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 an hour 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 3 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.

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').

1 Like

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.

November 26, 2023 at 10:14am

#1060145

Revisited: "So Much Winning"

Today we look at something from four years ago: December 1, 2019. Some context: I'd just had a very productive November blogging with 30DBC, and this was me going back to my default format of looking at some kind of link. After that December 1 entry, I would go on to write four more entries, then take a nine-day break while traveling—which was, though I didn't expect it at the time, or plan for it, my last break so far. That's right; I'm only about half a month away from four years of daily blog entries. I'm traveling again soon, but this time, if I miss a day without warning, that means something terrible, awful and/or horrible has happened to me, and you should mourn and/or celebrate. (Though keep in mind that an entry could happen at any time during the day, and it might be a short one.)

Enough context and digression. Here's the entry: "So Much Winning"

. It's commentary on an article that asserts that it turns out money can indeed "buy happiness." The linked article, from Vox, is still available.

Over the last four years, I've addressed the concept of happiness a few times, and, at least once, I've tackled the persistent myth that big lottery jackpot winners necessarily become miserable. I won't be digging those up today, so it's possible that I'm repeating myself, as is common because I never really remember what I wrote or when I wrote it.

First: I note that no one "liked" or commented on that entry, though I know people saw it (page views are provided to writers here). The surrounding entries received some responses; it was just this one provoking radio silence. I'm not salty about it, and I'm not complaining; I appreciate feedback, but I don't beg for it, and I don't get depressed if I don't get any. Just stating facts. And that's not going to stop me from writing about it again today.

Second: As noted, it's been four years, and there was at least one society-altering event between then and now. So my views have changed somewhat.

As it doesn't seem to be a popular topic, I'm only going to look at a few brief excerpts from that entry.

Gosh, when your sample set includes a significant number of people who are, by definition, not good with money, maybe there's some implicit bias going on?

In retrospect, this looks like I'm ragging on lottery players. I gamble in other ways (pretty sure I hared off to Vegas shortly after writing that), so no, I'm not ragging on lottery players in general. Being "not good with money" isn't meant to be an insult. I think I was just trying to point out the selection bias in studies like that. If you're not good with money, that doesn't make you a bad person, but getting a large windfall isn't going to suddenly turn you into someone who is good with money.

In my defense, I did note this in the following paragraph.

"Money doesn't buy happiness." "Money isn't everything." There might be grains of truth in such sayings, but mostly these homilies and others like them are designed to appeal to people who don't have a lot of money, and to keep them from rising up in revolt against the rich. We'll see how long that'll work.

At least four years.

And I maintain that anyone who seriously thinks that money doesn't buy happiness hasn't experienced the joys of drinking really good whiskey. That shit is not cheap, but it is a font of happiness.

I think my current iteration of this sentiment is: "Anyone who thinks that money doesn't buy happiness has never had really good scotch." Which, of course, is a personal thing for me; not everyone likes scotch.

That bit leads me to some musings about "happiness" in general, because what makes me happy isn't necessarily going to make you happy, or vice-versa. Happiness studies, in my view, have some major inherent and inescapable flaws.

One flaw is that they're either objective, or subjective. I know I've been saying life isn't binary, and it's not, but consider this:

An objective study has to use some metric to define happiness. But, newsflash, people are different, and any such metric would have to take that into account.

A study with some element of subjectivity would, perhaps, ask people to self-report their level of contentment with their life situation. Like a doctor's 1-10 pain scale, only for not-pain. Perhaps 1 is "completely unhappy" while 10 is "absolute nirvana" or whatever.

The problem is that, say you find two people in similar socioeconomic circumstances: same marital status, similar earning power, same number of kids (0 is a number), roughly the same level of health, etc. One might report a 3 because they've fallen from a higher standard of living, while the other reports an 8, as their situation represents several steps up from the hellhole in which they spent their early life.

As a further complication, some people simply cannot be happy unless the people around them are miserable, or at least, they can't consider themselves as having "won life" (ugh) unless other people lose at it.

Happiness, then, is subjective, situational, and personal.

Consequently, I don't know how you'd design a happiness study whose results would apply to everyone. Interview people in a bar, and you'll find that happiness is proportional to access to booze. Sample church attendees, and you might conclude that religion makes people happier (I'm sure it does in some cases, but not in others). Introverts are going to give different answers from extroverts. Ask married people, and... well, okay, in that case, you'll probably get a wide range.

Bottom line, in terms of today's featured entry from the past, is that a windfall will increase happiness for some, and decrease it for others. I just don't see how you can make broad, sweeping generalizations about what any given individual needs to do to help themselves find happiness.

So my last paragraph still holds true:

I do know that I wouldn't mind being a test case. You know. Just to be sure. I just can't bring myself to play the lottery with any kind of regularity (it's been many years since I actually bought a ticket), so I'll never find out.

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