Complex Numbers (Book)

Blog Calendar

◄ September ► 2020
S	M	T	W	T	F	S
		1	2	3	4	5
6	7	8	9	10	11	12
13	14	15	16	17	18	19
20	21	22	23	24	25	26
27	28	29	30
Archive				RSS

About This Author

Contact Me

Come closer.

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/

Signature for Honorable Mentions in 2018 Quill Awards

A signature for exclusive use of winners at the 2019 Quill Awards

Signature for those who have won a Quill Award at the 2020 Quill Awards

Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 3 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 5 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

September 4, 2020 at 12:01am

#992321

Once in a Blue Moon

I've done this before but I continue to see misinformation out there.

PROMPT September 4th

Write about a weird, cool, unbelievable, or interesting fact you know, but don’t think many other people do.

There is going to be a full moon on October 31.

That's not the salient fact for the prompt, though. The relevant fact is that this full moon will not, contrary to widespread reportage, be a Blue Moon.

The calendar we use, the Gregorian, is purely arbitrary, like most of our measurements of time. It's achieved worldwide acceptance, but it's not connected to anything real. Not the solstices, not the equinoxes, and certainly not the phases of the Moon. All it does is attempt to start at (nearly) the same point in the Earth's orbit every time, which is functional enough for a calendar.

Other calendars are in use, ranging from purely lunar (months based on actual Moon cycles) through lunisolar (a lunar calendar that occasionally adjusts to align with the solar calendar, like the Hebrew calendar). There are other proposed calendars, but I won't go into them here; the only relevant thing is that all of our dates are simply social constructs.

The lunar phase cycle is approximately 29.5 days -- that's how long it takes for our satellite to return to a particular phase. Our solar year, in contrast, is roughly 365.25 days. Divide the one by the other, and you get about 12.4; consequently, there are usually 12 full moons in a solar year, but sometimes the number is 13.

I use full moons because that's the most obvious (and awesome-looking) phase, but also because pre-technological humans were inclined to use full moons to subdivide time. Another thing they measured, as seen at sites such as Stonehenge, were the solar quarters: the solstices and equinoxes. Due to fuckery involving us having an elliptical orbit, the time between solstice and equinox, or vice versa, is not exactly the same for each quarter, but I'm going to call it a quarter of a year, or roughly 91 to 92 days.

So our ancestors, who were more inclined (and more able) to actually watch the sky because things like electric lights, Downton Abbey, and the internet had yet to be invented. And they cared deeply enough about the celestial clockwork to give all of these occasions -- full moons and quarterly solar transitions -- names.

The names themselves varied from culture to culture. Perhaps the most well-known full moon name is the Harvest Moon, but there were also things like Cold Moon, Wolf Moon, and so on. And all of these names were tied to the seasons; that is, solstices and equinoxes.

Now, look up these names online and what you generally find is misinformation. They'll tell you that the Wolf Moon occurs in January, the Flower Moon is in May, and so on.

Lies.

These names, whatever they were in different cultures, predated the Gregorian or even the Julian calendar. They have roots deep in natural cycles and folklore, not mechanical timekeeping devices or arbitrarily designated dates. Because, as I said, a season lasts around 91 days or so, and a moon cycle is 29.5, there are usually three full moons between solstice and equinox, another three between equinox and solstice, and so on around the annual cycle. And each one of those moons had a name, because the essential purpose of naming them was to mark the seasons with something more obvious to the farmers and herders than the zenith position of the sun.

But if you've been following along, you can see what the problem is: sometimes there are four full moons between solar quarters, which would throw off the naming conventions. This would happen, if I recall correctly (I'm writing this without references), approximately once every 2.5 years or so; in other words, relatively rarely.

And so, in a season containing four full moons, they inserted what we'd think of as a "leap moon," but was known as...

... the Blue Moon.

Technically, the Blue Moon is the third full moon in any season containing four full moons. All of the other full moons keep their original names.

