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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July 22, 2023 at 8:48am

#1052910

Smidgen Forge

It's been a while since I did a cooking article. Well, this is sort of cooking-adjacent, anyway.

How Much Is a Smidgen?
The history of measuring a dollop or a pinch.

I always figured a smidgen was what you get when you run over a pigeon. Turns out, I was wrong.

This article is basically an ad for a book, as is common around here, but uncommonly, it's a book I actually would like to read.

It also delves into more than just the "smidgen;" the first section goes into how "stone" became a unit of measurement:

For example, an English statute from around 1300 set a London stone at 12.5 lbs.; however, a stone for weighing lead was said to be 12 lbs., while a stone for measuring beeswax, sugar, pepper, cumin, almonds and alum was 8 lbs., and the stone for weighing glass was 5 lbs. The inconsistent and archaic use of stones continued in Britain for some time.

And, colloquially, they still speak of some weights in terms of (far more standardized) stones there. It took me a while, as an American, to figure out that one stone was 14 pounds, and even longer to convert that to kg (which isn't a proper conversion either, as kg is a unit of mass, not weight, but whatever).

An interesting side note is that, although the stone was not greatly used in America, in 1790 Thomas Jefferson suggested a new decimal system of coinage, weights and measures. His decimal currency was adopted, but his idea to introduce an American stone of 10 lbs. (with each pound weighing ten ounces) was rejected.

I think I did an entry a while back on how we managed to decimalize our currency as a newly-independent nation, but attempts to decimalize anything else failed miserably. The UK, in contrast, didn't use a decimal-based currency until almost 200 years later.

Smidgen, pinch, dollop, dash, and drop

I did promise a cooking-adjacent thing up there, and this is it.

Most people recognize that they refer to a small amount of something, but just how small is left open to interpretation.

In fairness, some recipes are more forgiving than others when it comes to quantities.

Smidgen is generally used to refer to an almost trace amount, a few grains or a tiny sliver.

Still wiggly, since it doesn't specify the size of the grains or how tiny "tiny" is.

To assist novice cooks it seems some American food writers have begun giving exact measurements (as fractions of teaspoons) to the traditionally inexact terms. A dash is said to be 1/8 of a teaspoon, a pinch 1/16 of a teaspoon, a smidgen 1/32 of a teaspoon, and a drop 1/64 of a teaspoon. You can now even purchase a set of measuring spoons for these tiny amounts.

Meanwhile, I wish we'd get rid of "teaspoon" and just measure everything in grams. Far more precise. Some ingredients don't fit nicely into a teaspoon, tablespoon, or even a cup. Ever try to measure 1/4 cup of brussels sprouts? No amount of precision in the manufacturing of the 1/4 cup measure can fix the problem.

There's a lot more explanation at the link, including (my favorite part) the etymology of some of these words. And, clearly, there's even more in the book.

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