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 2 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 4 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.

August 19, 2022 at 12:01am

#1036660

Now Lie In It

The best kind of article is the kind that confirms what I already believe.

Why You Shouldn’t Make Your Bed

Because it's an epic waste of time?

I'm kidding about the "best kind" thing, of course. But let's see what Art of Manliness says about this. No, it's not "make your wife do it." It's not promoting that kind of "manliness."

I have a confession to make: I don’t make my bed. I never saw the point in it; I’m just going to mess up the covers again that night.

Exactly.

Also, I don't like sleeping in a "made" bed. When I'm in hotel rooms, for example, the first thing I have to do in the bed is yank all the tucked-in covers out.

I don’t spend much time in my bedroom, and my guests don’t spend any...

Well, maybe if you made your bed, it would have guests.

I realize that not making your bed has a bit of slovenly shame associated with it.

Nope. No shame here. I don't have a drill sergeant coming to bounce a quarter off the sheets or whatever the hell they do in the Army.

In fact, when it comes to daily habits, it even has some cool cache. It’s the kind of foundational habit a four-star naval admiral could base a commencement address, and book, around, and even claim could very well help you change the world.

You know what you could be doing in the ten minutes you've wasted making a bed? Changing the world. Or, better, sleeping an extra ten minutes. Believe me, when I was working, that ten minutes was damn precious.

Given this cool, cleanly cache, I felt surprised (and a little vindicated) when I came across the following passage while recently reading A Bachelor’s Cupboard — a manual for young men on how to live independently published in 1906:

A woman who, as the mother of several sons, has many young men as guests at her large country house, says she can invariably judge a man from the care he takes of his room. A young man who has been well brought up, she says, never fails to turn back his bedclothes [sheets and blankets] upon arising in the morning...

So, hang on, don't make your bed, but do futz with the covers so they're all folded and whatnot? Still a waste of time.

So, the standard for neatness and cleanliness a century back was the opposite of what it is today: rather than pulling his sheets and covers back over his bed up to the headboard, a well-bred gentleman was supposed to drape his bedding over the footboard, leaving both the blankets and the sheet-covered mattress entirely open to the air. Such an airing out was thought to promote freshness and good health (hence why you would also place your pillows by an open window).

I will note that 1906 was before washing machines became a widely adopted thing. Back then, washing sheets and blankets would have been an all-day chore. Nowadays, you can even do it as often as once a year.

Kidding. But I did just have to buy a new washer/dryer because the old ones broke under the weight of my bed covers.

In 2005, a study was published which found that not making your bed may be better for you than making it.

"A" study? Not impressed. I don't not do it for scientific reasons, but out of sheer laziness.

More than a million dust mites live in your bed. These microscopic critters feed on the flakes of skin you slough off in your sheets, and thrive in warm, moist environments.

Remember that next time you're in a hotel room with their sanitized sheets (not so sure about the blankets, though). I'm always sneering at the clean-freaks who take black lights to hotel rooms and never once consider that they're sleeping in far worse conditions at home. Plus, then they go to AirBnBs, which don't have the big industrial-scale steam power washers.

Sleeping with dead, dehydrated dust mites may not seem significantly more appealing than lying with moist, live ones, but it’s their fecal matter (yes, dust mites are pooping in your bed) that trigger allergies and asthma.

As I don't have allergies or asthma, I'm pretty sure I'm doing it right.

Of course, to get the full, freshening effect of this, you should drape your bedding over the footboard of your bed, as The Bachelor’s Cupboard instructs, rather than leaving your sheets in a half-on/half-off rumple.

Who the hell has a footboard these days? Okay, me. I do. This is because I went through a period when I'd wake up with severe calf cramps, and pressing my foot against the footboard for a stretch is less disruptive of my sleep cycle than standing up to do it. So I bought a bed frame.

Anyway, point is, make your bed, or don't make your bed. I don't care. I'm far more concerned about how you treat other people than what you do in bed.

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