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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 7 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

December 24, 2024 at 2:28am

#1081447

Hot Shoulder

Posting early today because, like many people, I have stuff to do later. In my case, though, the stuff is completely unrelated to tomorrow's holiday.

I've written about Betelgeuse before, most recently here: "Betelgeuse 2"

. This is, however, a different article, more recent, from Big Think.

This is what we’ll see when Betelgeuse goes supernova
The closest known star that will soon undergo a core-collapse supernova is Betelgeuse, just 640 light-years away. Here’s what we’ll observe.

And already I have Quibbles.

1: "what we'll see." It's extremely unlikely that anyone alive as I write this will see it happen. I'm a gambling man, and I wouldn't bet on it, not unless some bookie was offering billion-to-one odds and I could bet, like, a dollar. The headline uses the same value of "we" as people do when they talk about when "we" will colonize distant star systems (hopefully not Betelgeuse).

2. "will soon undergo." As with "we," they're using a variant value of "soon." Best estimate I've seen is within 100,000 years. That's soon in cosmic terms. It's not soon in human terms. Hell, 100,000 years ago, we'd (entirely different definition of "we" this time) barely started using fire.

3. "640 light-years away." Yeah... maybe. For whatever technical reason (it's been explained to me, but it's over my head), B's distance has been tricky to pin down. Wiki claims 400-600 light years, and that's a huge margin which doesn't even include 640.

I should reiterate here that even if it's at the low end of that scale, astronomers expect no ill effects for any life remaining on Earth when it happens. Of course, astronomers have been known to be wrong, from time to time.

But, okay. Issues with the headline don't always transfer to the actual text. It's just that it's the first thing we see, so getting it right is kinda a big deal. I'm not saying that it's clickbait, but it is a bit sensationalized.

The stars in the night sky, as we typically perceive them, are normally static and unchanging to our eyes. Sure, there are variable stars that brighten and fainten, but most of those do so periodically and regularly, with only a few exceptions. One of the most prominent exceptions is Betelgeuse, the red supergiant that makes up one of the “shoulders” of the constellation Orion.

Hence the title of today's entry.

Over the past five years, not only has it been fluctuating in brightness, but its dimming in late 2019 and early 2020, followed by a strange brightening in 2023, indicates variation in a fashion never before witnessed by living humans.

It is necessary for a human to be living in order to witness anything (metaphysics and religion aside), but I think they mean it's weirder than it's been for the past 100 years or so.

There’s no scientific reason to believe that Betelgeuse is in any more danger of going supernova today than at any random day over the next ~100,000 years or so, but many of us — including a great many professional and amateur astronomers — are hoping to witness the first naked-eye supernova in our galaxy since 1604.

As unlikely as it might be, I've said before that it would be very, very cool if I got to see it. I'm just not betting on it.

Located approximately 640 light-years away, it’s more than 2,000 °C cooler than our Sun, but also much larger, at approximately 900 times our Sun’s radius and occupying some 700,000,000 times our Sun’s volume. If you were to replace our Sun with Betelgeuse, it would engulf Mercury, Venus, Earth, Mars, the asteroid belt, and even Jupiter!

See, those numbers don't hit very well with people, including me. Even comparing the size to our solar system doesn't give us a visceral idea of just how fucking huge that star is (not to mention I'd question the Jupiter orbit thing, because red giants like that just don't have a well-defined surface in the way that we think of the Sun as having one).

This image

might help with the size comparison.

Even when it transitions to the more advanced stages of life within its core, from carbon-burning to then neon and oxygen and eventually silicon fusion, we won’t have any directly observable signatures of those events.

Dude, people are easily confused. I get that stars, like people or cats, have a birth, time of existence, and death. As far as we know, though, stars themselves don't harbor life. Yes, the universe is weird, and it's fun to speculate that maybe they do, but "life" in this case is a metaphor for how stars change over time. Calling it life just begs people to misunderstand, deliberately or not, what's meant.

The article goes on to describe what whoever's on Earth when it happens can expect to experience when the event finally occurs. Not going to quote more, but it's pretty interesting, in my opinion. Pay no mind to the "it really happened 640 [or whatever] years ago" thing, though; it's irrelevant except as a way to acknowledge that information has a maximum speed.

Naturally, being science, everything there is based on our best knowledge at this point in time. I'd also expect surprises. But those surprises will only serve to advance the science.

Unless, of course, they're wrong about the "it won't irradiate and sterilize the Earth" thing.

Sleep tight!

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