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:

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').

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.

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

1 Like

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

March 1, 2021 at 12:03am

#1005530

Cry Wolf

Finally, the despicable month of February is over. To celebrate, I thought I'd do another Merit Badge giveaway; see below. But first a word about a word.

Never Say Wolf
How taboo language turned the wolf into a monster.

The article is fairly long, but intriguing to me because I've been fascinated by taboo words for a very long time.

The howl of the wolf, and the fear that accompanies it, sounds across millennia. Comparing the mythologies of cultures that descended from ancient European tribes, wolves loomed large in their minds. There are myths of heroes brought up by wolves, of great wolves who will devour the sun, of wolves guarding the underworld, and of warriors taken over by the spirit of wolves.

If you've ever seen a wolf -- or, on a cold winter night, heard one in the distance -- you can understand this.

Speakers had to find ways of referring to wolves without naming them. The word for wolf becomes taboo: It shouldn’t be said. Instead, the magic of summoning through a name can be tricked. By changing the sound of the word, by using another word, perhaps borrowed from another language, or by using a descriptive phrase rather than the word itself, speakers could talk of wolves, but avoid the dangerous word itself.

This article focuses on wolves, but there were similar taboos for other feared animals, notably the bear. If you can stomach another article about linguistic taboo, here's the story

of the words used to describe ursines.

What fascinates me, though, is that it's not just the feared that become linguistic taboo. The sacred and the profane both get the same treatment. Notably, the original Hebrew pronunciation of the word for "God" has been lost to time. At the same time, we're always coming up with new ways to say curse words to get around peoples' sensibilities (such as "Holy forking shirtballs" made famous by the hilarious series "The Good Place.")

Another example from modern times is the word "retarded." It's taboo in many circles now, but I have no problem using it. I understand that the word was coined as a value-neutral way of describing someone who's mentally challenged, or "slow," hence the Latin-root "retarded." Before we had "retarded," there were three words with precise definitions: idiot, cretin, and moron. I forget which is which now, but each one of those words were assigned to people who tested at certain points of the below-average IQ scale. Then, being assholes, people started tossing those words around with great abandon, using them to simply describe someone they didn't like. "That guy's an idiot." "She's a moron." That sort of thing. "Retarded" was meant to replace those words, but of course, being assholes, we started using it to describe people and situations we didn't like.

Once that fell out of favor for that reason, they started using the word "special" as in "special education," the idea being, I suppose, that you take a word with positive connotations and maybe people will stop being assholes to the mentally challenged. And so, of course, now you can't call someone "special" without implying, deliberately or otherwise, that they're a moron.

My prediction is that there is no word or phrase that we can come up with to describe that condition which will not eventually become stigmatized.

And, of course, there's the word that cannot be uttered, the one that was used to dehumanize an entire group of people.

I, personally, came up with certain taboos myself. For instance, I almost never name the seventh planet, you know, the one orbiting between the orbits of Saturn and Neptune. This is because assholes just can't help making stupid puns about it, regardless of how it's pronounced (you have a choice between "Yer Anus" and "Urine Us," unless you want to get all Ancient Greek and pronounce it Our-Ahn-oos, at which point no one will know what the hell you're talking about).

Anyway. Point is, "wolf" is a case study for taboo word modification. You have a word, everyone knows it, but you don't want to bring down evil spirits (or public outcry, in modern times) so you use a different word. That one becomes the default word, but you don't want to bring down evil spirits, so you change the word again. And so on until some hypothetical future time when people stop being assholes. Or superstitious.

Merit Badge Mini-Contest!

I thought about asking for other examples of taboo words, but then I remembered that this blog is rated 18+ so most of the fun ones are off limits -- another variant of taboo against certain language. And you can do that in the comments if you want, if you can think of one that won't break the rating. Or you can talk about wolves or bears or planets or whatever. But as it is a) no longer February and b) at some point over the last few days, this blog ticked past 100,000 views (a pleasant milestone and I appreciate the support), as long as you comment below, today, you have a chance. The comment should have *some* relevance (that is, just posting something like "I hope I win" won't count), and I'll pick a relevant comment at random and give the person a MB tomorrow.

As always, the deadline is midnight WDC time tonight, at the end of Monday March 1. And I'll do this again, so you'll always have another shot at it.

2 Like • Comment • View 4 comments

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