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

September 7, 2023 at 8:33am

#1055294

Less Than Super

Today's link, from Atlas Obscura, is now outdated, having languished in my queue for several weeks. It's just as well, because if it'd come up anytime before August 30, it would have led to a rant, and I ranted about that particular subject enough for this year.

Just How Super Is The Coming Supermoon?
July’s full moon will be thousands of miles closer to us than usual. Learn how to measure it with your hand.

I'm linking it despite its lack of timeliness, because it's got general information of value to skywatchers. And some dubious information that I, of course, am going to question.

The Moon is growing now, more and more of it visible every evening. Though it is technically full for just one night, it will look mostly full for a night or so before and after that.

Again, not current information. And yet, I'm going to pick on it. It's technically "full" for just one moment, and half the time, that moment occurs during daylight hours wherever you happen to be. Traditionally, in most cultures that observe this sort of thing, a full moon spans three nights.

The term “supermoon” is a relatively recent introduction to the astrological lexicon, but these events are nothing new; they occur several times a year.

Okay, fine, I'm going to rant anyway. "Astrological?"

I'm not even going to legitimize the sentence that follows that quoted one by pasting it here. Suffice it to say it made veins pop out in my forehead with its utter wrongness.

A supermoon takes place when the Moon is nearest to Earth in its elliptical orbit around our planet.

And that happens once per orbit: at perigee. It doesn't always coincide with a full moon. When it does, that's what they've taken to calling a supermoon. I don't object to the terminology here; what I object to is when people draw the wrong conclusions from imprecise language.

The Moon can look about 14 percent bigger during perigee, its closest approach to Earth, than during its apogee, its furthest point. It can appear 30 percent brighter, too. If you pay attention throughout the year, you might notice how much smaller or bigger the Moon can seem from month to month.

Fair enough, though the article starts out by comparing perigee to average, then compares perigee to apogee. Also not mentioned: the horizon moon illusion, where the rising or setting moon might appear larger to the eye. There's a very reasonable explanation for this illusion, and it has nothing to do with earth-moon distance.

Incidentally, the article doesn't do a very good follow-up on its promise in the subheadline, the "measure it with your hand" thing. There might be a way to do this, very roughly, but I suspect that individual differences in things like arm length and hand size overshadow the supermoon/micromoon difference.

Which, admittedly, is one I've never seen. I wouldn't know a particular full moon was a supermoon but for the websites I frequent, which all gush about it every time it happens. I know some people claim to be able to tell the difference, and I don't doubt them, but my perception just isn't that fine-tuned.

I make up for it by insisting upon pedantry.

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