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 8, 2023 at 9:07am

#1049353

Fair Morning

Last week, I did a bit on an article concerning early-birding: "For a Lark"

And then I found this one, from Popular Science (for whatever that's worth these days) on turning owls into larks:

Night owls can become early birds. Here’s how.
Fighting biology is hard.

My philosophical stance hasn't changed in five days: that we shouldn't be forcing or shaming chronotypes. But we live in the real world (well, most of us do), and sometimes you just have to fit your square peg into a round hole, or vice versa.

So this isn't about the desirability of waking up early, but rather how to go about it if you find yourself in a situation where you have to, as I did for most of my life. It should go without saying, but it won't, that one shouldn't make drastic changes to one's lifestyle based on internet advice, so don't blame me if you try any of this stuff and it goes badly.

Ah, Fall: slanting golden light, crisp mornings, and hitting the snooze button 15 times.

Yeah, the article is from September of 2021 and biased toward northern hemispherians.

Some research suggests that even the most stubborn night owl can indeed shift their natural sleep patterns.

Oddly, there's nothing here about larks switching to owl schedules. That's just not as important because it doesn't fit in with the Protestant work ethic. But some people have to do that, too: late-shift workers, for example.

Also, we're not "stubborn" any more than we're "stubborn" when we insist that, sometimes, we have to take a shit.

The first thing we need to realize is that in each of us, a clock ticks away. There’s a bundle of neurons buried deep inside the brain which regulates our day’s biological rhythms, including our metabolism, when we’re sleepy, and when it’s time to eat, says Gideon Dunster, sleep researcher at the National Institutes of Mental Health. “We call it the master clock,” Dunster says. But the master clock doesn’t always match perfectly with a typical 24 hour day. In fact, on average, it takes about 24.2 hours for people to complete this cycle, says Cathy Goldstein, a sleep neurologist at the University of Michigan Health.

I've heard different numbers for that average, but it's always more than 24 hours. I find this fascinating, as I would think that, if anything, we'd have internal clocks left over from the distant evolutionary past, when the length of the day was shorter, not longer.

Other factors, including environmental ones, seem to play a role. Light is the most important, and well-understood, of these cues, effectively delaying or advancing people’s master-clocks.

Fun fact: I sleep better when it's light out, but I don't sleep well at all with any artificial light around.

It wasn’t always so hard to wake up early. “Until 200 years ago, the Industrial Revolution, our biological clocks and environmental clocks were much more in sync,” Dunster says. Now the electricity that lights up our homes at night and powers our bright screens confuses our master clock, which mistakes it for daylight.

Right, blame progress, as usual.

Sorry, but the advances in laziness technology far outweigh the costs of any sleep disruptions.

Is there any hope for night owls to change their ways, especially as those early morning meetings and school bells begin to take over daily life? Maybe.

The article goes on to expand upon the "maybe."

Scientists in England conducted a trial on 22 “extreme” night owls in their late teens and early twenties.

Ah, yes, another study exclusively focused on college students, all in one industrialized country, who are probably not representative of the population as a whole, and with a minuscule sample size.

What I mean is, I wouldn't take the results of this study as settled science. (For the exact methodology, in plain English, the link is up there.)

At least they add this bit:

It’s our schedules that need to change to suit our chronotypes, not the reverse, Dunster argues. Alongside colleagues, he advocates for later school start times for kids. Other studies suggest that similar changes—work schedules catered to a person’s natural sleep and wake rhythms—could benefit adults, too. When shift workers were scheduled according to chronotype, they slept better and enjoyed their free time more, according to a 2015 study published in Current Biology.

While this lights up my confirmation bias more strongly, I still wouldn't take it as settled science.

One trick I've heard of before, but never really tried, is to adjust forward. That is, if you have the opportunity to do so, instead of forcing yourself to sleep earlier and wake up earlier, do the opposite: every day, go to sleep an hour or two later, and wake up an hour or two later. After a couple of weeks of this, you should find yourself aligned with your desired chronotype.

That's the hypothesis, anyway. I suspect that, like almost everything else, it would work for some but not others. And it does require a significant period of time where your schedule isn't being interrupted by external needs, like work or family.

Or, I suppose, you could try moving to a different time zone and let jet lag work in your favor.

But I still think the best way to handle this sort of thing is to make the world align with you, not the other way around. Good luck with that.

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