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🎶

about 22 hours 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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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."

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