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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Brandiwyn🎶

a day ago
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.

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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.

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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.

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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.

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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.

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Turkey DrumStik

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

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

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October 25, 2024 at 9:56am

#1078904

Pass da Pasta

Today in You're Doing It Wrong...

No, Salted Water Doesn't Boil Faster and 7 Other Pasta Myths
We asked an Italian chef about the eight biggest myths and misnomers linked to making pasta at home.

Most likely they asked "an Italian chef" because if they'd asked 10 Italian chefs, they'd have gotten a hundred different opinions.

If you think salting your pasta water is going to get dinner on the table faster, think again.

I have to wonder how that bit got started. Adding salt to water increases its boiling point, making it take longer to come to a boil. The higher-temperature salt water would then, presumably, cook the noodles faster. It would be pretty simple to do an experiment to see which effect takes precedence... and then you probably find that the amount of salt used in cooking makes no meaningful difference to boiling temperature.

To separate pasta protocol from the false and fabricated, we asked an expert about the biggest pasta myths, mistakes and misnomers that could be ruining your rotini and putting your pappardelle in peril.

Really, I'm just quoting this bit to illustrate how proud the author is of his alliteration.

Filippo de Marchi, is chef de cuisine at De Majo Restaurant & Terrace.

Stealth ad!

We grilled Marchi on nine of the top-circulated pasta cooking myths.

Tough to grill pasta.

"Cooking pasta isn't difficult at all. It's all about timing and the right water-to-pasta ratio," he says.

So, let me tell you what I remember of my mom's pasta-cooking technique. First, fill a small pot halfway with water. Then throw in the pasta (breaking the hell out of it if it's spaghetti). Then put it on the stove over high heat. Wait 45 minutes. Drain and serve.

My mom had no Italian heritage.

1. Throwing pasta against a wall to see if it sticks proves it's done

Chef's take: FALSE

Of course this is false. I've always known it to be false. And yet, the false information persists, as it usually does.

2. Adding olive oil to pasta water keeps noodles from sticking

Chef's take: FALSE

"The oil just floats on top of the water and doesn't coat the pasta effectively," says de Marchi.

Pretty sure there's more to it than that. Once the water boils, the oil gets stirred in more, though it's still going to be gloppy and not stick to the noodles.

3. Fresh pasta is always better than dry pasta

Chef's take: FALSE

Pretty sure that's a question of individual taste and opinion, not fact or fiction.

It's all about personal preference. Fresh, dry or frozen; chefs aren't here to dictate what your taste buds like and don't like.

Like I said.

4. Leave the pot covered while the pasta is cooking

Chef's take: FALSE

Of all the "myths" on the list, this is the only one I'd never even heard of. All pasta-making recipes I've seen demand open-top pots.

On the flip side, everyone's other favorite cooked starch, rice, requires a covered pot. Unless you cheat and get instant rice or whatever.

5. Adding salt helps water boil faster

Chef's take: FALSE

As I noted above, middle-school chemistry disproves this one.

"If you're cooking without enough salt, the pasta can end up tasting a bit bland," warns de Marchi, whose signature dish at NHC Murano Villa is a spaghetti alle vongole.

The stealth ad continues!

6. Drain pasta until it's completely dry

Chef's take: FALSE

Besides, "completely" is misleading, here. Presumably, the pasta started out dry; that's why you stick it in boiling water.

7. You should run cooked pasta under water before serving

Answer: FALSE

Yeah, I'm pretty sure there are situations where you do want to do that (creating pasta salad, e.g.), but for your hot spaghetti dishes? Never.

8. You should precook sheets of lasagna

Answer: FALSE

I have to admit, this one confused me for a long time. Most store-bought lasagna has pre-cooking on the box instructions, as I recall (I haven't made it in a very long time). It wasn't until I met my second wife, who also had zero Italian ancestry, who introduced me to the lazy wonders of cook-it-in-the-baking-pan lasagna.

Left out of the article: the other two cooking tips mentioned above (they said nine total), which presumably aren't myths. I guess we'll all have to go to this guy's restaurant and ask him ourselves.

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