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 6 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 8 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.

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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.

July 21, 2021 at 12:01am

#1014030

Plus C'est la Même Chose

More fun with "JAFBG"

[XGC]...

We look back on some historical practices with disbelief. What current practice do you think will be ridiculed in the future?

All of them.

Especially fashion.

You know all those fashion trends that we look back on and go, "What in the hell were they thinking?" Almost every one of them, at one time, was considered the pinnacle of self-decoration. Powdered wigs. Low-rise jeans. Feathered-back hair. Polyester disco suits. All of these things were the Hot New Thing when they came out, and all of these things are currently the butt of jokes (in the case of low-rise jeans, I mean that literally).

Each of them certainly had their haters at the time (or as I like to call them, "sensible people"), but they wouldn't have been trendy if the followers hadn't outnumbered the detractors.

It's not just clothing, though. I've had the misfortune of seeing those hardware store ads masquerading as reality TV, you know, the "buy a shit house and fix it up" genre. Some have legitimate aging problems, but others? "Ha ha, look at this green shag carpet; how blind were they in the 70s?" What they forget is a) At one point, green shag carpet was the In Thing, and b) In 40 years, that open-plan kitchen/dining/living/sitting/family/breakfast/study room with the granite countertops and burnt orange backsplash is going to seem ugly and stupid.

Those, though, are just the matters of taste.

100 years ago, the average life expectancy of someone in the US was around 50-55 years. (Source.)

Now, it's pushing 80. We look back on the medical practices of the past -- leeches, trepanning, whatever -- with horror and disgust. Our current medical practices seem bright and shiny now (hell, some of them seem like sorcery to me), but give it another 100 years and people be like "You mean they treated cancer with a combination of nasty pharmaceutical chemicals with horrible side effects?"

Obviously, what we consider right and wrong changes over time, as well. We're pulling slave-owner statues off of pedestals now. In a few decades at most, we of today will be cancelled by our robot overlords for having the audacity to own computers.

But hopefully -- assuming we make it that far, which is an assumption I'm not willing to grant -- the biggest thing people in the future will look back on with shock and anger will be: "They... burned... fossil fuels?"

But yeah. History backs me up on this: whatever the practices of a certain time period are, they will be mocked by those of a later time period (though to be fair, there might be a time of nostalgic references somewhere in the middle). Hell, even the "futuristic" architecture styles of the 50s and 60s don't say "future" anymore, but "past." The generation war now, marketed to be millennials versus boomers, is just the same shit all over again - back when the boomers were young, there was a "generation gap" and the then-younger generation (boomers) was going to Save the World. And when today's millennials and/or Zeds reach retirement age -- assuming they're still around and are able to retire, which, again, I'm not sanguine about -- they'll be pitted against whatever they'll call the younger generation, too.

I guess some things never change, after all.

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