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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Write_Mikey_Write!

about 10 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 12 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

February 7, 2024 at 10:47am

#1063694

Pinchers

I saved this one a few months ago, long enough so that I completely forgot, by now, what I thought I might say about this Lifehacker article. Did I save it to rag on it, or to agree with it? Let's find out.

Avoid These 'Money-Saving' Habits That Don't Actually Save You Money
Not all frugal habits will improve your bottom line.

One stereotype imposed upon my people is that we're all penny-pinching misers. When I was a kid, other kids would toss pennies down the school hallway and taunt me for it, because kids suck (I sucked, too). I picked up the pennies anyway; by the time the kids were old enough to keep their prejudices to themselves, I'd saved up enough for a pack of gum.

There's a similar stereotype for the Scots, but for some reason, with them it's called thrift and seen as a positive value.

My point being only that despite all stereotypes, there's a particular brand of US consumer, usually not of either of those ancestries, who really do pinch pennies; if a widget costs $20.99 nearby and $20.98 somewhere across town, they'll chase ass across town. This might be more a case of wanting to reward companies that provide the lowest price than actual miserliness, but it should be obvious to anyone who is not them that driving across town is going to cost them way more than they "saved." I put "saved" in quotes, because they don't actually need the widget, so they're not saving a penny; they're spending over $20.

Of course, we don't, and shouldn't, buy only what we need. There's benefit in budgeting for wants and even luxuries.

Anyway, the article itself mentions that one, and a few others. I'm not going to cover every point.

One part of frugality is finding the balance between the value of your time with the value of your money. Sure, you can spend hours searching for the best deal online or driving to different stores to buy items on sale, and you may actually save a few dollars at the point of purchase. But you may be spending money to do so, and you could be earning money during that time instead or investing in other non-monetary things you value, such as being with friends and family.

So yeah, it's not just that my example above involves spending more in fuel (whether gas or electricity), it's also the time involved. My time, for example, earns me nothing, but I still find it advantageous to pay for grocery delivery rather than waste time poking around the supermarket myself.

Making the rounds to different stores—such as several grocery stores—only to take advantage of a few small-dollar deals at each may seem like a smart strategy, but it may not be worthwhile after you factor in the time and money spent driving all over town relative to the dollars saved.

Again, sometimes, I think that certain people do this not to save money, but to punish retailers who have the audacity to charge more for something than other retailers.

Similarly, it’s tempting to drive out of the way to save a few cents per gallon when filling up your gas tank, and this may make sense if you only have to detour a few blocks or can pair the trip with a relatively efficient reroute or errands you would eventually need to run.

Another possible case of the same phenomenon.

I get it, though. If someone steals something from me, something that doesn't have some sentimental value, I get enraged beyond all rationality. It's not like I can't replace it; that's a mere inconvenience. It's that someone got the better of me, somehow, and I cannot abide that.

Not precisely the same thing, but close enough.

Cheaper versions of products—furniture, shoes, clothing, and tech, to name a few—that are of lower quality may wear out more quickly or break more easily, so you have to replace them more often and end up spending more in total than if you’d bought the better product in the first place.

This is actually one reason poor people stay poor; they have to pay less for lower-quality things. Terry Pratchett noted this effect years ago, known as the Sam Vimes 'Boots' theory of socioeconomic unfairness:

The reason that the rich were so rich, Vimes reasoned, was because they managed to spend less money. Take boots, for example. ... A man who could afford fifty dollars had a pair of boots that'd still be keeping his feet dry in ten years' time, while a poor man who could only afford cheap boots would have spent a hundred dollars on boots in the same time and would still have wet feet.

Pratchett was hardly the first to come up with it, but the point stands: paying more for quality, if you can afford it, saves you more money and time in the long run.

Especially with footwear.

So, I guess this is a rare case of me mostly agreeing with Lifehacker (we had another one fairly recently in here). Sometimes, pinching pennies just gives you hand cramps.

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