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🎶

about 22 hours 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

July 1, 2024 at 10:03am

#1073404

Pedantonyms

Today, from Cracked, we have an article that should be of interest to writers.

5 Evil Mirror Versions of Words We All Use
You’ve heard about sparkles. Now prepare to darkle

Yeah, people avoid me because of my darkly personality.

Words have opposites. That’s one of the first thing we learn about words, thanks to Grover demonstrating to us the difference between “near” and “far.”

The truly messed-up thing is, I remember that bit. I can't remember what I had for breakfast yesterday, but I remember Grover's running and shouting with near-perfect clarity.

Well. Probably not. Memories change. But at least I know what he's talking about (in the words of Captain America: "I understood that reference!")

Some words, however, have no opposites. Think about “lie,” for example. We’re not talking about lying down (though, that also has no exact opposite either, with “standing up” being just kind of an opposite). We mean the word for telling a lie. We don’t have a word for the opposite, a single word that means “tell the truth.”

As much consideration as I've given to words and language, I'd never thought of that, instead having to rely on a juvenile dick joke site to point it out to me.

To the limited extent of my knowledge, French is the same way.

Other words, meanwhile, have opposites that you’ve never heard of.

Or maybe we have, as in the case of "darkle" (or at least its poetic relative, "darkling"), but it's usually fun to go through these countdown lists anyway.

5 Forget Sparkles. It’s Time to Darkle

This appeals to me on a visceral level.

Darkling then gave birth to the word darkle. If something becomes concealed in the dark, it darkles.

And here I figured it was the other way around.

4 If You’re Not Ambidextrous, Be Thankful You’re At Least Not Ambisinistrous

Oh, but I am. Ever hear the expression "two left feet?" That's Waltz.

Having two of the same foot would ruin your dance skills, whether it’s two left feet or two right feet, but somehow, we never say someone has “two right feet.”

Hopefully, we all were already aware that "sinister" derives from a word meaning "left," revealing how important conformity is to society.

3 Euphemisms Smoothen the Conversation. Dysphemisms Make it Worse

If "dysphemisms" weren't already a word, we'd have made it up (well, all words were made up; it's just a matter of how long ago they were). Consider how they coined "dystopia" as the opposite of "utopia," for instance, the latter being a pun of "eutopia."

Over time, it’s quite possible for euphemisms to become dysphemisms. Originally, the words “moron” and “imbecile” were objective ways of categorizing someone by I.Q. When those mutated into insults, we came up with a euphemism that was more acceptable: retard. In time, retard became a dysphemism more offensive than any earlier synonym, and almost more offensive than any other word period.

I've noted this before, which is exactly why I don't accept the taboo around the word "retard." (Besides, it's a perfectly good word in French, albeit pronounced slightly differently, meaning simply "delay.")

We replaced "retarded" with "special," and now you can't call someone special without them thinking you're slyly calling them an idiot (which was another precursor to "retarded"). No matter what words we come up with for people in the slow lane, it will inevitably become a general insult, and then we have to think of something new and value-neutral, which will quickly become an insult in turn, and we have to find yet another word.

Sure, it would be nice (a word that used to mean "foolish") if we stopped insulting people altogether, but that ain't gonna happen.

2 Serendipity’s Great. Watch Out for Zemblanity

Okay, one I've never heard of or would have guessed.

George Carlin had a joke about how we call it a miracle when someone barely escapes death, but it’s actually just as miraculous when someone who’s almost certain to escape death dies anyway. We need a word for when the world seems to conspire so events pan out in just the worst way. Let’s call that zemblanity.

I've just been calling it "the normal state of things." Or maybe "exactly what I expect." Pessimism pays dividends on those rare occasions when things actually go your way.

1 Natural Gas Replaced Unnatural Gas

Except that nothing's really unnatural, but okay.

“Gas” refers to any number of substances that happen to occupy a certain state of matter. You’re currently surrounded by oxygen gas and nitrogen gas, for example (at least we hope you are, because if you’re not, you have just moments left). “Natural gas,” however, refers specifically to cooking gas, which consists mostly of methane, along with a few other trace ingredients.

I've known about chemistry from a very young age. I'm pretty sure I memorized the alkane series (methane, ethane, propane, butane, etc.) before I learned the alphabet. So it has bugged me almost my entire life that we (in the US) also use "gas" as an abbreviation for "gasoline," which is obviously a liquid (albeit a volatile one). I'd call it "petrol," but then people would think I'm trying to be British.

The reason natural gas got that name is it had a predecessor, which wasn’t nearly as natural. Back in the 19th century, we used to make gas — out of coal.

Which is also natural, but whatever.

I won't go into the cooked-up controversies surrounding natural gas use right now. Another time. This one's about words, and I've reached the end of mine for today.

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