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!

a day 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

a day 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.

November 22, 2019 at 12:09am

#970185

Missed Connection

PROMPT November 22nd

Write about a missed connection.

I'm convinced that USB connectors exist in four-dimensional space.

You try to plug one in and it doesn't work, so you give it half a turn and try again. It still doesn't work. So you give it another half a turn - rotating it 360° from its original orientation - et voilà! It slides right in, no problem.

A 360° rotation in our normal, everyday universe is the same as a 0° rotation. But in the subatomic realm, you get weird things like electrons and other particles with "spin ½," which only return to their original orientation after a 2*360=720° rotation. One hypothesis for this weirdness is that it's partially rotating in a fourth (or fifth) dimension, a property limited to fermions - and USB connectors. "But you only rotated the USB connector by one and a half turns." Yes, but the first time it didn't work, remember? Another half-turn and it would go back to not working, which was its original orientation.

I'm going to be traveling next month. It's only for six days, but the complication here is that, usually, when I travel, my housemate is here to take care of my cats - and vice-versa. Well, this time, she's going to be away as well, so we needed to arrange for cat-sitting. All my other friends live too far away now, so I opted (this became my responsibility) to go with a professional pet care service. So I did extensive research (translation: I chose the third result on Google Search, not trusting the first two to be anything other than a Search Engine Optimization scam, which is my advice for the day) and called a local place that does that sort of thing.

I spoke with the nice lady and we arranged for a meeting at chez moi so she could meet the cats. This is harder than it sounds because all the cats tend to be shy around strangers. So on the arranged date, which was Wednesday, I took the time to ensure that all the cats were at least inside where there was some chance she'd be able to get to know them, and that their litter boxes weren't too bad. The time for the meeting came - and went, without a sign of the pet sitter.

I called her and, lo, she had utterly forgotten about the meeting. This does not bode well, of course, because what if she forgets when the cats need their food and water? But it's the accursed holiday season, and at this point it's probably too late to pick someone else, so when we finally did connect, yesterday, I wanted some reassurance that someone would actually show up to feed the cats. She assured me that, yes, it was on the calendar, and she'd contact me every day after the deed was done.

My reasoning for going ahead with this was simple: she messed up once, and when you're running a business and mess up once, you do everything in your power to recover a good impression. I thought about installing a few nanny cams for insurance, but then I remembered that I would also get notifications when someone (dis)arms the security system, so I'd have verification that at least someone showed up.

So yeah, sometimes you have to rotate something by more than 360° to get it to fit, but then it runs as smooth as anything.

5 Like • Comment • View 4 comments

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