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').

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

November 19, 2020 at 12:01am

#998678

Strangers in the Night

Really, isn't everyone a stranger? Philosophically speaking, I mean.

PROMPT November 19th

Share a time when an interaction with a stranger had an affect on you.

I walked into a brewpub and talked to a stranger. The stranger gave me beer and took my money. That had a wonderful effect on both of us: I got beer and she got money.

On the other extreme, one time a stranger pulled out in front of me while I was driving. My car hit his, I went to the hospital, we went to court. Those weren't great interactions, but they had an effect.

Incidentally, I wasn't seriously injured, it was entirely his fault, and I ended up with a small settlement. Car was totaled, though. Was he injured? I don't know and I don't give a shit.

In writing this, I've been trying really, really hard not to unleash my inner editor.

I have failed.

People mix up "affect" and "effect" all the time. It's something I notice when it's wrong, but I don't end up thinking less of whoever does it, not like I do with people who can't keep "its" and "it's" straight, or "there," "their," and "they're." Or how I pass judgement on people who type "loose" when they mean "lose," or "opps" when they mean "oops." Those last two are generally the same people, and I think they follow some sort of Law of Conservation of Os -- though that doesn't explain where they get the extra P from. Probably from drinking too much beer.

Anyway. As I was saying. "affect" and "effect" are often confused, because frankly they're confusing. You can affect a person, or someone can have an effect on you. Someone once said, "'affect' is a verb but 'effect' is a noun." And in those common uses, they're right... but here's the confusing part:

Affect can also be a noun. And effect can also be a verb.

Those uses are less common, it's true, but anyone following the "affect-verb, effect-noun" rule is going to be caught off guard, like when you realize just how many exceptions there are to "I before E except after C."

Example of affect as noun: "The patient's affect was consistent with schizophrenia." (yeah, this version of the word is mostly just used in psychology)

Example of effect as a verb: "With perseverance and tenacity, and a whole lot of money, we can effect change in the world."

Isn't English fun? I think it's fun. I have a great deal of affection for it, even when I'm effectively putting on an affectation of pedantry.

Speaking of "loose" and "lose," there's a similar problem there: "loose" is normally an adjective, but it can be a verb: "He loosed an arrow at the target." This is, of course, distinct from "He will lose the archery competition because he mixes up 'lose' and 'loose.'"

"Only loosers care about spelling and grammar." "You mean 'losers.'" "Your mother is a looser." "Yeah? Well, yours is looser."

And, for the record, I know these things because a series of strangers taught me English grammar and spelling when I was in school. I guess they had an effect on me.

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