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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Writer_Mike

31 minutes ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 17 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 20 hours ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

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Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

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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