studied our own biosphere. If we knew to look for different signals indicating a different biology (or technology), you bet we'd be looking for them, too.

Also, I didn't miss the inherent carnivorous metaphor in the above quoted bit: "carves nature at the same joints as we do." I find it amusing in a text that, while not explicitly pro-vegan, could certainly nudge a few people in that direction.

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