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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 12 hours ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
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🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
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?

July 22, 2024 at 10:25am

#1074266

Waxy Scaley

It's been a while since I had a bit about cooking here, partly because I haven't found many lately, and partly because I haven't been doing much cooking, opting instead for convenience. But here's one from bon appétit:

How to Measure Sticky Ingredients Without Losing Your Gosh Darn Mind
I spent 30 years measuring sticky ingredients like a fool.

Amusingly, if you preview the URL at the link above, it provides a handy spoiler answer to the clickbait headline.

There used to be exactly one thing I didn’t like about baking. I called it the Peanut Butter Problem and it happened every time I made my favorite three-ingredient cookies.

Are you going to share the magical recipe with us along with the sorcerous solution to the problem? No? Oh, well, there's always Google, if you can stand scrolling through pages and pages of anecdote before getting to the actual recipe.

The Peanut Butter Problem was, as you may have guessed, my best attempt to get a precise amount of peanut butter from a measuring cup into a mixing bowl, and it involved multiple butter knives, my pointer finger, and swearing.

From my lofty perch of having figured out two different solutions to this problem many years ago, I could only laugh at this imagery. It's like those commercials that show people failing at different tasks before presenting the One True Solution, Only $19.95 While Supplies Last CALL NOW. Do they still have those? I avoid ads. You'll be getting my solutions for the low, low price of free.

And then came TikTok.

hu... hu.... HURK

The trick? Simply oil your measuring spoons and cups before measuring sticky ingredients.

I can't trust any other content from this author, because, like I said, I figured that out decades ago.

There are two ways to do it: The first is to dip a finger in oil (hereby referred to as your “oily fingie”) and swoop it around the inside of a measuring spoon or cup, using your oily fingie pad to grease the bottom like you’re coloring it in.

And also because she finds "oily fingie" somehow amusing.

Anyway, I know a better solution. This whole "measuring ingredients by volume" thing is imprecise. With or without "oily fingie," two cooks measuring the same ingredients with the same tools can come up with different quantities, because of how void spaces get filled or not. No, for the absolute best method, and one that requires fewer steps, is to measure ingredients by mass, not volume.

Yes, this does require you to figure out the proper number of grams or ounces in a standard volumetric measure of, say, peanut butter. But you only have to do this once. A quick Google search yields 256 grams for 1 cup. Now, set a sheet of wax paper on a kitchen scale and tare it. Add peanut butter until you reach 256 grams. If you go over, put a dab back into the jar. Then dump the ingredient into your mixing bowl or whatever.

For ingredients that are sticky but spread out more, line a bowl with wax paper and tare the scale with that.

Better yet, find recipes that give quantities in mass/weight (really, as long as you're on Earth, it doesn't matter) in the first place.

This has the other important advantage of requiring less cleanup. Just chuck the wax paper into the trash afterwards. Sure, it contributes to landfill problems, but at least you waste less water.

But then you won't get to give it a cutesy name like "oily fingie." Unless you're cool with "waxy scaley."

5 Like • Comment • View 3 comments

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