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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Brandiwyn🎶

a day ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

August 25, 2023 at 8:13am

#1054570

Symbol Minds

Writing is all about symbolism. Even if you don't use it consciously, it'll show up. When your work is being discussed in a graduate-school literature class, they'll find it, though you swear that wasn't what you meant.

Failing all of that, every one of our letters and punctuation marks is also a symbol. So there.

Today's article, from Cracked, takes us out of the realm of writing and into consensus reality.

5 Everyday Objects With Hidden Symbolism

To brag a bit, most of these, I already knew. I just never figured it was worth an article. That's why they get paid the big bucks over there, I guess.

Symbolism might mostly seem like a word overused by English teachers and people who don’t understand movies as much as they think they do, but it’s nevertheless both omnipresent and essential.

The Venn diagram (a symbol) of "English teachers and people who don’t understand movies as much as they think they do" is a pair of concentric circles (probably).

Weirdly, too, some symbols become so commonplace that while their intended meaning holds up, their origin can get lost to time.

And sometimes their meaning completely shifts; the swastika, for example, started as a symbol for the divine.

Here are five bits of everyday symbology you might not know about…

Or, as in my case, maybe you do know about them, but you never know what new information you might find. Of course, standard disclaimer: this is from a comedy site, not a scholarly research paper.

5. Barbershop Pole

I have vague memories of asking my parents about this one when I was a kid. What I have no memory of is how they might have answered it in a way that Kid Me could have understood. But I eventually got the pole truth.

The fact is that it has absolutely nothing to do with haircuts, and more to do with something a little more morbid. It comes all the way from the middle ages, when barbers also performed basic surgery, most notably bloodletting.

They had the right tools at the time. And yeah, this reminds you of Sweeney Todd for good reason.

4. Eagle on the Dollar Bill

The fact that these bits of cotton and linen are packed tip to tail with symbolism shouldn’t be new to anybody.

Almost all money is rife with symbolism. Even Monopoly money. It's like, "We have a flag, but there's just not enough symbolism on it, so let's put it on the money."

Most notably, the eagle seal on the back is repeatedly, almost obnoxiously obsessed with the number 13. Which is very spooky, if you are not aware of how many original colonies there were.

And if you're not, and you're over the age of 13, then you probably don't know enough to bother with symbolism, anyway.

It would have been much more amusing had they gone with Franklin's proposal to make the turkey the national bird. Which, I think, was one of ol' Ben's jokes, but it could have happened.

3. The Bluetooth Symbol

Not to toot my own flute here, but yeah, I knew this one too. Didn't even need to have it pointed out. Well... to be completely honest, I misinterpreted the symbols, but I knew it was based on Norse runes, at least.

One symbol that I would bet almost no one would be able to crack without help, though, is the symbol for Bluetooth.

Hi! I'm almost no one.

Well, the bluetooth symbol is actually a rune, because Bluetooth is ancient magic that was found in a Viking tomb. Which is only half bullshit. It’s not magic, but it is a rune with Viking connections.

I'm still not sure about the sobriquet "Viking." I've done other blog entries about that. People ascribe the runes magical powers, but the facts are: they existed and were used for writing both magical and mundane. I'm not real clear on their connection to Greek and Roman alphabets (we use a modified Roman alphabet, of course), but there's too much similarity in some of the symbols to be entirely coincidental.

The symbol itself is a combination of the runes hagal and bjarkan, or “HB” for Harald Bluetooth.

The rune names have variations, too, because they weren't limited to what's now Norway. To make matters more complicated, hagal, or hagalaz, has a more common variation that looks kind of like an H with two slanted crossbars instead of the one horizontal one. The one used in the Bluetooth rune is more like what we call an asterisk, with six radiating arms, four of which are embedded in the bjarkan, or berkana, rune (the "B" looking part on the right).

For full disclosure, then, I didn't see it as hagalaz/berkana, but as a reverse kenaz with a berkana. It's a connection I should have made, but didn't.

Kind of like when I try to use Bluetooth: it should make the connection, but usually doesn't.

2. Cartoon Heart

If you’ve ever read an anatomy book or committed a horrible, grisly murder, you probably know that the human heart looks a whole lot less cute than the version we get on Valentine’s Day cards.

Few internal organs are "cute."

So how did we end up with the cute little peach-shaped symbol? Well, there’s two prevailing theories...

I propose a third, which is that it vaguely resembles certain sexual organs, in certain circumstances.

1. The Middle Finger

Okay, this one, I can't claim prior knowledge of, except some vague idea that it resembles certain other sexual organs. It's important to learn new shit, especially about such a grand and noble gesture.

Have you ever been walking around and found yourself in a confrontation with a fellow human? Maybe you’ve been in an argument with a parent, sibling, friend or partner. However you got there, you may have noticed them proffering a single, extended middle finger in your direction.

Sadly, this article stops short of actually explaining the origin of the New Jersey State Bird, so I had to go to that equally questionable source, Wikipedia:

"The gesture dates back to ancient Greece and it was also used in ancient Rome. Historically, it represented the phallus. In the early 1800s, it gained increasing recognition as a sign of disrespect and was used by music artists (notably more common among actors, celebrities, athletes and politicians; most still view the gesture as obscene)."

In the UK, it is, or maybe was, more common to use two fingers. I don't know why that is, either (there's a story people tell involving longbowmen at the Battle of Agincourt, but I'm pretty sure that's the hand-gesture equivalent of folk etymology).

There exist myriad symbols beyond these, many of whose origins are still shrouded in mystery. Perhaps another time, I'll delve into some of them.

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