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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10 Most Recent Blog Comments:

Write_Mikey_Write!

about 5 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 7 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.

April 7, 2024 at 12:57am

#1067846

Revisited: "Social Greases"

I'm getting this done early (really early) because I don't know when or if I'll have a chance to get online later today. And the next couple of days' entries might be short and also at weird times, because I'll be traveling.

Today, we're going back in time again, but not very far—February of last year. I have a self-imposed one-year blackout on Revisited entries, and this one makes the cut by less than two months: "Social Greases"

In it, I discussed an article from New York (not The New Yorker). Unsurprisingly, the link

is still valid as of today. I'm not sure I can say the same for a lot of the "rules" they discuss for "living in a polite society" (I'm increasingly unconvinced that we do).

From the article:

We sparked office arguments and made messes and ended up with a guide that we hope will stand the test of at least a bit of time — until the next great exciting social upheaval.

So, a year later, what's held up and what hasn't? You might need to read the article and decide for yourself. Me, I'm just going to rehash some of my own comments.

4. When shopping with a friend, don’t cut them in the rack.

Someone needs to explain this to me. On second thought, no, don't. I never shop with friends.

I still haven't figured out what in the hell this is supposed to mean. I could have looked it up, but it's so far removed from my lifestyle that I simply haven't bothered. Not to mention I promptly forgot it until re-reading it just now.

6. Never wake up your significant other on purpose, ever.

Never wake them up accidentally, either.

You know what's worse than either of those things? Passive-aggressive waking. You're mad that they're still snoozing, so you do the dishes as loudly as you can without breaking them, or run the vacuum, or something equally obnoxious, productive, and designed to make them feel bad that their lazy ass is still in bed.

Their lazy ass is still in bed because they need the sleep. Never mess with that.

Yes, I've been on the receiving end of that.

Another thing sure to make me grumpy? Sleep-shaming. "Oh, look who decided to join the living." Not everyone's a morning person, dammit.

28. Don’t ask people how they got COVID.

29. Or why they’re wearing a mask.

No, please ask me why I'm wearing a mask. Depending on my mood, I might answer with "to mess with facial recognition," "because I get compliments on my cat mask but never on my face," or "to keep you from running away screaming."

This one probably aged less well than most of the others.

30. When casually asked how you are, say “Good!”

Valid only in English-speaking countries. And not even then; if someone asks me that, I assume they really want to know.

Corollary: Don't ask if you don't want to know the truth.

34. Actually, it’s great to talk about the weather.

No. Talking about the weather is just going to start an argument about climate change. It's now a topic more fraught than sex, politics, or religion.

I've had people try to strike up conversations with me by bringing up sports. I'm happy to talk to people, despite what I said in that entry (or others), but the only thing I need to know about sports is that they involve way too many commercials for me to even think about watching them.

38. Always wink.

What the bullshit is this?

Another one I had no idea about then, forgot about until now, and still have no idea.

84-91. There are new rules of tipping.

This might irritate or confuse you, but the reality is there are new social expectations around what deserves a tip.

These have, on the other hand, been on my mind. I've started to follow a simple rule: the more work I have to do, the less I tip. Full-service restaurant? Full tip. Delivery? Same. Pick up at restaurant? If I have to do the driving, I tip less. Uber? Always. (That's kind of a pun.) And if I have to order at a counter, wait, pick up my order, and bus my own table, they get nothing. I despise self-checkout, so I never use it, but I've heard some of them have now added tip begging to the checkout screens. Never. Not in a billion years. If I'm doing all the work, shouldn't I be getting the tip?

I'm considering implementing another one: if the screen suggests something like a 50% tip, they get nothing and I never go back. Already did that for cabs in Vegas. That's when I signed up for Uber.

Some people suggest memorizing who gets lower wages and tip all of them. I guess you can do what you want, but I still think it should be based on actual service, not intricate economic knowledge. I'd prefer a system where people get paid by their actual companies, and not the customers. Raising prices to compensate would make little difference in total amount spent.

Obviously, these tipping tips are only valid in the US.

Enough of that. The "rules" that really raise my hackles? There are some pearls of advice that suggest that one should do the opposite of what you're asked. Like, "Even when a kids’ party says 'no gifts,' you’re supposed to bring a gift." I've lived my entire life in the South, with frequent trips to New York (whose population this article was obviously aimed at). Doing the opposite of what someone says has always been a Southern thing, and I don't like it here, either. In New York, this "I know I asked you to not to being a gift, but what I meant was bring a gift" or similar would be like traveling to Italy and seeing everyone speaking in perfect, unaccented English without moving their hands. It's not right. It breaks my entire conception of how the world works.

Doing as asked is a mark of respect and courtesy. Anything else requires one to memorize a maze of rules, and I don't have the bandwidth for it. On the flip side of that, it's courteous to state your desires plainly. Don't be coy. "Don't bring a bottle of wine" means "don't bring a bottle of wine," not "bring a bottle of wine."

It's not that I try to be discourteous. I'm simply not a mind-reader, and some of society's expectations are opaque to me.

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