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:

tj-Merry Mischief Maker

about 20 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

a day ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

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.

5 Like • Comment • View 3 comments

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