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 8 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
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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

about 22 hours ago
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Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
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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"

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WakeUpAndLive️~Happiness

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

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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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November 18, 2023 at 8:49am

#1059681

Big Mistakes

Sometimes, science progresses because of happy accidents—scientists not intending to find something, but finding it anyway. Penicillin, for example, or X-rays.

But then there are the... not very happy accidents, for which Cracked has us covered.

5 Scientists Who Brought Unimaginable Horrors Upon Themselves

Obviously, the upside is, if you can crack the case, you’re likely to have your name etched into the history books. The downside is, depending on your particular area of expertise, that same mysterious field of research might be destroying your body in ways you haven’t even learned about yet.

Oh, sure, that's great comfort as you die a painful death from something we didn't even know existed.

The usual disclaimer: I'm too lazy to fact-check these, but they track with what I already knew. Don't get your science (or history) from a dick joke site alone. Or from me, for that matter.

5. Louis Slotin

Remember that "upside" I quoted above? Well, I didn't know this guy's name, and science is something of an interest of mine.

Louis Slotin was one of the world’s top experts on nuclear weapons and the associated risks, having been a contributor to the Manhattan Project.

And it's not as if I hadn't heard of other people working on that. No, I didn't go see Oppenheimer. But I doubt anyone's going to make a movie called Slotin.

Slotin was working with a 14-pound hunk of plutonium that’s since earned the nickname “the demon core” due to its death count.

I'd even heard of the "demon core."

Already, it sounds like something you’d make sure to have the proper equipment for. Instead, when Slotin lowered a beryllium tamper over the core, he elected to prop it up with a flathead screwdriver.

Now that I think of it, I have indeed heard mention of that particular incident. Just not Slotin's name. I believe he was referred to as "some idiot." After all, it's not like they didn't know plutonium was somewhat hazardous (that's an example of understatement), or what would generally happen if neutrons couldn't escape confinement.

Over the next nine days, Slotin died a slow and horrendous death from acute radiation poisoning, including radiation burns covering his internal organs.

Idiot or not, no one deserves that. Well, almost no one, but politicians don't generally handle plutonium. Hopefully, his demise was documented and added to the corpus of knowledge surrounding "how people die from radiation."

4. The Fluorine Martyrs

Nobody would be surprised that people messing around with plutonium might have ended up six feet under. But in this case, we’re talking about an element that, in some form, is added to most drinking water and toothpaste: fluorine.

Sigh.

There are enough people running around spouting unfounded conspiracy "theories" involving fluorine. The stuff in your toothpaste or water? Fluoride. Fluoride is a fluorine atom with an extra electron, usually borrowed from some atom on the other side of the periodic table to make an ionic bond. In that form, it's both safe and beneficial, in pretty much an analogous way as sodium chloride is necessary for our health, while pure chlorine gas is poison.

The linked article kind of comes at this sideways. I just don't like adding fuel to the conspiracy "theory" fires.

Thus, the nickname for those who were poisoned, blinded or killed by their work with fluorine: the “fluorine martyrs.”

Unlike "some idiot" above, the full extent of the potential hazard wasn't likely known. Hence, martyrs, not idiots.

3. Carl Scheele

So what’s the unfortunate bit? Well, Scheele famously had a particular penchant for a certain test on any substance that came his way. A procedure that you’ve probably performed on a carton of milk before: the sniff and taste test. Scheele treated his chemistry lab like a Baskin Robbins, and though the exact effect of his forbidden treats isn’t recorded, he likely had a less than stellar bill of health while cycling heavy metals through his stomach.

And yet, if you look up most any naturally-occurring element in reference material, you'll sometimes find notes about its smell and taste. Those aren't theoretical. To use the fluorine example, Britannica

states: "At room temperature fluorine is a faintly yellow gas with an irritating odour."

Somehow, that bit's left out of the Wikipedia entry, maybe because "irritating" is a bit of an understatement, much as how I described plutonium as "somewhat hazardous." Or maybe because Wiki gets more views, and they don't want the liability when someone inevitably goes, "Hm, I wonder if it really smells all that bad."

Point is, someone had to be a self-selected guinea pig for that kind of thing. Someone like Scheele.

2. Galileo

Beyond any doubt, the most famous person on this list. But he really doesn't belong here.

I'm sure everyone is aware of his offense: contradicting the Catholic Church's dogma. However... it wasn't exactly accidental, and his punishment wasn't "unimaginable horror" but, as the article notes, house arrest for life. Other Church victims fared far worse. Giordano Bruno,

e.g.

That’s like discovering hot sauce and being sentenced to eat nothing but unflavored oatmeal for the rest of your life. It’s also a pretty backwards punishment for someone that you’re worried is thinking too much: “Your ideas are too crazy and threatening! Go home and do nothing but think for the rest of your life!”

My parents, too, thought "go to your room!" was a legitimate punishment for me, an introvert, and I tried really hard not to disabuse them of this notion.

1. Thomas Midgley

This one is less pitiable and more infuriating, as the person in question didn’t experience any of the horrors he’d discovered. He died in 1944, probably with a smile on his face, thinking about how much everybody loved his excellent inventions.

I knew about this guy, too. The only defense I can muster for him is he couldn't have been aware just how potentially apocalyptic his two most famous inventions were: leaded gasoline, his solution for stopping engine knock (premature ignition of the gasoline/air mixture in an internal combustion cylinder); and CFCs.

The former worked, but put enough lead in the atmosphere to poison the entire world. The latter worked, too (its thermodynamic properties makes it a very efficient coolant for heat pumps such as refrigerators and air conditioners), but also punched a hole in our protective ozone layer.

Oh, and the F in CFC? Fluorine.

Of course.

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