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 15 hours ago
In response to "Landmarks"

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

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Wordsmitty ✍️

about 18 hours ago
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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

a day 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
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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
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"

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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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August 9, 2023 at 10:26am

#1053868

Planet Ix

The random number gods bring us back to the solar system today, but this time we're talking about a planet that only exists in educated speculation.

The Hunt for Planet Nine
What will it take to find the biggest missing object in our solar system?

Before Pluto achieved dwarf planet status, that would have been Planet Ten. I will admit, Planet X sounds better than Planet IX. But maybe I'll start calling it Planet Ix.

At 9,200 feet, there is 20 percent less oxygen than at sea level, enough to take all the air from my lungs after just three steps.

Is it bad that I understood immediately why one would open an astronomy article with this sentence? I had to walk very slowly when I was at the peak of Haleakala, where a bunch of telescopes lurk.

This wasn't Maui, though, but that other island. And the actual telescope is a couple thousand feet higher.

We were at base camp on the dormant volcano Mauna Kea, on the big Island of Hawaii. The pair were here to use one of the most powerful telescopes in the world, called Subaru.

Which has nothing to do with the car company; both take their names from what we call the Pleiades.

Tomorrow night, December 3, marked the start of their sixth observing run and their next attempt to find the biggest missing object in our solar system, called — for the moment — Planet Nine.

Yeah, I'm gonna call it Ix.

The article is from Longreads, so... well, it's right there in the name. I won't be quoting a lot more of it, but I wanted to highlight a few relevant sections.

As a theorist Batygin feels that he’s already mathematically proven its existence. But it’s generally accepted that for a planet to be considered discovered in the field of astronomy, the theory must also be accompanied by a photograph.

I know I've pointed this out somewhere before, but a while back, a planet inside Mercury's orbit was "mathematically proven." They figured one had to be there based on perturbations in Mercury's orbits. Some people even claimed to have images of it, taken during eclipses or whatever. So they called it Vulcan, after the forge god (not to be confused with, well, you know). You think it was bad when Pluto was designated a dwarf planet? Imagine renumbering all the planets. Earth becomes Planet 4, for example.

Anyway, short version, there was no Vulcan; the images were false and Mercury's orbit was fully explained by Einstein.

I'm not saying the data pointing to Planet Ix is nothing but gravitational anomalies, but there's still a lot we don't understand. Which is why we do science.

Part of their job was first to try to find a solution less extreme— like a passing star or a galactic anomaly — than a giant undiscovered planet far off in the depths of the solar system, because, a hidden planet? That was absurd.

See, now, to me, a passing star or a galactic anomaly would be more weird than a hitherto unobserved planet. But that's why they get paid the big bucks.

(I'm joking; astronomers don't get paid the big bucks.)

After the observing run was complete, I asked the pair if they ever felt that trying to find Planet Nine was ridiculous, that the whole notion of a giant missing planet and the efforts they have gone to to find it ever make them feel defeated. They both gave me roughly the same response: no. Their answer brought to mind the French philosopher and writer Albert Camus. He thought a lot about the myth of Sisyphus and plucked his unfortunate mythical backstory away from the root of his actions, the eternal task of pushing a boulder up a mountain only to watch it fall back down again.

Right? Because the telescope is located at the summit of a mountain, and the astronomers are like Sisyphus, climbing and descending the mountain over and over again.

Like I said, a long read, and a lot of it is, pun intended, atmosphere. But I found it fascinating. Life is barely possible at the summit of that volcano; it has, as the article notes, barely enough oxygen to survive, let alone move around and do stuff.

When I went up Haleakala—from which you can see Mauna Kea off in the distance—I'd brought a bottle of Coke with me. When I finished it, up at the summit, I twisted the top back on and left it in the car, because I didn't want to add to any trash at the peak. After enjoying the cold, thin air for a couple of hours, we headed back down the mountain. Back near sea level, in Kehei, I went to toss the bottle into a bin, and saw that, despite the cap being tightly screwed on, it had crumpled, as if squeezed by a fist; this was physical proof of the vast difference in air pressure. And Haleakala is "only" at about 10,000 feet.

My point here is that outside of a thin membrane of habitability, proportionally thinner than an eggshell, the Universe is actively trying to kill us. And yet we continue to push that boundary to learn everything we can about this vast, uncaring expanse.

Whether we find proof of Planet Ix or not, there's something to be said for the lengths we'll go to in order to try.

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