Complex Numbers (Book)

Blog Calendar

◄ September ► 2022
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:

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

September 10, 2022 at 12:02am

#1037565

More Time

Ever wish you had more time in a day? Well, stop that.

The length of Earth’s days has been mysteriously increasing, and scientists don’t know why

Of course, the headline (and some of the text) has to get all sensational about it. We're talking tiny fractions of a second here, a period of time our brains are incapable of registering.

Our brains, but not computers.

Atomic clocks, combined with precise astronomical measurements, have revealed that the length of a day is suddenly getting longer, and scientists don’t know why.

Solar day or sidereal day? Because if it's the latter, that has implications for the length of a year as well.

This has critical impacts not just on our timekeeping, but also things like GPS and other technologies that govern our modern life.

It is true that GPS requires highly precise timekeeping—so precise that it has to take into account relativistic effects due to height and speed of the satellites involved.

Over millions of years, Earth’s rotation has been slowing down due to friction effects associated with the tides driven by the Moon. That process adds about about 2.3 milliseconds to the length of each day every century. A few billion years ago an Earth day was only about 19 hours.

You know what else that does? It causes the moon to move further away, also very slowly. (the reason why is pretty basic physics having to do with conservation of angular momentum in a system). We happen to live in an epoch where the apparent size of the moon is about the same as the apparent size of the sun. As far as anyone who doesn't dabble in mysticism can tell, it is pure coincidence that this is the case during an era when there are curious creatures around to observe it. This is why the moon appears to just cover the face of the sun during a total solar eclipse. Well, usually. Orbits are elliptical, so sometimes you get an annular, or ring of fire, eclipse. A billion years from now, all eclipses will be annular. Assuming, of course, that we manage to avoid blowing up the moon, the Earth, or the sun during that time. Which is an assumption I'm not prepared to make.

For the past 20,000 years, another process has been working in the opposite direction, speeding up Earth’s rotation. When the last ice age ended, melting polar ice sheets reduced surface pressure, and Earth’s mantle started steadily moving toward the poles.

That's the bit I hadn't heard of.

And this process shortens each day by about 0.6 milliseconds each century.

Like I said, tiny fractions of a second.

But there’s a surprising reveal once we take away the rotation speed fluctuations we know happen due to the tides and seasonal effects. Despite Earth reaching its shortest day on June 29 2022, the long-term trajectory seems to have shifted from shortening to lengthening since 2020. This change is unprecedented over the past 50 years.

Sounds like it's just reverting to the mean. But I'm just a spectator here.

The reason for this change is not clear. It could be due to changes in weather systems, with back-to-back La Niña events, although these have occurred before. It could be increased melting of the ice sheets, although those have not deviated hugely from their steady rate of melt in recent years.

Again, I'm sure they had to have thought of this, but what about rising sea levels and/or thickening atmosphere due to general global warming? Hotter things expand, and the only way to expand is up, away from the axis. Conservation of angular momentum would require the rotation to slow down slightly from these effects. Hell, I once read that they can measure the change in the length of a day that results from damming a river to fill a reservoir (I wonder if some of the more recent speeding up is due to drought lowering lake levels, like with Lake Mead). This happens because a lot of water collects at a higher elevation than before the damming happened. Again, the effect is tiny, but atomic clocks are incredibly precise. So precise, in fact, that if you stack one on top of another, you can measure the difference in time

due to relativity in a gravitational field. Well, someone can, anyway; you probably don't have an atomic clock, let alone two of 'em.

If Earth were to shift to even longer days, we may need to incorporate a “negative leap second” – this would be unprecedented, and may break the internet.

Sounds like an echo of the Y2K problem—which people scoff at now, but the reason it turned out to be a giant nothing is that people actually worked to make it a giant nothing. But if the general, long-term trend is to longer days (which it is), it seems to me like someone should get right on this instead of freaking people out with sensationalist reporting.

And no, none of this means that "time is an illusion." (There's another article in my queue that talks about this; I'll get to it eventually.) If something we can measure and predict that precisely is an illusion, then everything is an illusion, which would call into question the definition of "illusion" itself. Reality is complicated and chaotic, and we certainly don't know everything about it—but that's half the fun.

Oh, and speaking of time and fun, tomorrow is my 18th anniversary here on Writing.Com. Doesn't seem that long, but the calendar rarely lies (time isn't an illusion, but our perception of it can be wonky). At 18, one is eligible to vote in the US and, theoretically, be drafted into military service. (Still males only; can we get rid of that sexist nonsense already?) Please don't conscript my account; I know the pen is mightier than the sword and all that, but my account is a pacifist and claims an exemption on that basis.

8 Like • Comment • View 1 comment

© 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.