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 3, 2021 at 12:02am

#1016645

Oh Yeah, I Vaguely Remember "Dates"

Here's a Bloomberg link that's not nearly as pants-on-head stupid as the last Bloomberg article... though that's not a high bar to clear.

The Economics of Dining as a Couple
Sure, protectionism would guard your pommes frites. But free trade ensures that the best goods reach both sides of the table.

Actually, it's rather amusing, and I do like to see complicated concepts made relatable. I will point out that the article is five years old, but I don't think that matters here.

Marriage counselors tell us that couples frequently tie the knot without discussing the core matters that can cement or sunder their marriage: finances, children, religion. Well, let me add one under-discussed biggie to the list: restaurant dining.

Well, I guess it does matter. A lot of restaurants in my area never reopened. Some of those that have are operating under reduced hours and/or capacity, and workers are fed up with dealing with assholes while not getting paid enough to deal with assholes, so they're understaffed.

Still, you remember the concept of "restaurants," don't you?

But. To get back to the article. I'm sure some couples "forget" to discuss finances, children, and religion before getting hitched, though... I mean... why? However, I'd venture that most couples go on at least one restaurant date before getting married. If they don't, chances are they're not restaurant people to start with, so it won't matter.

I am eternally astonished to find not only that many couples I know failed to discuss this key area before they marched up to the altar, but also that many of them still have not developed a joint dining strategy even after 10 or 20 years together. This is madness.

This... is... Sparta!

Sorry, knee-jerk reflex.

As a romantic economist might put it in a wedding-reception toast, couples have the chance to jointly move to a higher utility curve.

You'll probably have to look that one up. I did.

There are basically four strategies that a couple can practice at a restaurant.

I should note here that this is something I'll personally never have to deal with again, but I do have an interest in economic theory, so I read on.

1. Autarky. This is when a country is closed to imports or exports, and produces everything it consumes. In the restaurant context, it means that you are each independently deciding on what to eat, with no input from the other person, and then solely consuming what you ordered.

Admittedly, this was always my preferred strategy. I was an only child, and while I'm happy to share a lot of things, my fries aren't one of them.

An economist will tell you that autarky is terrible. You’re missing gains from trade!

What gains? You steal my fries and offer me a bite of salad? Get bent.

2. Individual production with trade. Under this model, you both order whatever you want, and allow the other person to take a few bites in exchange for a few bites of their food. This is how the world economy works, and it is much better for dining than autarky.

The above notwithstanding, if we have an international trade agreement in place before we order, I'm open to negotiation... but only if you don't order something I'm going to hate, like any vegan dish, or anything with eggplant.

3. Individual property rights, with option trading. Now we’re moving toward a more centrally planned economy. The menu is individually consulted, and then the two parties state their preferences.

Which is what I was talking about above, Still, no, I don't care if it's someone I kiss all the time, her taking a bite out of my hamburger is completely out of the question.

4. Full food communism. A communist economy is a terrible idea. A communist dinner table, on the other hand, truly is a bounteous paradise.

No. No it is not. If I ordered a steak, I expect to eat the whole steak, and not trade some for your oysters (which are only technically food anyway).

Anyway, just a bit of fun here. I was especially amused, though, by the standard disclaimer at the bottom of the article:

This column does not necessarily reflect the opinion of the editorial board or Bloomberg LP and its owners.

Obviously, because it has the word "communism" in it. And rich people don't share. That's how they became rich.

And now it's time for me to drink some Romulan Ale and watch Lower Decks. No, you can't have any; make your own. I'd have watched it earlier, but Shang-Chi opened yesterday and of course I had to go see that. Consequently...

One-Sentence Move Review: Shang-Chi and the Legend of the Ten Rings

I enjoy martial arts flicks and MCU movies; this one combines good stuff from both, though the Marvel tie-ins aren't nearly pervasive enough to shut out anyone unfamiliar with those films, and I feel like it can stand alone -- though one might not get the full effect from some of the in-universe references, there's plenty of action, fights, SFX, and even car chases to keep your mind off what you might be missing.

Rating: 4.5/5

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