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:

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').

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.

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

1 Like

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 14, 2021 at 12:02am

#1017367

This Isn't Nam - There Are Rules!

Well, after two days of heavy shit, how about a good old-fashioned Bloomberg roast?

After 45 Birthdays, Here Are '12 Rules for Life'
Don't lose friends over politics. Don't lose a spouse over pickles.

The headline made me laugh. The subhead made me stop laughing. But of course, the article is from early 2018, before politics became life-or-death, and pickles were cheaper.

Yesterday was Jan. 29, meaning that Oprah Winfrey and I are each a year older: 64 and 45.

Of all the crappy reasons to name-drop, sharing a birthday has got to be on top of the list. Why not brag about being an Aquarius while you're at it?

Forty-five is somehow a very definite year; there is no question that you are middle aged.

Average age of death is roughly 80 years, being generous and also noting that the writer is female (statistically longer-lived). Half of 80 is 40. So 31-50 is what should be considered middle age. Sorry, Megan, you're now (three years after this article) two years away from being an old lady.

At 45 one takes stock.

I think it's the rule that any Bloomberg article has to have the word "stock" in it.

The building years of your life are over, and what you are now is pretty much what you are going to be. Soon it will be what you were.

Bite my ass.

You can no longer tell yourself that you might move to Lisbon, learn Portuguese, and take up the guitar. You cannot learn Portuguese at your age. You can’t remember new words anymore; you can’t even remember where you have left your keys.

Bite my ass hard and then kiss it better. And you may want to go get a cognitive test. I started learning French two years ago, at much older than 45. Sure, it's taking me longer than it would some kid, and my memory is far from perfect, but I'm not ready to stop learning yet. J'apprends ce que je veux apprendre.

Oh, and I know exactly where my keys are (Je sais où sont mes clés). And my phone (mon portable), and my glasses (mes lunettes). Not sure about my cat (Je ne sais pas où est mon chat), but they tend to wander about on their own. (I'm not trying to pretend I know a lot of French, and I'm certainly not fluent, but I am making the point that she's wrong about the language thing.)

So it seems a good opportunity to do two things. First, to wish Oprah Winfrey a happy belated birthday.

She's not going to sleep with you.

And second, to address this “12 Rules for Life” meme that you young whippersnappers have got up to on the social medias.

Did I forget the "12 Rules for Life" meme? Oh, no, wait, I didn't give a shit or follow les réseaux sociaux so whatever.

I am probably more than halfway through my life now; I ought to have some rules.

If you're lucky, yeah, but again, might want to get that memory checked out. And no, it works exactly the other way around: the older you get, the less you need rules, and the more you can cheerfully break the ones society has laid out for you. To demonstrate this, I'mma destroy this punk's "rules."

1. Be kind. Mean is easy; kind is hard.

While it doesn't do to be a bully, sometimes you just have to kick someone's ass (metaphorically; I'm not into actual violence unless it's absolutely necessary). Someone says something stupid? Call 'em out. To be fair, that might be a lesson learned since early 2018.

2. Politics is not the most important thing in the world... If you have to choose between politics and a friendship, choose the friendship every time.

Yeah, no, I've cut off more than a few freakin' idiots who are on the wrong side of the aisle. Especially over the past 18 months or so.

3. Always order one extra dish at a restaurant, an unfamiliar one.

I'm all for trying new things from time to time, but dammit, commit. Just order the new thing. I hate taking home leftovers and I'm not ordering (and therefore eating) two meals in one sitting.

4. Give yourself permission to be bad. You know what you’re really good at? Things you’ve done many times before. Mastery is boredom.

Typed like someone who's never mastered anything.

5. Go to the party even when you don’t want to.

Okay, I'm an introvert and I've never not wanted to go to a party. Leave early? Sure, sometimes. I guess it's because I get invited to parties, like, once a year. Less now. So I have to take advantage of it when I am.

6. Save 25 percent of your income.

Yeah, right, this- no, wait, I can't argue with this one. She's right.

7. Don’t just pay people compliments; give them living eulogies.

Yeah, if you're a chick you can get away with this. If you're an old guy with a beard, you get labeled "creep."

8. That thing you kinda want to do someday? Do it now. I mean, literally, pause reading this column, pick up the phone, and book that skydiving session. RIGHT NOW.

Before-Time-like typing detected.

9. Early modernist critics used to complain about the sanitized unreality of “nice” books with no bathrooms. The great modernist mistake was to decide that if books without sewers were unrealistic, “reality” must be the sewers. This was a greater error than the one it aimed to correct.

And so we ended up with post-modernism, in which nothing matters, everything has to be dissected and deconstructed, and nothing can be enjoyed for what it is, only blasted for what its implications are.

Incidentally, I finally got to the point in my chronological Trek watching that I heard a reference to a bathroom. I think it was on DS9. It took until the mid-90s.

10. Don’t try to resolve fundamental conflicts with your spouse or roommates.

Don't have fundamental conflicts with spice or roommates. Better yet, don't have a spouse or a roommate if you're going to be picky about things. I mean, like... I have this thing about sticks of butter. They have to be cut square, or the rest of the stick is ruined. Fortunately, my housemate has her own butter stash, so it's not an issue.

11. Be grateful. No matter how awful your life seems at the moment, you have something to be grateful for. Focus on it with the laser-like, single-minded devotion of a dog eyeing a porterhouse.

Okay, I like that image, but still...

12. Always make more dinner rolls than you think you can eat.

Look, I'm a big fan of bread, and I like to eat, but I'm older than this lady, and there are limits to how much I can stuff into my dough-hole.

So of course, I had to come up with my own 12 Rules for Life. They don't parallel hers. But these are things I've learned with my far greater life experience:

1. Never go grocery shopping while hungry.
2. Don't pretend to like (or dislike) something just because everyone else likes (or dislikes) it.
3. Eat what you're craving (just maybe not too much of it).
4. Sleep when you're tired.
5. Sex is overrated.
6. Don't interrupt my writing unless something is on fire, or you want to be.
7. Always have a backup for critical systems (e.g. keep a paper map in your car for when your GPS craps out).
8. Make sure there's always booze in the house. (Or, you know, whatever your favorite drink is.)
9. Own, don't rent. (This isn't just about housing.)
10. Pay extra for quality shoes.
11. Money isn't everything, but it can buy you anything. Maybe it can't buy happiness, but it can buy beer, and that's good enough for me.
12. Automate everything you can.

Maybe you have your own Rules (or Guidelines). Which means it's finally time for another...

Merit Badge Mini-Contest!

Got your own Rules for Life? Doesn't have to be 12, but say a minimum of 5. Post them in a comment on this entry. It's okay to use one of the rules above (hers or mine), but put your own spin on it if you do so. The comment I like best will earn its author a Merit Badge tomorrow (Wednesday). As usual, the deadline is midnight WDC time at the end of today, Tuesday, September 14.

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