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 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.

September 5, 2020 at 12:02am

#992410

The Force Is Strong in This One

Maybe I already have a superpower. How would you know?

PROMPT September 5th

Imagine you have a superpower of your choice. However, no one knows about it! What does your superpower enable you to do? Do you confide in anyone? What happens?

If I confide in someone, that violates the "no one knows about it" requirement. Just saying.

Here's the thing about actual comic-book style superpowers: you'd better hope you have an extremely powerful one, or once somebody finds out about it, you're going to disappear for a very long time while they run tests on you.

So you just have to make sure that nobody finds out.

It's kind of like winning the lottery jackpot. I mean, being rich is kind of a superpower by itself (one I wouldn't mind having). But the point is, if you go public with your lottery win and stand up there grinning and holding that gigantic fake check in front of cameras, it will be approximately 15 microseconds before your mobile starts lighting up with scammers, long-lost relatives, and long-lost scammers. And then there's the IRS.

So, you know, you discover you can fly, and you swoop around for a while and people see you, and then next time you land you'll find yourself netted, tagged, and rushed off to Area 52 for observation, tests, and vivisection. Yes, Area 52. Area 51 is too public.

Or say you can Jedi Mind Trick people into thinking these aren't the droids they're looking for, and as soon as it wears off, they'll come for you wearing earplugs so you can't convince them of anything.

Super-strength is another obvious and popular one. Also very useful. And maybe you could pass it off once or twice as adrenaline or whatever, but eventually they'll catch on and boom - underground in a vault made of 6-foot-thick titanium walls.

Then there's invisibility. You might think, "well, they can't Gitmo me if they can't see me," but really, invisibility is trivially easy to get around. Blow some flour into the room, and then your outline becomes powdery-visible. Also, you're naked. Seriously, though, invisibility wouldn't be all that great - in order to see, light has to get focused by your eye lens and absorbed by your retina, and if those aren't visible, you're blind, which effectively makes everyone and everything else invisible to you. Better would be if, instead of actual invisibility, you have the power to make people not notice you at all. This happens naturally when you hit about 50 years old, though, so you could just wait.

Worst of all are the powers that rely on external objects, like Green Lantern's ring or Ant-Man's suit. (I'll spare you the science of why Ant-Man could never be an actual thing.) Someone steals it and poof, no more powers.

So if I could choose a superpower, it would be one that would be useful in multiple situations, easy to conceal, and has a chance of getting me out of Area 52 should I slip up and get captured by dudes wearing dark suits and sunglasses. Telekinesis, say. Being able to move objects with mere thought is more useful than most people give it credit for. Especially since I wouldn't have to, you know, get up from my chair to get a beer. Or hold it in my hand to drink it.

What, you were expecting me to use it to fight crime? Or commit it? Nah. Too much work. Can't be arsed.

(Edited to add) oh yeah, Merit Badge.

Today's goes to Lilli 🧿 ☕ because I want to see a video of this:

Might be a good night to dance in the street wearing my witchy apparel, while sipping cabernet from my crystal skull goblet.

Still in Birthday Week, so another one of my free MBs will go to a commenter of my choice tomorrow!

11 Like • Comment • View 10 comments

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