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:

Kåre เลียม Enga

about 5 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

about 22 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

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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