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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 2 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 4 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').

1 Like

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.

August 8, 2022 at 12:01am

#1036224

Ah, les œufs

Oh, look, a cooking article from The Guardian.

One box of eggs – 17 delicious ways to use them, from french toast to huevos rancheros

I know, I know—you're probably thinking the same thing I was: eggs come in dozens, not 17-packs.

It is a time of scarcity, but also of glut; a time when you might buy 18 eggs at the supermarket, just because the last time you tried, they didn’t have any.

Yeah, the article's from 2020; how did you know?

This could present an opportunity for creativity, but 18 eggs is enough to induce a failure of imagination in anyone: you boil them, fry them, poach them – then what?

Uh, hard-boil the lot of 'em so they last longer and become a quick, handy source of protein?

Here are 17 delicious and slightly further-afield recipes to use up your eggs, most of them employing simple ingredients you may already have.

Look, I don't eat eggs every day, but I still manage to get through a dozen before they go off. I get the feeling the "use the whole box" thing is just the hook here. But even if so, why stop at 17? What happens to the lone, leftover egg from the dozen and a half?

Also, I counted nine dishes here, not seventeen. Maybe my script-blocker cut something off; I don't know.

Mayonnaise

I get way too much credit for making my own mayonnaise – it just isn’t very hard.

Ah, the rare sighting of the wild Humblebrag.

Everyone should make their own mayonnaise.

Why, when it's right there in a jar?

French toast

Call it what you want, but as an American, I find it hard to say the words “eggy bread” without pulling a face.

Yeah, I'm with him on this one.

Having said that, the recipe is more or less the same. Beat some eggs with milk, dip in slices of slightly stale bread (almost any kind will do, from crappy supermarket white to brioche), fry on both sides in butter until light browned and … voilà is possibly not the right word to use here.

No. No, it is indeed not. If you're gong to murder French, at least call the stuff le pain perdu.

That said, I haven't made it in a while, but my preferred French toast involves challah bread, and I use cream instead of milk because fuck you, heart (look, even though yesterday's article was all about not worrying about full-fat dairy products, it's still calories). Also: maple syrup and vanilla extract.

Still, it's almost impossible to create le pain perdu avec seulement un œuf... sorry, with only one egg, unless you're using only one really small piece of bread.

Thomasina Miers offers a version with salted caramel pumpkin puree (and tequila).

I would die. It would be worth it, though.

The monte cristo

This is more an artefact than a meal, a forgotten fossil of the American diner experience. The monte cristo is basically a ham and cheese sandwich (the cheese is usually swiss – emmental, say) that has been dipped, in its entirety, in some beaten egg and then fried as a solid mass.

hurk

Soups

OH COME ON

Skipping a couple of Spanish-themed ones here, not because I don't find them appealing, but because I don't have much to say about them.

Eggs benedict

Okay, look. I love Eggs Benedict. I would never even consider creating it, usually because when I'm craving it, I'm too hungover to cook anything.

What generally stops people making eggs benedict at home is the hollandaise sauce, which is fiddly and prone to splitting and curdling. But without it, we’re just talking about poached eggs on toasted muffins.

Yep. Screw making hollandaise. Also keep in mind that because even though the writer is American, the article uses British names for foods, including simply "muffins" for what we'd call "English muffins."

Soufflés

Nope. Too much work.

Crostata di limone

When this lemon tart appeared in the first River Cafe cookbook, it fascinated the amateur: it looked easy, if highly improbable. There is still something highly daunting about a recipe requiring six whole eggs and nine additional yolks.

And there goes 15 of your 18 eggs right there. No way is that "easy," especially the bit about separating nine of them. "Easy" is cracking a couple of eggs into a pan and frying them.

Also, and no one has ever answered this to my satisfaction (remember, my mom was a crap cook): what the hell do you do with the other parts of the egg when you separate them? Are you obligated to, I dunno, make a meringue or something, or do they just go down the drain?

Anyway, someone who is not me might try some of these recipes. The only one of these that I have done, or would do, is the French toast. In my defense, it is very, very good French toast.

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