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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Brandiwyn🎶

about 22 hours ago
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.

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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.

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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.

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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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

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Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

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.

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