Hopefully you can see the logic behind this: it is not tied to any human-made calendar; it's completely independent of January, February, etc., or the numbers of the days therein.

An old issue of, again if I recall correctly, Sky and Telescope magazine from the late 1940s created our error in nomenclature: misinterpreting some information in the Farmer's Almanac or something, that magazine confidently asserted that a Blue Moon was the second full moon in any given calendar month.

Again. Lies. I mean, not deliberate lies, but falsehood, at any rate. But somehow, like many falsehoods are wont to do, it stuck. And so we get what we're inevitably going to see over the next eight weeks: "The full moon on Halloween is a Blue Moon!" No. No, it vehemently is not. (As an aside, a full moon on Halloween is a rare coincidence that's pretty cool for other reasons.) There are no Blue Moons for the rest of this calendar year. There's the Harvest Moon after the equinox in early October, the one on October 31, and one in late November... and then comes the winter solstice. Three. Not four. No Blue Moon.

Why does this matter?

Well, for one thing, I hate seeing mistakes perpetuated and then treated as fact. Bad enough in politics, but now you're messing with folklore and natural cycles.

For another, this leads to what we had a few years ago: two full moons in January, none in February, and then two in March. And anything that happens twice in the span of three calendar months should never be associated with the phrase "once in a Blue Moon."

And, finally, it's disrespectful to cultures that don't use the Gregorian calendar.

Okay, that last one might be a bit of a stretch, but the point is, the false definition of Blue Moon could only happen in a purely solar calendar such as the Gregorian. It can never happen in, say, the Hebrew calendar (which uses leap months every so often so, say, Pesach doesn't get observed in the Northern Hemisphere fall or Hanukkah in the spring). And the true definition of Blue Moon is tied to actual things happening in the sky, the relative positions and orientations of the Earth, Sun, and Moon -- which will be the same for the foreseeable future, whereas human-created calendars change.

Is the false definition easier to compute for the average person? Sure. But that's no excuse.

And, believe me, I've heard counterarguments. "But Waltz, definitions change. Language itself changes over time." True, but irrelevant; the great calendar in the sky hasn't changed appreciably in all of human history, and will continue to not change appreciably well past human and civilization time scales. "But Waltz, I first heard the twice-in-a-calendar-month definition, and any contradictory information just makes me double down on that." Yeah, I heard that one first, too, but when I hear something that's later contradicted by the truth, I change my knowledge to fit the data. "But aren't the full moon names themselves arbitrary?" Yes, but their positions in time, relative to solstices and equinoxes, are not.

So this is my crusade. This is the hill I have chosen to die on. Like I said, I've explained all of this before, maybe even right here in this blog. Certainly in a newsletter a while back. Definitely in my travel blog.

Respect our shared human heritage, and embrace the true definition of Blue Moon.

Just... don't drink that beer. It's pisswater.

Oh, and I appreciated all of the comments from yesterday, but especially Apondia

's:

The interesting thing I found when studying math in a college setting was that I love writing words. I can express things no one ever bothered to listen to about me before. I love reading because I found myself in so many horrible situations in stories I read. Then, when I took math from an older teacher who wanted me to be able to learn, what she was teaching we discussed math vocabulary. I got really happy reading a math dictionary!! And, lo and behold numbers began to fall into equations like they belonged there because I understood that words and numbers are part of each other. I'm not the smartest mathematician but, at least now I get it.

Because it acknowledges that math is mostly just another kind of language (alternatively, language is actually a really complicated form of mathematics), and because -- relevant to today's discussion and that of a couple of days ago -- a person can change their beliefs, and thence succeed. So one MB coming your way -- and since we're still in Birthday Week, I'll give out another one tomorrow, to someone who comments here today.

10 Like • Comment • View 11 comments

© Copyright 2024 Robert Waltz (UN: cathartes02 at Writing.Com). All rights reserved.
Robert Waltz has granted InkSpot.Com, its affiliates and its syndicates non-exclusive rights to display this work.