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.

3 Like

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.

2 Like

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.

1 Like

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.

1 Like

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.

3 Like

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.

2 Like

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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