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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Write_Mikey_Write!

about 11 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 13 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

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Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

May 30, 2022 at 12:01am

#1033039

Egg Zactly

You can't make an omelet without breaking eggs. But you can make hard-boiled eggs.

We Tried 7 Ways to Hard-Boil Eggs and Found a Clear Winner

It's pretty rare that I'll boil eggs, but when I do, I think my method came from Joy of Cooking (because you really, really do not want to know how my mother attempted to do it; suffice it to say, green stuff happened): Boil water. Lower cackleberries in. Turn off heat. Cover. Let sit for 11 minutes. Prepare a bowl of ice water during the interminable wait. Remove eggs (not with your fingers, idiot) and give them an ice bath for like 3 minutes. Peel under cold running water. Swallow whole.

So I guess technically, they're not boiled for more than a couple of seconds, because you turn the heat off. Whatever. It works. The ice bath makes them easy to shell. Even that simple process, though, is more work than I usually want to do for eating, especially something that is, for me, about a 6 on the 1-10 scale from "nasty" to "delicious."

Now, let's see the science:

Hard-boiled eggs are one of those kitchen staples I always have in the fridge. They’re great for out-of-hand snacking, they instantly add a protein boost to salads, they form the basis of a super-speedy breakfast, and — with just a little extra effort — they become deviled eggs, the perfect cocktail snack.

"A little extra effort?" Spoken like a non-lazy person.

If you go online to find the best way to hard-boil eggs, you’ll find umpteen different methods, all purporting to be the best (many of which don’t even involve boiling).

No, if you go online for that, you'll find this article that I'm quoting. Also, I'm not claiming my way is the best. Just that it works.

The article includes a helpful photo of the resulting eggs, halved, so you can compare what each yolk looked like.

I was delighted to find that most methods worked quite well. A couple gave less-than-stellar results, and some received higher ratings from me because they produced great eggs with less effort.

Now, see, that's useful to me.

Method: Bake in a Muffin Tin

I'm no cooking expert, but my first reaction was "no way would this work out."

This method took the longest from start to finish than any other, and the results were undesirable—the whites were overcooked, and the eggs didn’t peel cleanly.

Good to know my suspicions were close to correct.

Method: Bring to a Boil, Then Remove from Heat

Aha!

For testing purposes, I used the straightforward instructions from Martha Stewart.

HAH! No.

Although this technique was pretty quick and yielded nicely cooked eggs, I would not use it again because the eggs were so difficult to peel.

Huh. Like I said, it works for me. Maybe it's because she's using fresh eggs. I never have fresh eggs. In fact, one of the few reasons I have to hard-boil eggs is when they're approaching expiration and I want to extend their pseudolife just a little longer.

Method: Simmer

Maintaining that brisk simmer was very tricky. I kept having to adjust the heat up or down, so this method involved a lot of babysitting.

That alone is enough to make me go "next!"

Method: Pressure Cook

I have nothing against pressure cooking. I like the science behind it. I'm told, and this article verifies, that you can do it with an Instant Pot, but it just seems like another gadget in a crowded kitchen to me. When I was a kid, we had the giant cast-iron kind with the railroad whistle steam valve. My mother was afraid of it, and only my father got any use out of it. We didn't have a microwave, so this was the next best thing to get things to cook in a hurry.

Method: Boil

Following Bon Appetit’s method, you lower eggs into a large saucepan of boiling water, then adjust the heat to maintain a gentle boil for 10 minutes for an “ever-so-soft center to the yolk” (aka the jammy egg).

I don't understand the difference between "simmer" and "gentle boil." As I noted above, I'm no cooking expert. I just like to eat, so I learned some basics.

Method: Steam

Too much equipment again, but the author claims this works.

Method: Boil Then Simmer

Oh, wait, I guess this is closer to my method than the one above, although I don't simmer. I just turn off the heat entirely. It's not like a covered pot of water loses that much heat in 11 minutes.

The claim is that starting the eggs in hot water helps prevent the whites from sticking to the shell, thus making peeling easier.

Which is, as I noted, important, but is there actual science to that?

I was a little dubious about using 3 quarts of water for just 6 eggs, but López-Alt explains that this volume ensures that the timing and temperature are spot-on.

Yeah, I use way more water than you'd expect for my method. My reasoning is that a larger volume of water loses heat less rapidly. Basic thermodynamics and volume to surface area ratios. The tradeoff is it takes longer to boil. But if you're going to simmer them, why would this matter? Whatever.

So that's it, then. None of these are precisely my method, which, again, I'm pretty sure I stole from Joy. But I do like reading about the trade-offs between ease of cooking and quality of the result, because I'm always looking for the easiest, fastest way to achieve a good outcome. It's a matter of optimization, you know.

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