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:

Kåre เลียม Enga

about 5 hours ago
In response to "Extro Special"

I probably fit omnivert best. I'm temperamental.

Words change... definitions change... we change.

1 Like

tj-Merry Mischief Maker

about 23 hours ago
In response to "Sensational"

They say carrots can improve your eye sight, but it's been proven that drinking alcoholic beverages can make you see double.

After stating this, I find myself wondering why hasn't anyone tried fermenting and distilling carrots and marketing it as a vision tonic?

2 Like

🌖 HuntersMoon

a day ago
In response to "Sensational"

OK, look around you and observe (where possible) the reality of the newly revealed "facts." It's pretty clear that "Smartphones make you a hell of a lot of folks stupid."

2 Like

Elisa: Snowman Stik

a day ago
In response to "Sensational"

Thank you for reminding me to check on Retraction Watch.

https://retractionwatch.com/

3 Like

tj-Merry Mischief Maker

a day ago
In response to "Gray Matters"

I wonder how many nine year old people have gray hair? I also wonder how much stress most nine year olds have?

3 Like

Wordsmitty ✍️

Dec 26, 2024 at 1:07pm
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

3 Like

🌖 HuntersMoon

Dec 26, 2024 at 10:00am
In response to "Gray Matters"

I've always believed "better gray than gone away..."

2 Like

tj-Merry Mischief Maker

Dec 25, 2024 at 3:02pm
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

Dec 24, 2024 at 5:16pm
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

Dec 23, 2024 at 4:55pm
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

March 27, 2024 at 11:05am

#1067027

Celestial Spheres

Unless you've been hiding under a rock or on a remote arctic island, you know there will be a total solar eclipse visible from parts of North America in a couple of weeks. I've known about it since 2010 or so, when I decided I wanted to see an eclipse, even if it meant traveling, and I saw that there would be not one, but two solar eclipses within the following 15 years visible from places that didn't even require me to put up with the discomfort, indignity, and humiliation of air travel.

Of course, this foreknowledge (eclipses can be predicted to a very high degree of space-time accuracy even several centuries out) didn't stop me from procrastinating the procurement of travel arrangements until very nearly the last minute, but the fault there lies not in the heavans, but in myself.

Point is, stuff about the upcoming eclipse is all over the news. Where to see it. How to get the proper viewing glasses. Hell, I wouldn't be a bit surprised if someone didn't have a "what shoes should we wear for the eclipse" (sponsored, of course, by a shoe manufacturer). I hate to add to that cacophony; perhaps you're sick of eclipse pieces by now, and I wouldn't blame you.

But I couldn't let this egregious pandering from a formerly distinguished and reputable source (BBC) go unremarked.

Move over, solar eclipse: Scientists predict a once-in-a-lifetime nova explosion in the coming months

"Move over?" Look. There is, objectively, no sight more spectacular than that of a total solar eclipse. None. Oh, you may think "except for the birth of my child," but no, babies get born thousands of times a day and they all come from the same place and look exactly the same. A partial eclipse doesn't come close. A lunar eclipse is cool, but ooh, the moon turned red; now what? I thought Avengers:Endgame was a fun movie with great CGI, but it still can't touch an eclipse on the grandeur scale. Photos, no matter how good, cannot capture the experience. No writer has ever been able to pen words that convey even ten percent of the awesomeness. The only thing I can even conceive of that might approach the spectacle of a total eclipse is if there were a nearby (but not too nearby) supernova; those things can outshine entire galaxies and can be visible during the daytime. But even that, while astronomers and cosmologists would absolutely have orgasms over it, couldn't possibly match the unique experience of a total solar eclipse.

So this nova had better come with multicolored flashing lights engineered by sentient aliens saying "We did this!" for it to even be in the same galactic quadrant of gloriousness as a total solar eclipse.

The nova T Coronae Borealis explodes about once every 80 years.

In other words, it's somewhat predictable, but not nearly to the accuracy of eclipses.

While the world's attention has been focused on the total solar eclipse that will occur later this spring, the distant Corona Borealis binary system – which contains one dead white dwarf star and one ageing red giant star – has been busy gearing up for its own moment of glory: a spectacular nova explosion.

I'll say this for the article: it does a pretty good job explaining, based on the best science we have, how and why this happens.

So, how will this event overtake the eclipse in the race of awesome?

The T CrB star system normally has a visibility magnitude +10 in terms of brightness, according to Nasa. But when the upcoming T CrB nova eruption takes place, the visibility will jump significantly, up to what's known as a magnitude +2, which is far brighter than a +10. To put that into some context, a +2 is a similar level of brightness as the North Star, Polaris.

By the time that happens, T CrB will be visible to the naked eye.

