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

about 9 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

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Wordsmitty ✍️

about 12 hours ago
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Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

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Annette in France

about 23 hours ago
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Bienvenue en France.

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WakeUpAndLive️~Happiness

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

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

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WakeUpAndLive️~Happiness

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

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

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Elisa: Middle Aged Stik

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

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

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WakeUpAndLive️~Happiness

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

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

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WakeUpAndLive️~Happiness

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

Je suis sure, vous avez aimez la ville!!!!

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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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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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