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:

Write_Mikey_Write!

about 8 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 10 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.

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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