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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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 8 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').

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.

March 17, 2024 at 9:27am

#1066413

Revisited: "Power"

When I do these Sunday retrospectives, I can usually find a lot of things in common with the guy who wrote the earlier blog entry. The further back in time I go, the less I find in common, but such is life.

This personal update, though, from February of 2008, is tough for me to relate to, and difficult to re-read: "Power"

The wind has been crazy here, today. It started sometime during the night, whistling between the houses and threatening to spin my attic vents right off.

While I still live in the same house, I've replaced the roof since then, and goodbye old-fashioned spinning attic vents.

When I woke up, at around the crack of noon, it was still going on.

I'm always a late to bed, late to rise person, but, with age, my ability to sleep until noon has vanished.

I figured I'd grab some breakfastlunch and go write this week's Comedy newsletter.

Well, I'm still writing Comedy newsletter editorials, at least. I'm not sure if this is a good thing or a bad thing.

That, of course, is when the power went out.

I've since purchased and had installed a whole-house generator, an automatic start one hooked up to the city gas line. While I justified it to myself and others with my medical devices and a basement sump pump that tends to be most necessary at the times when the weather's lousy enough to cut the power, the real reason is so I never lack internet access.

I'm sure the neighbors appreciate the blast of a nat-gas engine cranking up and running like a combination leaf blower / lawnmower every time we lose electricity, but if anyone ever said anything to me, I'd absolutely claim medical necessity.

Now, I don't have my truck this weekend. A friend of mine is using it to move.

I don't drive pickups anymore, either. And that's one of the reasons why: someone inevitably asks to borrow it or, worse, me.

My wife was out in her car, so I was stuck.

No longer married, either.

I worked some crosswords for a while, but then started feeling antsy about getting the newsletter in.

I miss old-style paper crossword puzzles. Probably they still make them, but there's plenty to do on the internet.

So I called my wife to drive me to my office.

Nor do I have an office. Or a business. Or a job. (My choice.)

I booted up my computer, checked some email, started looking for items to include in the newsletter (funny things about romance, mostly - it's the Singles Awareness Day issue of the Comedy newsletter), and the power went out.

Well, I continue to rag on Valentine's Day most years in the Comedy NLs. Again, good or bad thing? I don't know. With my memory the way it is, I'm sure it gets repetitive.

Before we went home, though, we visited the hospital, where my dad's still being treated for his UTI.

Dad would go on to pass on the following month. Not of the UTI.

When the doctor called back, she said, "I don't know too much about that patient. UTI. Tested positive for influenza. Going to have to keep him at least a couple more days. Did they make you wear a mask in the room?"

As a reminder, this was 2008, long before masks became a political statement and were merely an effective means of helping to prevent disease transmission.

"Oh, and there's some question about whether we can send him back to assisited living or if he'll have to be moved to a nursing facility."

Wow, a rare misspelling. Eh, it happens. And I'm certainly not going to edit out that wart at this late date. Anyway, as I recall, they eventually moved him back to the assisted living facility (which specialized in Alzheimer's care).

He didn't die of the flu, either. Which would have been an irony I would not have appreciated. See, he had been born in New Orleans in 1917. A year later, his mother, quite a young woman at the time, dropped dead. The family never told me more details than that; when I was a kid, I just accepted that dying is what grandparents do.

It wasn't until much later in life that it dawned on me: New Orleans. 1918. Death of a young person. Oh... right. Spanish Flu.

Not going to leave you in suspense. Official cause of his death was "complications from Alzheimer's."

But I got home and did the Comedy newsletter anyway. Sometimes you just have to be funny, even if there's nothing left to laugh at.

And that philosophy, folks, seems to be the one thing about me that does not change.

There's a kind of power in that.

2 Like • Comment • View 1 comment

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