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 11 hours ago
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

a day 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

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

🐕GeminiGem🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
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?

July 29, 2024 at 11:52am

#1074540

Unpredictable

Because of the way I usually choose topics here, this BBC article was irresistible to me.

The search for the random numbers that run our lives

Our world runs on randomly generated numbers and without them a surprising proportion of modern life would break down. So, why are they so hard to find?

Most importantly, random numbers are used for gaming. Can't play D&D without 'em. We game online now (and we use a different game system, but that's irrelevant). It's important that when it simulates the roll of a d20, there's about a 5% chance for each integer from 1 to 20.

The friends did their best to explain what they were looking for – a really crackly radio. You know, the kind that blares a hail of static between stations as you turn the dial. The shop salesman didn't know what to say.

Because of the headline, I already figured where this was going.

"I think he thought we were quite crazy," recalls Mads Haahr. It was 1997, and he and his pals were nosing around a Radio Shack outlet in Berkeley, California.

Ah, yes, Radio Shack. They used to be cool. Before 1997.

Haahr and three of his friends had been working on gambling software – digital slot machines and blackjack games that they wanted to host online.

Another important use of random numbers. Assuming you want to run a fair game, of course. For various definitions of "fair;" if the House didn't have an edge, they'd have no incentive to run the games.

If these things weren't random, the digital casino wouldn't be very fair and players could even try to beat the system by looking for predictable patterns in the games.

The patterns thing sounds farfetched, but there have been instances of people winning lotteries because the lottery was using pseudorandom numbers with a difficult but discernible pattern.

And so the four friends were seeking a source of randomness that they could all agree was trustworthy. Something that, by definition, couldn't be biased. The idea had hit Haahr like a tonne of poker chips – the hiss of a dirt-cheap analogue radio.

Still not entirely random, but technically chaotic. And chaos gets you close enough, usually.

The noise blurted out by such a device is actually a messy signal shaped by lightning and electromagnetic activity in the Earth's atmosphere.

If I recall correctly, some of it is actually caused by cosmic rays from the beginning of the Universe. Which is pretty cool, for something we've worked so hard to eliminate.

People have long sought external sources of randomness as the basis of random number generators. In this search for true randomness, they have looked practically everywhere for chaotic phenomena that can't be predicted or manipulated.

The more precision you measure something with, the more unpredictability you get in the last decimal place.

In the end, the little grey radio didn't make Haahr and his friends rich after all. The online gaming business was too much in its infancy back then for the young entrepreneurs to make a killing out of it. But the random number generator they built was, they reasoned, still useful. So Haahr made it public at random.org, where it has been churning out random numbers ever since. It gets a lot of visitors.

I've used it. I don't use it to pick articles to discuss here, because that's hardly mission-critical. Mostly, this page

is basically throwing darts at a map. It does have the problem that longitude lines are further apart near the equator than near the poles, so the average separation of lots of random points would be closer together near the poles than near the equator. Again, though, I consider it just a fun toy to play with, so it hardly matters.

But I do wonder how one would design a random point generator where every point on the surface of a sphere is equally likely. One could, I suppose, treat the Earth like a giant d20 (icosahedron), roll a number from 1-20, then further subdivide that triangular face into arbitrarily small but equal areas, pick one of those at random, and project that onto the sphere's surface. But I suspect there would still be some clumping. I can't be arsed to figure it out further.

One man even says he turned to it to help him choose which discs from his 700-strong CD collection to put into his car each week.

See, that sounds like something I would do. I'm always listening to songs on shuffle (which probably doesn't use strong randomization, but whatever).

Cloudflare, a tech firm that provides cloud security services, uses a lot of random numbers at its data centres. The company has sought some eye-catching ways of generating randomness – including a collection of lava lamps.

"Why do you have a thousand lava lamps?" "Random number generation."

There's some discussion at to whether true randomness really exists anywhere but we can leave that to the theoretical physicists.

You mean the folks who keep arguing about whether everything's an illusion or not? No, thanks. I'd rather ask a mathematician. They at least know they're working with abstractions.

Of course, the articles I post here aren't truly random, either. If I find something interesting enough to have comments about it, I add it to a list. It's not like I run random numbers on every webpage in existence (I'm not even sure that's technically possible), but choose from a select list.

Most random numbers we encounter are like that: with boundaries. A standard die may roll a number from 1 to 6, but never a -5 or a 42. A lottery may draw winning numbers between 1 and 50 or whatever, but one hopes that each of those has an equal probability, and that the selection is entirely unpredictable.

Because life would be a lot less fun if everything were predictable.

1 Like • Comment • View 1 comment

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