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 3 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 5 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.

April 6, 2021 at 12:02am

#1007848

You Call That Easy?

I certainly don't want to turn this into a food blog, but, well, everyone eats occasionally, so sometimes food articles are relevant. So is "easy," because this is me we're talking about here.

Low effort, huge reward: readers recommend 10 very easy, supremely tasty recipes
A delicious, hearty meal needn’t take much time to make. Here are some startlingly quick options for big taste sensations – from midnight spaghetti to fried green plantain

Well, that sounds good; I'm all about easy food, and...

Put enough spaghetti for two in a pot of heavily salted water and bring to the boil. Finely slice four garlic cloves, a red chilli and the stems of four leaves of rainbow chard (saving the leaves for later) and add to a frying pan with a generous glug of olive oil, then heat gently until golden. Add a couple of teaspoons of capers and a lot of ground black pepper. Roughly chop the chard leaves and a couple of handfuls of parsley. Add the chard leaves to the frying pan along with the almost-cooked spaghetti and 4 tbsp of pasta water. Turn off the heat after a minute or two, add the parsley and stir through, then serve with grated parmesan.

Are you fucking kidding me?

Maybe the Brits just have a different definition of "very easy..."

Put small pieces of vegetables (such as broccoli, pea pods, thinly sliced carrots and thin strips of cabbage), tofu cubes and/or prawns, and wakame seaweed (optional) in boiling water then simmer for 3-4 minutes until slightly tender. Turn the heat off and stir in miso paste to taste (do not boil the miso), then top with scallions [spring onions] and furikake. You can serve with rice on the side or add quick-cooking noodles with the veggies if you want.

Nope. That's a faux-miso soup recipe from someone in Minnesota (I can only assume the Guardian editors translated "shrimp" into "prawns.")

Now, look, most of these dishes look delicious; I'm not arguing that at all. But I'm not inspired to read on, because they've already complicated the first two dishes.

I mean... here's a true quick and easy recipe:

Put ham and cheese between two slices of bread along with your choice of condiments.

Here's another one:

Dump Chef Boy-Ar-Dee into a pot; heat until hot.

And a third:

Boil spaghetti; drain. Mix with tomato sauce.

And my actual definition of easy:

Remove pretzel from package.

But no, this article goes on to describe things that are the polar opposite of easy.

Top a ready-rolled sheet of puff pastry with cream cheese or soft goat’s cheese loosened with a dollop of creme fraiche (mix in some garlic, black pepper and fresh herbs if you have them), then top with tomatoes or other vegetables – roasted peppers, courgettes, aubergines, whatever you’ve got. Bake for around 30-35 minutes at 180 degrees and voilà!

Aubergines aren't food, but at least they're optional here.

And here's one from Ecuador:

Peel the plantain and chop into slices the width of your index finger; you should get between 10 and 15 slices per plantain. Fry them in hot oil, turning once or twice. Remove from the oil and leave to stand on a plate for a minute or two. Now squash them with a cup or glass so they spread into something twice their initial diameter. Return to the hot oil for another minute or so. Drain on kitchen roll, sprinkle with salt and eat. Great served with queso fresco, chilli sauce or avocado.

Sure, I have plantains sitting around.

Like I said, most of these look delicious. But by the time I've turned on ovens, operated the pressure cooker, chopped and cleaned and whatever, hell, I could have walked to Taco Bell and back. Not to mention the planning it takes to acquire the specialty ingredients.

I should turn this into a Merit Badge Mini-Contest, but I can't be arsed to do the formatting tonight. So if you've made it this far: Give me an actual "quick and easy" recipe that doesn't involve fancy, privileged ingredients or too much prep/cleaning. Let's see if we can beat the Guardian here. My favorite will get a food/cooking merit badge. As always, you have until midnight, end of the day Tuesday WDC time.

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