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 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 20, 2020 at 12:02am

#993759

Patterns

Yay! MadLibs!

PROMPT September 20th

Complete this sentence:

The next time I ___________, I will __________.

The temptation is there to plug in random verbs, so I found a random verb generator and cranked it up.

The next time I settle, I will whisper.

The next time I put, I will hope.

The next time I revise, I will award.

The next time I teach, I will travel.

The next time I interpret, I will assist.

Hm. Maybe if there were an option to limit to intransitive verbs. But, you know, I don't build these things; I just wreck 'em.

So I also tried a sentence generator, pulling entire verb phrases out of the randomly generated sentence. For example, my first result was: "She had a difficult time owning up to her own crazy self." Plug in the phrases and:

The next time I have a difficult time, I will own up to my own crazy self.

This is fun, if maybe a little too on the nose. Let's try more:

The next time I should have been a time and a place, I will be it.

The next time I crash on the shore, I will be a lovely sight.

The next time I play with Legos, I will be at camp.

Okay, so not all of them work. Still fun, though.

Oooh, found another one where you can specify more parts of speech.

The next time I abhor Hades, I will disengage mousse. (I changed the verb tenses, but that's about it)

The next time I immobilize medication, I will cache vise.

Nah... maybe not.

The links I used are here:

https://randomwordgenerator.com/sentence.php

http://watchout4snakes.com/wo4snakes/random/randomphrase

Be careful with that second one. I don't fully trust it. There's also one I found that didn't quite do what I was looking for, but someone might be able to do something with it:

https://www.michaelfogleman.com/phrases/

Anyway, yeah, some of these are nonsense, but nonsense can be fun. Hell, poems -- entire novels -- have been written with nonsense, and people drool all over them looking for meaning (I'm talking about YOU, Ulysses.) Because that's what we do... we look for patterns. Usually this serves us well, especially if we're being stalked by a lioness. Sometimes it leads to impressive accomplishments. Other times, it's just cool -- people looked up at the stars and decided they made little connect-the-dots sketches, and named them, and so we have the constellations.

And then, sometimes, things get weird, like with pareidolia -- seeing a face in a mesa on Mars, or Jesus on a piece of toast. But then, that's also how we get a "man" (or in some cultures, a rabbit) in the moon.

According to Wiki (look, I can't be arsed to check every reference, especially when I'm just playing like I am today), Pareidolia

"can be considered a subcategory of apophenia

. So of course I had to be reminded what apophenia was, and at that entry I found this:

Apophenia... is the tendency to mistakenly perceive connections and meaning between unrelated things.

And this:

In contrast to an epiphany, an apophany (i.e., an instance of apophenia) does not provide insight into the nature of reality nor its interconnectedness, but is a "process of repetitively and monotonously experiencing abnormal meanings in the entire surrounding experiential field". Such meanings are entirely self-referential, solipsistic, and paranoid—"being observed, spoken about, the object of eavesdropping, followed by strangers".

As that article explains, taken to an extreme, it's a disorder or an indication of a disorder. And it can certainly lead to fallacies. But... that's what I do as a writer: I find or imagine connections between disparate things. Hell, a metaphor can be considered an apophany by this definition.

Metaphors are what humans do. It's what we're good at. Like... when airplanes were invented, we called their fixed appendages "wings," even though they're not exactly the same as the flappy wings of the birds which inspired them.

It's important to not take these connections too far, to mistake symbol for referent. This sort of thing is, however, what creativity in writing is all about - at least to me. And I mean, hell, isn't a realistic painting just a case of intentional pareidolia? Take La Gioconda, for example, better known as Mona Lisa -- da Vinci intended it to represent a woman's face, and that's what we see. We don't actually see a person, of course, just a bunch of paint splotches, but our minds are tricked into seeing some long-dead Italian.

Randomness can lead to imagined connections, and imagined connections are the heart of creativity. So...

The next time I write a story or poem, I will be more conscious of apophenia.

9 Like • Comment • View 8 comments

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