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 25, 2022 at 12:01am

#1041003

Not in Our Stars

Wrong time of year for this, but that's never stopped me before.

The Ancient Math That Sets the Date of Easter and Passover
Why don’t the two holidays always coincide? It is, to some degree, the moon’s fault.

And yes the headline uses the M word.

Passover is a springtime Jewish festival celebrating the early Israelites’ exodus from Egypt and freedom from slavery. Jews observe it by hosting a ritual dinner, called a seder, and then by abstaining from eating all leavened bread for about a week.

Hey, it's the yeast we could do.

Easter is a springtime Christian holiday celebrating the resurrection of Jesus Christ and freedom from sin and death. It is preceded by a series of holidays commemorating Jesus’s path to the cross. One of these holidays is Maundy Thursday, which, aside from being a great name for a holiday, is a remembrance of the Last Supper, which was a seder. In the United States, many Christians observe Easter by attending a ritual meal between breakfast and lunch, called a brunch.

That part cracked me up.

These holidays have a lot in common: They share themes of liberation and triumph; they both involve buying a lot of eggs; they were both a pretty big deal for Jesus.

To acknowledge that I'm writing this in late November, just when the winter holiday marketing season is gearing into overdrive and rolling coal at our collective Priuses, I'll note that there is something of a parallel here with Christmas and Hanukkah. There is, however, one incredibly important difference: while Easter was built off of Passover (and both holidays stole from Pagans), Hanukkah and Christmas (also stolen from Pagans) have fuck-all to do with each other, apart from generally happening when it's way too bloody damn cold in the northern hemisphere.

Without going into detail, Hanukkah isn't "Jewish Christmas" (my friend likes to call it "Blue and Silver Christmas"). But, like the holidays in the article, sometimes they happen to overlap (like this year), and sometimes they don't.

In the Gospels, the existential drama of Easter happens against the backdrop of Passover. Yet about 15 percent of the time, the two holidays actually occur a month apart.

Those are good years for me. See, my cousin usually wants me to travel for Passover. Which is fine. Except when Passover falls on Easter, in which case traveling up the Northeast Corridor is the First Circle of Hell.

Anyway, the rest of the article goes into the differences between the Hebrew lunisolar calendar and the Christian solar calendar.

During the month of Adar (which directly precedes the Passover month of Nisan), the ancient rabbinical court would decide if it was springy enough outside for Passover. If spring seemed to be on track, Nisan could occur. But if it wasn’t warm enough outside yet, the rabbis would tack on another month of Adar. They called this leap month Adar II.

Early Rabbinical Judaism was very creative with names.

Today Roman Catholics and most Protestant traditions now celebrate Easter after March 21 on the Gregorian calendar. But the Eastern Orthodox Church uses the older version of that calendar, known as the Julian, to determine the date of Easter and other festivals.

So it's not just different religions' calendars that cause issues, but that of different sects of the same religion. This in no way surprises me.

Thanks to tiny wobbles in Earth’s orbit, some years are a second or two longer or shorter than others. So every year, the International Earth Rotation and Reference Systems Service announces whether to add a leap second in order to align Earth time with solar time.

Yeah, it looks like they're abandoning, or at least pausing,

this practice.

And as it happens, the first night of Passover can never fall on Maundy Thursday, even though that holiday commemorates a seder. That’s because Passover can never begin on Thursday, ever. “The calendar is rigged so that [seder] can fall only on certain days of the week,” Dreyfus told me. “If Passover started Thursday night, it would push Rosh Hashanah the following year to start on Saturday night.” And neither Rosh Hashanah nor Yom Kippur, the two High Holidays of the Jewish year, can fall the day after Shabbat.

Just in case you thought it wasn't complicated enough.

But no, to directly address the article's subhead, it's not the moon's fault at all. Nor is it the sun's. "The fault, dear Brutus, is not in our stars, but in ourselves, that we are underlings. Might you become master of your fate through choice—no matter what the stars say?" When it comes to holidays and observances, we're all servants of the calendar, and the calendar—any calendar—is purely arbitrary. I could argue that the Hebrew calendar, tied as it is to lunar cycles, is somewhat less arbitrary than the Gregorian, which is tied to nothing (well, sort of, mostly). But it's still arbitrary: Hebrew calendar months start on new moons. Why not full? Or half?

Sometimes I think we should just abandon the whole thing and adopt the Tranquility Calendar.

Other times I go in the complete opposite direction and want a calendar more obviously tied to actual astronomical observations: solstices, equinoxes, moon phases.

Hardly matters, though. Inertia favors the Gregorian. We can't even agree on stopping this Daylight Saving Time nonsense; calendars ain't gonna change.

Well. Eventually they will, because even the orbits of the Earth and Moon are, over a long enough timescale, chaotic. Or we could disappear, along with our calendars. But we're stuck with these complicated things for the foreseeable future. Fortunately, you don't have to make any of the observations and computations yourselves; someone else will tell you when it's time to celebrate whatever.

Because yesterday was Thanksgiving, I had nothing better to do than go to the movies.

One-Sentence Movie Review: Bones and All:

A meaty movie with a side of mashed metaphors and symbolism sauce, this story of two fine young cannibals in the 1980s was filling Thanksgiving Day fare for me.

Rating: 3.5/5

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