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 12 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 14 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').

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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.

May 25, 2020 at 12:09am

#984274

Superpowers

A friend of mine once told me -- I think he got it from a movie or something -- that everyone has a superpower.

PROMPT May 25th

What is the most useless skill you have? The most valuable?

I was describing to another friend once the way rabbits act when threatened. They're prey animals with few defenses apart from speed, so they get et a lot. They make up for this with their famous reproductive strategy: make a lot of bunnies and maybe some of them don't get et, and those go on to make more lots of bunnies.

Point is, with a reproductive strategy like that and a relative lack of defenses, bunnies are pretty much Russian fatalists when it comes to being about to get et. A wolf comes sauntering along, baring its teeth, and if the lagomorph can't escape, it'll just stop its own heart so it dies quickly instead of by being bit a lot.

My friend said, "Worst superpower ever."

Yes, of course I know it's "eaten," not "et," but some words are just too funny not to use.

Anyway, the friend who told me about the superpower thing told someone else that my superpower is punning. I don't know. I mean, I'm pretty good at it. But I think my real superpower, the one that almost never fails me, is being able to ignore rules about which drinks can be drunk before which other drinks. You know, the whole "liquor before beer, never fear" thing. Or is it the other way around? I honestly don't know because it doesn't apply to me. I can drink -- and have done so, on many occasions -- any type of alcoholic beverage, in any order. I might start with beer or whiskey, then move on to tequila or wine, maybe have some cider, throw in some rum... whatever. Sure, I get hangovers like anyone, but no combination can make me sick, only excessive quantities. Naturally there are mixtures that I stay away from; mixing wine and beer is an unholy abomination against all that is beautiful and right with the world, like carpeting your bathroom or drinking orange juice after brushing your teeth. But drinking one and then the other? Not a problem.

This practice does, however, tend to suppress my punning abilities, which is why I call it my true superpower. And it's mostly useless. I mean, it's nice to have because I drink for the taste, not to get drunk, and I like to sample different products. Getting drunk is usually a result, not a goal. Usually. Sometimes these days I read a news story and getting blotted becomes the goal, but not often. Anyway, I don't know if it's a "skill" or not, because it's not something I get better at with practice (and I've had quite a bit of practice), but it was the first thing that came to mind for "most useless skill."

As for the most valuable, I'd say, technically, the ability to design stormwater management systems. I mean, that's the one with the most earning potential back when I was using it. I expect I could still do it, so it counts as a skill, I suppose. But it's boring. There's also writing, which I think is valuable but no one else does; and finance, which keeps me from having to design stormwater management systems for a living.

"Valuable," of course, is a value judgment, by tautological definition. My mind went to "what skills can make me the most money?" But others might find that skills that produce non-monetary rewards to be the most valuable. Relating to other people, for example, or being a good parent, or the ability to dress in a matching outfit, none of which are in my skillset at all. I only care about money.

And booze.

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