Magnitude 2? Magnitude 2? And you have the audacity to compare the experience of seeing this to witnessing the moon eat the sun, belch corona, and shit the sun out again?

Despite what a certain pop song asserts ("You are as constant as the Northern Star, the brightest light that shines") Polaris is neither all that bright, nor constant. Not only is it a Cepheid variable, which means its absolute brightness fluctuates over time, but apart from its useful position near the north celestial pole, there's nothing distinguishing it from surrounding stars. Hell, the only way I can ever find it in the sky is to locate the Big Dipper (hard to miss) and follow the two "bowl" stars across the blackness to Polaris, which is the tail star of the Little Dipper. Further, due to precession, it wasn't the North Star in the past and won't be in the future. So much for "constant."

Don't get your scientific facts from popular music. Well, except maybe They Might Be Giants. And Schoolhouse Rock.

The dimmest stars you can see on a clear, cold night in the desert, if you have decent eyesight, are magnitude 6. From a city, you can usually see magnitude 2 stars, but they lack context. It's a logarithmic scale, like with earthquake magnitudes or decibels.

I'm a big fan of astronomy and love looking at the night sky, but if a new mag-2 point of light shows up in a small, obscure northern constellation, and I didn't know there was a nova, I'd never even notice it.

Those hoping to see the nova display should look in the sky for the constellation Corona Borealis, or the Northern Crown – a small, semicircular arc near Bootes and Hercules, says Nasa.

Right, because everyone can recognize Boötes and Hercules at a glance. (The article does helpfully provide constellation illustrations.)

Now, look. I'm not downplaying how awesome the nova is from a scientific viewpoint. We certainly have better instruments to observe it with than we did last time it flared, and I bet they'll do some great research on the thing. And I'll do what I can to take a look at it, myself. But from our everyday perspective? It's not going to outdo a total solar eclipse.

I think the BBC is just salty that we Yanks get two solar eclipses seven years apart, while they haven't had any in a while.

So, speaking of celestial sights, I recently found out that there's a comet

visiting us here in the inner solar system, and that it might—just might—become visible during totality.

My first thought upon reading that was "That would be so fucking awesome." And then, like a few hours later, it hit me:

People are, in general, prone to superstition; and both eclipses and comets have been considered portents of DOOM through most of human history. Having both in the sky at once? Damn, that would be spectacular. Except... idiots.

Think I'm being too harsh on the people of the land? Well, consider this article from Atlas Obscura: Why Doomsayers Think the Eclipse Will Bring Disaster to Illinois

“If you lived forever, and you never moved from where you are today, on average, you would have to wait 400 years for a total eclipse to come across where you are,” says Frank Close, Professor of Physics at Oxford University and a Fellow of Exeter College. The likelihood that you could experience two total solar eclipses in one place in the space of seven years is miniscule. The chances are so low, that some believe something special is going on in Carbondale. In particular, conspiracy theorists believe that a seismic event will be triggered when the eclipse arrives in this part of the state, known as Little Egypt, killing hundreds of thousands of people.

Back when I was preparing for the 2017 eclipse, I noticed that the track took it right over Yellowstone. Pretty much everyone knows about the Yellowstone Supervolcano (though the actual danger there may be, pun intended, overblown), so I tried to start a rumor that the combined gravitational forces of the sun and moon would aggravate the magma underneath Yellowstone, causing the doomsday eruption.

It's a stupid enough idea that I was genuinely surprised when it didn't gain traction.

Now, let's be clear. Something bad is going to happen. Something bad often happens, like just the other day when that container ship destroyed the bridge near Baltimore. There's usually a reason for it, but that reason is never a) we're sinners or b) celestial events (unless the celestial event is, you know, a nearby supernova or a giant meteor hitting the planet, in which case it's still not because we're sinners).

But I'm glad my idea didn't take off, even though it logically made more sense than the Carbondale "theory" (and by "more sense," I really mean "slightly less nonsense"). Because shit like that often becomes a self-fulfilling prophecy. Like when it's about to snow, and you think the grocery store is about to run out of milk and bread, so you rush there along with everyone else in town to buy out the entire stock of milk and bread.

If you want to contemplate cosmic coincidences, the one to ponder is this: the sun and moon appear to be about the same size in the sky. Because of elliptical orbits, sometimes they seem slightly bigger or smaller, which is why you sometimes get annular (ring of fire) eclipses instead of totality. But there's no known scientific reason for this coincidence. In the distant past, the moon was closer and appeared larger. In the future, it will be more distant and all eclipses will be annular. There will, consequently, at some point, be a last total solar eclipse.

And that makes me ineffably sad.

Me? I plan on going to central Indiana. It's fairly close to here, with some good hang time for totality. Because I'm me, though, I fully expect thick obscuring cloud cover. That would suck. But at least I saw the one in 2017.

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