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 9 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 10 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').

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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.

February 8, 2022 at 12:04am

#1026268

Rounded By a Sleep

I saved this one to share because a) my sleep is biphasic and b) it's another example of "Yeah, about that..."

Sleep however you feel like

This comes from a biologist whose blog I follow, but here he's quoting another external source.

A while back, there was an idea that swept through social media that there is this thing called biphasic sleep.

I had no idea it was on "social media," because other than Untappd (social media for drunks), I don't follow any of it. Still, I saw some articles about it, though I don't think I linked them here. If you're not sure what I'm talking about, here's a Wikipedia link.

The particular kind Myers is talking about here, though, is when one goes to bed after sunset, sleeps for a few hours, wakes up, takes care of some writing or whatever, then goes back to sleep. Some recent articles have been touting that as an ideal from pre-industrial societies.

Hold it right there, not so fast: it’s probably not true. There isn’t one way you’re supposed to sleep.

And of course someone has to throw a wrench into it.

There is no evidence that sleep was universally segmented, and there is also little evidence that segmented sleep is better. A 2021 meta-analysis of studies on biphasic sleep schedules found that segmented-sleeping subjects actually reported “lower sleep quality … and spent more time in lighter stages of sleep.” One reasonable takeaway is that biphasic sleep is like anarchical foraging: Both might have well served some ancient populations some of the time, but neither of them offers a clear solution to modern problems.

Back to Myers:

I find that even more reassuring. Don’t hold people to the one true way they’re supposed to do something, everyone is different, different cultures lead to different behaviors, and there’s nothing wrong with being a bi or a mono-sleeper. Individuals can even change! The only norm is that we have diverse patterns of activity.

And that's a conclusion I can get behind. We're not all alike. I'm not a morning person; I never have been a morning person, and I still resent the conditions that pretty much forced me into being a morning person for my entire working life. I thrive by following a sleep pattern that involves me waking up after the accursed daystar is already well clear of the horizon.

Someone else might prefer to wake up when I'm going to sleep, and they've already passed out just when I'm starting to get energized (which is right around now). And that's okay too.

The problem comes in when the assumption is made that there is only One Approved Sleep Cycle, and everyone is forced into it, like it or not.

I've seen evolutionary psychology hypotheses for why we don't all have just one sleep cycle, but I'm lukewarm at best on evo-psych; a lot of it reads like Kipling's "Just-So" stories. But this one actually makes sense: that before agriculture, before industry, it was beneficial for survival to have some tribal members sleeping at different hours, to be able to stand watch for predators. Obviously I don't know if it's true or not, or whether it's correlation or causation, but it does make a kind of sense. We're not all the same, and we shouldn't all be made to follow the same patterns.

The article I linked above also touches on cultural differences -- not differences in sleep patterns among members of a particular culture, but how different cultures approach sleep. Which is interesting.

But I do have to take issue with Myers' conclusion:

Although “like a baby” is a pretty good ideal.

Really? It's been a long time since I've been a baby (and I don't remember much of it), and I've strenuously avoided having any of my own, but everything I've seen makes "sleep like a baby" sound like anything but an ideal. They wake up in the middle of the night demanding food and/or diaper changes. Once you do get one of them to sleep, every adult within a ten-mile radius is forced to shh, be quiet, shh, the baby's sleeping, because it'll wake up at anything louder than a pin drop.

Still, I think the general advice is sound: use the sleep cycle that works for you, to the extent possible given other responsibilities. If that's one solid nine-hour sleep, great. If you only need six, fine. If you'd rather make it three hours in the afternoon and six at night, go for it. I just think that healthy sleep is sleeping when your own rhythms urge it.

So, no, I don't want to sleep like a baby. I just want to sleep the way my body tells me I ought to be sleeping.

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