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 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.

May 3, 2023 at 9:01am

#1049152

For a Lark

Because, obviously, the only true measure of self-worth isn't money or power or friends or a Pokemon collection, but productivity...

What Happened When I Forced Myself To Wake Up At 5 A.M. Every Day For A Month
Some early risers might get more done, but it turns out there are times when getting up earlier can make you less productive.

I have another article in my queue about the joyous benefits of waking up at 5 am. I'll get to it eventually.

But okay, let's see how this author fared with rising at about the time I prefer to be going to bed:

Early risers get a lot of good press: They are supposedly more productive and possibly better problem solvers.

I'm not going to reproduce the hyperlinks from the article here. Suffice it to say that people still seem to have an issue confusing correlation with causation, and also with trying to generalize conclusions from a single data point.

But after a month of forcing myself out of bed at 5 a.m., I learned that getting up early isn’t always the best thing for you.

Or, and bear with me here, because this is a complicated assertion: people aren't all the same, and what works for one person won't work for another.

I’m a morning person...

Hey, I'm a morning person, too! Midnight to six a.m. is "morning."

...and most days I’m out of bed by 5:45 a.m.

Seriously? I had to wake up at that time (or even earlier) in the summer when I had an outdoor job that required sunlight. I managed. I never enjoyed it.

I usually have 15 minutes before the rest of my household starts to wake, and I use this time to enjoy a cup of tea as well as the stillness of the morning. I look forward to this time so much that I wondered, What would happen if I expanded the 15 minutes to an hour?

Society reinforces the "early to rise" thing as something to admire, something aspirational. What do you think would happen if I wrote an article about the productivity gains you'd experience if you lived alone, unbothered by unnecessary distractions, not for an hour, but for 24 of them?

The second day I decided to meditate, a practice I’ve wanted to do but never seemed to have the time for. Unfortunately, I fell asleep in my chair.

Duh.

As the month went on, I used the time to get a head start on work, but by 9 p.m., I was exhausted and would head to bed.

Again: duh. You're cutting off a strip of blanket and sewing it back on the other end.

Consider: we're supposed to aspire to 8 hours of sleep a night. Again, people are different, and some need 9 while others need only 7. If you consider this variation, and the fact that "head to bed" implies you're brushing teeth, changing clothes, and whatever other nightly rituals you have, the author was still getting less than 8 hours of sleep (9 to 5 is 8 hours whichever is am and pm). And that's not even getting into the time it takes to perform her marital duties.

(That's supposed to be a joke.)

The article goes on to quote a purported sleep expert, someone who literally wrote the book on the subject. I won't copy most of it here.

And not having a strong plan doesn’t help, says Stevenson. “If you don’t have a reason to get up, and your body wants to rest, forget about it,” he says. “You need something that will fill that space that is compelling.”

I can wake up early if I have to. I do it on road trips because my schedule doesn't mesh with hotel schedules. I used to do it to adhere to a work schedule. But I have to have to. Getting up just to meditate or exercise won't cut it for me. It may work for other people, and that's fine; I'm not knocking it. I'm just saying that this whole Puritan ideal of waking up early isn't the only way to get through life.

At the risk of overwhelming my readers with links, here's one

from a different productivity-porn publication that is closer to my own views on the subject.

I should note that I have issues with that article, too, but I'm not going to discuss them here. I'll just point out that it's entirely possible that Ben Franklin was trolling everyone with that "early to bed, early to rise" crap. He was an epic troll, and still my favorite Founding Father.

Back to the title article:

Being the proverbial “early bird” has its advantages, says Shanon Makekau, medical director of the Kaiser Permanente Sleep Lab in Hawaii.

I'm going to go out on a limb here and assume that, based on the surname, that person is at least part native Hawaiian—a culture not really known for strict adherence to Puritanical ideals of sleep/wake schedules. Or worshiping productivity. But I don't want to fall into a trap of stereotyping; I'm sure Hawaiians, like all other groups of humans, have different chronotypes.

“Morning people have been shown to be more proactive, which is linked to better job performance, career success, and higher wages, as well as more goal-oriented,” she says. “These people tend to be more in sync with the typical workday schedule, versus night owls who may be still be waking up at around lunchtime.”

The problem, here, is, again, correlation vs. causation. Owls trying to become larks are often doomed, as are larks attempting to stick to owl schedules.

Early-morning hours also tend to be more productive because there are fewer distractions.

This is, once more, a lifestyle choice. I always found I'm more productive and creative at the end of a day than at the beginning. This might be changing, though; as you may have noticed, I don't usually do midnight blog entries anymore. Age affects preferred sleep schedules, too. I will emphasize, however, that I don't consider one schedule inherently superior to another. I'm also still biphasic, but the timing of my first sleep seems to be changing as well. One of the many advantages of being retired: I go to sleep and wake up whenever I feel like it, for the most part. And yet, according to my CPAP, I usually get the standard 8 hours of sleep per 24-hour day, in total.

The point is, even if you have family or whatever, you can still get the benefits of "fewer distractions" after everyone has gone to bed as opposed to before everyone wakes up.

As for the author's self-described experiment:

Unfortunately, my experiment didn’t produce long-lasting results. When my month was over, I immediately returned to my normal 5:45 a.m., which felt like sleeping in. I even slept until 10 a.m. on weekend mornings–a very rare occurrence for me. I feel more productive now that I’m back to my normal routine.

And in the end, at least one of the experts quoted seems to agree with my take on it:

“The jury is still out regarding whether or not simply shifting one’s wake time earlier is enough to garner all of the positive benefits of the early bird,” says Makekau. “It may be that one’s internal tendency toward productivity is inherent or, more importantly, is tied to the congruency between the internal sleep/wake clock and one’s external schedule. Night owls could be just as productive as long as they are allowed to work on a delayed schedule.”

Shaka, brah.

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