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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10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 9 hours ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

a day ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

September 9, 2024 at 9:39am

#1076518

Time Out

Back before the welcome Birthday Week diversion, I featured an article from Ars Technica on how they're coordinating time between Earth and Moon: "Moon Time"

Then, I wrote: "I have a couple of articles about this, from different perspectives." Well, today, the other article popped up. Being from Atlas Obscura, this one may be a little more approachable, but it also gives me more opportunity to be pedantic.

Moon Time Is a Thing Now—Here’s Why It Matters
Physicists recently created Coordinated Lunar Time, a time zone for our Moon.

On the Moon, there is no normal. There is no wind, no rain, no erosion.

Leaving aside that these things are "normal" for the moon, there is some wind (the solar wind) and some erosion, though that process works a good bit more slowly there.

Nothing flies overhead, and nothing green strains toward the sky.

...yet.

There are no daily rhythms like those we experience on Earth—no chirping of crickets, no sunset breeze.

I could live with the former, very easily. But there's also no air, so I wouldn't live very long.

A lunar day and a lunar night each last two Earth weeks.

Roughly. This corresponds with the phases of the moon, and is the result of the Moon always showing the same hemisphere to Earth, and its orbit around our planet. It's all very Newtonian.

What’s more, seconds tick by slightly faster than they do on our home world.

That effect is relativistic.

To understand why Moon time is so strange—and why scientists recently created a new and unique time zone just for the Moon—we have to spend a moment with Einstein.

People think of Einstein, they think of the famous equation. But time dilation is another thing entirely. He figured out a bunch of different shit; that's why they called him a genius.

Special relativity explains how different places in the universe perceive time and space differently; time seems to move more slowly in a larger gravitational field, relative to a lower gravity environment. And a clock moving through space ticks more slowly than a stationary one.

I have no quibbles here. It's just really hard to grok, because it's outside our everyday experience.

In the Hollywood interpretation of these theories, a character traveling in space for a long time, or under the gravitational influence of a black hole or other very large object, will age much more slowly than his loved ones on Earth.

As usual, Hollywood tends to exaggerate for the spectacle. You have to be traveling pretty damn close to light speed to experience that effect without precise measuring tools. And by "large," the author here means "massive." Black holes are generally pretty small, on a cosmic scale.

GPS satellites have to account for it in order to work. So every time you get directions on your smartphone, you experience the practical effects of special relativity.

I've noted this before, but it's relevant here.

In fact, all navigation is really about time. We invented longitude for this reason.

And I've talked about this, as well. Latitude is pretty easy to figure out. Longitude is a lot trickier.

First, we needed to invent a prime meridian, an imaginary line drawn from pole to pole along the planet’s surface. Then, for navigation, you need to figure out where you are relative to that fixed line. The simplest way to do this is to use a time scale that is the same at both locations. So longitude is really about clocks.

There's been a lot of talk lately, among some science video channels I occasionally watch, about how the Greenwich Meridian, the 0 degree longitude line, is in the "wrong" place. But ultimately, it doesn't matter where we put it, as long as everyone agrees on it. An earlier Prime Merdian went through Paris instead of a suburb of London, and I'm sure the French are just pleased as punch that everyone went with the English definition. So pleased that, even though France is mostly south of England, they use the next time zone to the east.

The only relevant thing here is that one needs to define a Prime Meridian for the Moon, which I'm pretty sure they've already done.

I say it should have gone through the center of Armstrong's first footprint, but no one listens to me.

Anyway, the article goes into how GPS accounts for relative time for Earth.

But we can’t just port this system over to the Moon. Clocks on the lunar surface actually tick faster than Earth clocks by almost 58 microseconds per day. It’s not much in a given day, but over the months it will add up, and it’s enough to disrupt the precise timing of GPS.

And thus, the calculation of latitude/longitude.

“It’s like having the entire Moon synchronized to one ‘time zone’ adjusted for the Moon’s gravity, rather than having clocks gradually drift out of sync with Earth’s time,” NIST physicist Bijunath Patla said in a statement earlier this month.

As good a summary as I've seen.

The physicists say their efforts to develop Coordinated Lunar Time could be applied to other places in the solar system, simply by adapting the clock system for any other world’s gravity.

Yeah, not really "any." The world would need to have a definable surface, which some of the larger planets don't. But we're not about to live in Jupiter's atmosphere anytime soon, so, fine.

It’s about time.

Thanks, now I can't use that pun in this entry's title.

2 Like • Comment • View 3 comments

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