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

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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

December 29, 2020 at 12:02am

#1000975

Cold Moon

"30-Day Blogging Challenge ON HIATUS" [13+]:
29. December Full Moon (aka Cold Moon)
Write something about the full moon.

Dammit.

We really, really need to stop associating full moon names with Gregorian calendar months.

The concept of naming full moons evolved separately from the Gregorian calendar and its predecessor, the Julian. The folklore associated with moon phases was tied to seasons; that is, equinoxes and solstices.

The Cold Moon - or whatever it's called in whatever culture - was the first full moon after the winter solstice in the Northern Hemisphere (hence "Cold"). This has an approximately 1/3 chance of occurring in our December, and about a 2/3 chance of occurring in January.

I know I've ranted about this sort of thing before, in connection with the "Blue Moon." It's as if the internet just doesn't pay attention to a single word I say...

...which is fair, I suppose, since I'm basically nobody.

It's still the hill I choose to die on.

Once more, with feeling: there will usually be three full moons between equinox and solstice, or vice versa. On those rare occasions when there are four, the third of the four is the Blue Moon. "Rare" meaning about once every three years or so; hence the expression "once in a blue moon."

There was a time, some few years ago, when we had two full moons in January, none in February, and two again in March (January and March having 31 days each by fiat). This all by itself should have been sufficient to stop this "second full moon in a calendar month is a Blue Moon" nonsense, but apparently that was not the case, as the wrongness persists to this day.

And again, the method of naming moons relative to equinox/solstice is independent of whatever calendar is being used, so it is more universal.

An astronomer I'm acquainted with, Phil Plait, accepts that either method is "right," because like it or not, the Gregorian calendar is what we're stuck with, at least for now. I have a great deal of respect for Phil; I spent a week in a cabin in Colorado with him and his wife and a few other people. I think I understand his reasoning for that; as with a dictionary, it's a matter of descriptive versus prescriptive, and from his point of view anything that gets people interested in looking up at the sky is a Good Thing.

I agree with that bit: if you can be arsed to get out of your warm house in December or January (again, I'm being Northern-Hemisphere-centric here) to look up at the Moon in all her glory, that's great. Do it. Or if you spend a week in Colorado looking at the sky, preferably during a time when there is not a full moon because without the competition, and also preferably in the company of a professional astronomer who can show you shit through a telescope that's even more amazing, the rest of the sky can be even more glorious.

Where I disagree with Phil is that I believe we can't just abandon the folklore; trying to fit it into modern timekeeping is, to me, a bit like trying to hammer a round peg into a square hole. Or maybe it's the other way around; whatever. Doesn't matter; point is it's not going to fit right no matter what you do.

And honestly, I'm of two minds regarding calendar reforms. Personally, I think we should either go back to a purely astronomical one, where solstices and equinoxes mark the seasons but the moons are tracked as per one's chosen folklore; or abandon the concept entirely in favor of something like the Tranquility Calendar

.

Still. Go look at the full moon, if weather conditions allow. I know it happens every 29 days or so, but it's still worth noting.

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