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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Writer_Mike

32 minutes ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 17 hours ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

about 20 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

2 Like

Aiva Raine

Sep 20, 2024 at 10:13pm
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.

July 11, 2023 at 9:47am

#1052470

If Only Things Were Different

I would accompany this article from The Conversation with a significant amount of sodium chloride, but it's still interesting speculation.

8 billion people: how different the world would look if Neanderthals had prevailed

In evolutionary terms, the human population has rocketed in seconds. The news that it has now reached 8 billion seems inexplicable when you think about our history.

No, it seems inevitable when you think about our history.

There's a graph at the link showing population over time. Unsurprisingly, it's a classic exponential hockey-stick.

Neanderthals were more inclined to stay in their family groups and were warier of new people. If they had outcompeted our own species (Homo sapiens), the density of population would likely be far lower.

For homework, explain why this is a fallacious argument.

Hell, I'll save you the trouble: to outcompete sapiens, they would have had to be more social, and thus increase their population density.

But okay, let's play pretend.

The reasons for our dramatic population growth may lie in the early days of Homo sapiens more than 100,000 years ago. Genetic and anatomical differences between us and extinct species such as Neanderthals made us more similar to domesticated animal species. Large herds of cows, for example, can better tolerate the stress of living in a small space together than their wild ancestors who lived in small groups, spaced apart. These genetic differences changed our attitudes to people outside our own group. We became more tolerant.

I know this is going to raise objections from the down-on-humans brigade, but remember, "more tolerant" doesn't have to mean "very tolerant." We still have in-groups and out-groups.

The article then goes into what it says in the headline, speculating (based on data, not wild guesses) how things might be different. I don't find this particular speculation useful, but it still sheds light on the differences between the two human species... assuming, of course, that inferences of their behavior based on archaeology and other disciplines are correct.

As an aside, "species" can be a fuzzy concept. Usually it's meant to describe non-interbreeding populations. But, clearly, sometimes sapiens and neanderthalensis did interbreed. The whole designation thing is over my head, but for general purposes, calling them different species is good enough for communication; I'll let biologists hammer out the details.

The more technology humankind develops, the more our use of it harms the planet. Intensive farming is draining our soils of nutrients, overfishing is wrecking the seas, and the greenhouse gases we release when we produce the products we now rely on are driving extreme weather.

And yet, despite discussing both population and technology, the article stops short of what, to me, is a blindingly obvious connection there.

The vast majority of human population increase has occurred over the last 200 years, roughly coinciding with the industrial revolution. But that's not a coincidence. Advances in science have increased average life spans and reduced infant mortality, while at the same time fouling the environment. On the surface, reduced mortality seems like a good thing, but the increase in population requires extraction and processing of ever more resources, feeding back onto the "fouling the environment" part. "Intensive farming" is needed to feed the 8+ billion. So is "overfishing." It's not "the more technology humankind develops, the more our use of it harms the planet;" it's "the more people there are, the more we need that technology to survive and make more people."

So far, the benefits of technology have outweighed the downsides. But I doubt that will persist.

I feel like the article veers way too close to romanticizing the Neanderthal, kind of a "noble savage" conceit. Sure, maybe the ecosystem would be in better shape had they been more successful, from an evolution perspective, than we were, but that's like speculating about whether dinosaurs would have ended up building rockets if that asteroid hadn't fucked things up for them.

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