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

about 10 hours 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.

1 Like

QPdoll

a day ago
In response to "Landmarks"

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

2 Like

Wordsmitty ✍️

a day 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

Sep 21, 2024 at 7:34am
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

Sep 21, 2024 at 6:46am
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

Sep 21, 2024 at 6:16am
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.

February 23, 2023 at 11:17am

#1045405

Alt-Earth

I wanted to share this one from Aeon. Not because I completely agree with it, though it does raise interesting points.

There’s no planet B
The scientific evidence is clear: the only celestial body that can support us is the one we evolved with. Here’s why

"Earth first! We'll strip-mine the other planets later."

The essay begins with what could be a decent elevator pitch for a science fiction series:

We built powerful astronomical telescopes to search the skies for planets resembling our own, and very quickly found hundreds of Earth twins orbiting distant stars. Our home was not so unique after all. The universe is full of Earths!

And that is science-fiction, make no mistake. You might have heard that we have discovered earth-size, or even earth-like, planets out there. The former is true. The latter is not. The kind of telescope it would take to find planets that are definitely capable of supporting our kind of life would be much bigger and much more advanced than the JWST.

This futuristic dream-like scenario is being sold to us as a real scientific possibility, with billionaires planning to move humanity to Mars in the near future.

Well, that's a non-sequitur. Getting to exoplanets is billions of times harder than getting to our solar system neighbor, which is, by an astronomer's definition, earthlike... but not capable of supporting our kind of life.

Many of the highest-grossing films are set on fictional planets, with paid advisors keeping the science ‘realistic’.

That should read "with paid advisors who are almost completely ignored to service plot and the rule of cool." There's nothing inherently wrong with servicing plot and making things cool, but we should all be aware that this is what's happening.

I know a guy, an astronomer, who got paid to be a science advisor on a minor TV series. Very first episode, they got the astronomy wrong. And I don't mean "mistaking a Wolf-Rayet star for a hypernova" or anything like that, but "being able to see Sirius from Boston in the summer." Basic stuff that anyone who cares to can look up and disprove.

And that's not even getting into the stupid cliffhanger they ended the series with before it got canceled.

At the same time, narratives of humans trying to survive on a post-apocalyptic Earth have also become mainstream.

To be fair, those can be fun.

Given all our technological advances, it’s tempting to believe we are approaching an age of interplanetary colonisation. But can we really leave Earth and all our worries behind?

No.

No.

See?

Astronomical observations and Earth’s geological record are clear: the only planet that can support us is the one we evolved with. There is no plan B. There is no planet B. Our future is here, and it doesn’t have to mean we’re doomed.

No, it doesn't have to.

But it does.

There is no planet B. These days, everyone is throwing around this catchy slogan.

Yes, and I wish they'd stop. Not because they're wrong, but because it's a pun. A pun loses all impact when it's repeated.

However, no one actually explains why there isn’t another planet we could live on, even though the evidence from Earth sciences and astronomy is clear.

Maybe because the explanation wouldn't fit on a placard, in a tweet, or even in a blog entry hastily done so I can go back to playing Pathfinder:Kingmaker. Or even in an Aeon essay.

While human-led missions to Mars seem likely in the coming decades, what are our prospects of long-term habitation on Mars?

I'm all for space exploration, and visiting other worlds. With our current technology and near-future prospects, though, terraforming is an opium dream (literally, in the case of some science fiction authors). Which is not to say it can never be within our means, but not before Earth becomes an uninhabitable shithole. Which we'll have to terraform.

The essay goes into some technical reasons why we can't make the surface of Mars habitable, and they track with my prior knowledge, but I won't paste it here. I'll just say that they didn't address one of the major barriers: radiation, which our magnetic field protects us from.

How many Earth-sized, temperate planets are there in our galaxy? Since we have discovered only a handful of these planets so far, it is still quite difficult to estimate their number.

This is complicated by the limitations of our observations. For one thing, most exoplanets are found because their orbital plane happens to be edge-on to our view, and the planet transits the star from our perspective. This severely limits what we can actually find. For another thing, a lot of small planets can be missed.

The essay then attempts to predict, with this limited data, how many potential alternative worlds we have out there. I'm not faulting the math, but it is a wild-ass guess... though much less of a wild-ass guess than before we started finding evidence of exoplanets.

And then there's a long discussion about how we wouldn't have been able to survive on Earth for the vast majority of its existence. Again, worth reading, but I'm not copying it here.

I'll just quote one more part:

Earth is the home we know and love not because it is Earth-sized and temperate. No, we call this planet our home thanks to its billion-year-old relationship with life. Just as people are shaped not only by their genetics, but by their culture and relationships with others, planets are shaped by the living organisms that emerge and thrive on them.

And to me, that's the crux of the issue. Science fiction (a genre I love) can make the assumption that evolution on other planets works parallel to ours, and that we could fit right in to different ecosystems.

But this is obviously not the case.

Consequently, we're doomed.

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