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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QPdoll

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WakeUpAndLive️~Happiness

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a day ago
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a day ago
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WakeUpAndLive️~Happiness

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

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Aiva Raine

a day ago
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.

Sum1's In Kalispell Montana

Sep 20, 2024 at 9:08am
In response to "Get Baked"

I have long advocated using only the oven for baked potatoes, and am guilty of punching holes in the tops of one. I don't use Aluminum Foil either. But my other half likes to nuke them first, or only nuke them. It's the speed thing for her. I prefer to take my time when I cook, and do things the way I do them, not her way. But, sometimes, ,ya gotta keep the peace in the house. She won't buy Russet potatoes, but those are the only ones to use for this. I may have to exert a little effort an buty Russets just for Baked Potatoes. She loves the English Chips (really enjoyed those in England when I was there, so maybe she'll like the link I sent her (from you!))

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September 20, 2023 at 8:36am

#1055978

Labor Saving... Not

I fail to see how this is better.

Save money by making your own dishwasher tablets
Keep ’em clean, keep ’em shiny.

We may earn revenue from the products available on this page...

Popular Science used to be a credible publication.

Washing dishes is awful.

Well, sure; any chore is. But I can name a whole lot worse.

Thankfully, a woman named Josephine Cochrane, who was really concerned about her fancy china getting chipped while being hand-washed, stepped up and invented the first dishwasher.

Yes, I've referenced her before, here: "Pretty Petty"

. I would like to reiterate that she invented it so she wouldn't have to do the dishes herself because her servants were incompetent. Presumably, she'd chip the china too, and then who was she going to blame? Eh, probably the servants, anyway.

Now, other than the brainpower and Tetris skill you need to load the machine, you’ve only got to put some detergent in the soap compartment and press “start.”

Brainpower? Tetris skill? I like a good hyperbole, but it's really not that difficult.

Sure, Cochrane’s invention saves you time, but you still have to buy detergent.

The horror.

Make your own dishwasher tablets, though, and you’ll save some money.

The draw of a dishwasher is that it saves you time. It lets you be lazy, or at least concentrate on working on something else, if you're one of those weirdos who has to be productive all the time. Making your own dishwasher tablets defeats the purpose, much as making pancake batter from scratch defeats the purpose of buying frozen microwavable pancakes.

It’s easy...

Having read the instructions, no, it's not.

...makes cleaning up a bit more exciting...

A number can be less negative, but still negative.

and will leave your most likely not-so-fancy china shiny and smelling of fresh lemon…

Yep, that's the goal: have your plates and silverware, with which you're going to eat, smelling like fruit or flowers. Not.

Time: 20 minutes (with a minimum 24-hour drying period)

In my experience, you have to triple the time (at least) of any recipe, food or not.

Material cost: $27 ($5.40 per batch—or $0.14 per pod)

Similarly, the given cost is always underestimated.

Difficulty: easy

Lie.

So, the rest of the article is the actual recipe, presented in the same style as food recipes. I'll give them this, though: the intro wasn't 20 pages long, though some of the steps have meta-information to make up for the brevity of the intro.

Considering that it's finicky, and involves a lot of cleanup in itself (all the equipment needed, plus the inevitable powder spills on your counter), I'm not convinced you're saving anything. You get bragging rights, maybe, but doing all that work defeats the purpose of having a dishwasher in the first place.

At least they seem to have gotten the science right, which, considering the source, is a bare minimum.

7 Like • Comment • View 2 comments

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