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

about 9 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 12 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.

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Annette in France

about 23 hours ago
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Bienvenue en France.

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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

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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

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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

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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/

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WakeUpAndLive️~Happiness

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

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

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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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November 28, 2023 at 12:58am

#1060236

Upward Spiral

I'll be on the road for some unknown amount of time tomorrow [edit: I meant today. I'm writing this before I go to sleep, so tomorrow is today and today is totally tomorrow, while yesterday is but a fading memory as usual], stopping who knows where, so I figured I'd get in this last entry before traveling.

Speaking of traveling, here are some places none of us will ever go (probably):

In the Milky Way’s Stars, a History of Violence
Our galaxy's stars keep a record of its past. By reading those stories, astronomers are learning more about how the Milky Way came to be — and about the galaxy we live in today.

The amazing thing is that we can figure any of this out at all. It's been barely a century since we learned that there's more to the universe than our familiar galaxy, the discovery of which leads the linked article.

Late in the evening of October 5, 1923, Edwin Hubble sat at the eyepiece of the Hooker telescope at the Mount Wilson Observatory, atop the mountains overlooking the Los Angeles basin.

Visiting that telescope was the highlight of my trip to L.A. last year.

One might think, "What use is a telescope in L.A.?" Well, one, it's on a pretty high mountain and, two, I don't think the area was quite as smoggy or light-polluted a hundred years ago.

With the cleaving of the cosmos into a home galaxy and a larger universe, the study of our finite home — and how it exists within that universe — could begin in earnest.

I mean, sure, it was a paradigm shift much like the heliocentric model of the solar system centuries earlier, and a Big Deal, but it's not like we hadn't been studying the stars.

The latest results, amassed over the past four years, are now painting a picture of our home as a unique place, at a unique time.

To recap, the history of our thinking about our place in the universe went something like:

1) We're special and everything revolves around us.
2) We're not that special and not the center of the universe.
3) The universe is unimaginably big, and we're a rounding error.
4) Wait, maybe we are special.

We have been lucky, it seems, to live near a particularly quiet star on the calm fringes of a middle-aged, oddly tilted, loosely spiraling galaxy that has been largely left alone for most of its existence.

As regular readers know, I'm not discounting the power of luck, but if those conditions hadn't been relatively stable, we wouldn't be here to congratulate ourselves on our luck.

From the Earth’s surface — if you are somewhere very dark — you can only see the bright stripe of the Milky Way’s galactic disk, edge-on. But the galaxy we live in is so much more complicated.

It's getting harder and harder to find a place where you can see that. Oh, incidentally, it's called the Milky Way because of its appearance to the eye: a pale streak, like a road ("way.") It's translated from Latin's "via lactea," which, like a lot of things in Rome, came from Greek: galaxías kýklos. Hence "galaxy." Point being that the word, now applied to collections of stars all across the universe, shares a root with "milk" words like lactose and lactate.

I mention this simply because I find it amusing, if horribly mammal-centric.

A supermassive black hole churns at its center, surrounded by the “bulge,” a knot of stars containing some of the galaxy’s oldest stellar denizens.

Good to know it's not just middle-aged people who develop a "bulge."

Another digression: black holes have a scary reputation, like they're going to eat everything around them. And they do, but only up to a certain distance. Past that distance, they work just like any other massive object; you could orbit a black hole for a very long time, in the same way the Earth has been orbiting the sun for a very long time without showing any signs of falling in.

Next comes the “thin disk” — the structure we can see — where most of the Milky Way’s stars, including the sun, are partitioned into gargantuan spiraling arms.

One thing I've never been really clear on is how the spiral arms formed or are maintained.

While the article doesn't shed much light (pun intended) on that, it does go into the latest observations and theories about our galaxy's structure and how it got here.

And it's all very fascinating, at least to me, but I don't have much to comment on the rest of the article except to note that, given all the paradigm shifts we've already seen in cosmology, I'd expect more. And I look forward to it.

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