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.

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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.

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.

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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.

2 Like

March 11, 2023 at 11:14am

#1046235

Cache Is King

Today's financial advice is brought to you by Lifehacker. Lifehacker: You're Doing It Wrong.

Here’s How Much Emergency Cash You Should Keep at Home
Having some cash salted away in your home is a really useful thing.

Wait, wait... we're supposed to have cash that we haven't spent yet?

In many places, cash is no longer king. While you still can manage to live a cash-only lifestyle if you want to, few of us do.

Yeah... not really. Depending on what you mean by "lifestyle."

Some of us don’t even carry cash at all, relying on our phones and credit cards, tapping and swiping at stores and using apps like Venmo or Cash App for personal transactions that would once have involved passing a wad of crumpled bills to someone.

Beggar in the median the other day: "Can you help out?" Me: "No cash." Beggar: "I have Venmo."

Look, I'm not trying to rag on the legitimately homeless here, but these median beggars generally aren't.

Even when we talk about the necessity of an emergency fund, we’re talking about money in a bank account, not cash stuffed under a mattress.

If I were robbing a house, under the mattress is the first place I'd look. Followed by sock drawer. You should express some creativity in cash caching, like, say, the toilet tank. (Note: I do not keep cash in my toilet tank; this was just an example.)

Money in a bank will earn interest, be insured against robbery, and will be accessible in some way no matter where you are.

This comes up in my queue the day after a bank fails. In fairness, the depositors' money was insured and the FDIC did its job. Shareholders are boned, but that's the risk they take.

But cash can still be useful. And in an emergency, cash can even be necessary. That’s why you should definitely keep a modest amount of cash at home. But how much?

I've said this before and I'll say it again: Come the collapse of civilization, bitcoin will be worthless (well, more worthless). Gold will be worthless, because other than being shiny, it's useless in an apocalypse. Cash, though, will still have value, because they won't be making any more of it. But your best bet is to stash away coffee, chocolate, and cigarettes.

As for how much cash, well, to some, a modest amount is a million dollars. If you have that stashed away, please let me know. I'll check your toilet tank.

It’s true that in most day-to-day operations no longer require cash, so keeping a stack of bills in a home safe or hidden away somewhere might seem like a waste.

Just be aware that there's an opportunity cost involved.

When it comes to cash, it’s best to have enough to get you through a week or so of paying for fundamentals like gas for your car and generators, groceries, necessary prescriptions, and maybe a night or two of emergency housing like a hotel or motel room.

I wasn't aware that you could even use cash at a hotel.

Anyway, the article goes into specifics. I'm not actually snarking on the idea; it's a good idea. It's just that, these days, for many people, the concept of "extra cash" might as well be "rainbow unicorn" or "pixie dust."

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