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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Write_Mikey_Write!

a day 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

a day 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').

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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 28, 2019 at 12:45am

#966907

Universal

PROMPT September 28th

Today’s prompt is from Cass--Autumn Spirit

!

Write about what fascinates you about space and/or the universe.

Are you kidding me?

Regular readers will know that the answer to this question is "everything."

There's a lot of everything. I mean, sure, space is mostly empty... well, not really, because of vacuum energy.

It's way above my pay grade, but from what I understand, it turns out that everywhere, all the time, the universe is creating particle-antiparticle pairs that then proceed to annihilate each other within a time period so tiny that it's hard to imagine. Well, most of the time, that is. Sometimes, based on what Stephen Hawking hypothesized, such a pair pops into existence right at the event horizon of a black hole. The black hole sucks one of the pair into its... whatever happens in there, while the other now exists in our universe. It's more complicated than that, of course, but that's the gist of it.

Point is, even "empty" space is interesting - though I probably wouldn't feel the same way if I were stuck on a spaceship plowing through it; that would become very boring very quickly, once the whole "holy fuck I'm on a spaceship" excitement faded.

Want to know something cool? We've seen a black hole tearing a star apart. No shit. Here's Phil to explain it.

Well, a black hole ate a star. (!!!) From the data, it looks like it was a supermassive black hole, about six million times the Sun’s mass, sitting in the center of that distant galaxy. A star got too close, and the gravity from the black hole ripped it apart.

This part cracks me up: These are called tidal disruption events, or TDEs, perhaps the most prosaic and mundane name for one of the Universe’s most violent and catastrophic events.

Tides are what astronomers call the effect of gravity changing with distance. As you move away from a massive object the gravity you feel from it gets weaker. A black hole has ferocious gravity. A star is a big object, and so if it happens to have a particularly unfortunate trajectory that takes it close to a black hole, the side of the star closer to the black hole feels a far stronger force of gravity than the side of the star facing away from the black hole. The overall effect is a stretching, literally pulling the star apart like taffy.

So yeah, that happens. It doesn't happen a lot - the thing people miss about black holes is that past a certain distance from them, they act just like any other source of gravity, so you have to get really lucky (or from the point of view of the star, unlucky) - but given that we can see something like 200 billion galaxies, and these events give off so much energy that you can see it happen from far, far away, it's inevitable that we'd see one eventually.

Also, no, it's probably not going to happen in our own galaxy anytime soon. They've imaged a lot of the stars around our own galaxy's central black hole, and they're pretty sure it's not about to dine on one of them. If it did, though, I'm not sure if 23,000 light years is far enough away.

That's just today's news, by the way. Or, okay, technically yesterday's, and if you want to get relativistic about it, it's news from 375 million years ago, but the point is this is only the most recent thing I read about shit going on in the universe.

As fascinating as it is that we can collect data from such an event and still have a pretty good idea what's going on, there's something even more fascinating: all the stuff we don't know. And all the stuff that we're wrong about, which gives us the opportunity to improve our understanding.

Space is, as Douglas Adams once wrote, big. “Space is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space.” It is, in fact, so big that there's no way to see it all. It's a little bit like my quest to visit all the breweries in the US: even if you manage to do it, by the time you think you're done, new stuff has popped up and old stuff is gone, so you just have to keep trying.

That's the truly amazing thing about space: never-ending discovery. Almost certainly, at some point, we'll find evidence for life (not necessarily sentient life, but life) beyond our biosphere. That will be cool, and it will teach us a lot about our own biology. Then we'll discover more, somewhere else, and that'll teach us something different. And so on. Things we can't imagine, even though we do our best.

Yes, there are things about our own planet we don't fully understand, depths we haven't plunged. Nor do we fully understand ourselves; it's possible that we cannot do so, but I don't think we've hit the philosophical limits yet. There are over seven billion of us, though, and I'm confident in our ability as a species to multitask - and what we discover "out there" could also shed light on what's going on "down here."

I'm going to end this now, or I'll be writing here all night, and no one wants that. I've said a lot more about space in other posts, and as long as I stay alive I'll certainly have more to say in future installments. I'll just note one more thing, something that's been said before by way smarter people than me: we are "space." Every atom in our bodies (save for hydrogen) was forged in the heart of a star somewhere, sometimes in its death throes. We are, in a literal sense, made of star material. To understand us, we have to understand our environment - and our environment, ultimately, is the entire universe.

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