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').

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.

August 8, 2019 at 12:42am

#963883

Dedicated Follower of Fashion

Of all the idiosyncrasies of human nature, one that remains elusive to me is the concept of "fashion."

https://www.technologyreview.com/s/613034/the-hipster-effect-why-anti-conformist...

The hipster effect: Why anti-conformists always end up looking the same
Complexity science explains why efforts to reject the mainstream merely result in a new conformity.

I used to joke that the "Fashion" merit badge here on WDC would be the last one I'd get, so of course, once, someone awarded me with one as a joke. What the hell; I'll take it. Somehow, I managed to get a "Family" one, too, for similar reasons - which is amusing if you read yesterday's entry.

So I'm going to take the time to say I'll probably never get a "Consistent" Merit Badge.

You’ve probably seen this effect—perhaps you are a victim of it. You feel alienated from mainstream culture and want to make a statement that you are not part of it. You think about wearing different clothes, experimenting with a new hairstyle, or even trying unconventional makeup and grooming products.

"Victim?" Hardly. I wear what's comfortable, with some thought to appropriateness. When I go out, I wear one of my most elegant t-shirts, along with a Hawaiian shirt and black jeans. When I don't, I wear a less ritzy t-shirt and sweats. As for makeup and grooming products, well, I haven't worn makeup outside a stage production (though I don't object to it for men on cultural grounds, it's simply too much trouble), and my grooming products consist of an electric razor and a comb.

The razor is mostly to keep my cheeks relatively smooth and my goatee relatively symmetrical and non-frizzy. Due to some genetic fuckery, I can't grow a full beard; hence the goatee. Why a beard at all? I don't know; I just think I look better with a goatee.

And yet when you finally reveal your new look to the world, it turns out you are not alone—millions of others have made exactly the same choices. Indeed, you all look more or less identical, the exact opposite of the countercultural statement you wanted to achieve.

Apart from people deliberately emulating The Dude, I rarely encounter people who made the same choices. Not "never," of course; there are about three billion men out there, and certainly some of them have longish hair and goatees.

This is the hipster effect—the counterintuitive phenomenon in which people who oppose mainstream culture all end up looking the same. Similar effects occur among investors and in other areas of the social sciences.

This actually came to my attention when I was a kid and hippies were still a thing. My dad pointed it out. "Let's display our individuality by all looking the same!"

And his conclusion is that in a vast range of scenarios, the hipster population always undergoes a kind of phase transition in which members become synchronized with each other in opposing the mainstream. In other words, the hipster effect is the inevitable outcome of the behavior of large numbers of people.

Or it could be an inevitable outcome of having large numbers of people and finite fashion choices, combined with the Blue Car Effect, also known as confirmation bias. The idea is before you own a blue car, you rarely notice them, but when you do own one, you see them everywhere. But what do I know? I'm not a college professor and I know dicksquat about fashion. I do own a blue car, however.

I have noticed an odd thing about fellow beer geeks, though. The male ones, anyway. The vast majority seem to have beards. This is, to me, counterintuitive; if one of your primary recreational activities involves pouring delicious, foamy malt beverage into your maw, it seems like having a beard (or at least the mustache part) would be contraindicated; it's a lot easier to wipe beer head off your upper lip if it's bare. But since I have upper lip whiskers, and it doesn't bother me, I don't really care.

This simple model generates some fantastically complex behaviors. In general, Touboul says, the population of hipsters initially act randomly but then undergo a phase transition into a synchronized state. He finds that this happens for a wide range of parameters but that the behavior can become extremely complex, depending on the way hipsters interact with conformists.

I guess it becomes a form of social signaling, like putting bumper stickers on your car, or even your choice of car. "I am in this tribe, and not the others." Whenever I see a dude with a full beard, it's a good bet he's a beer guy and we can have a conversation. Still not sure how to identify female beer geeks in the wild; if I knew that, I might not be single.

Hipsters are an easy target for a bit of fun, but the results have much wider applicability. For example, they could be useful for understanding financial systems in which speculators attempt to make money by taking decisions that oppose the majority in a stock exchange.

Those are called "contrarians," and, pretty much by definition, they're not generally speculators. Warren Buffett, arguably the greatest investor alive and once the richest man in the world (still in the top five) as a result, once said something like (can't be arsed to look it up) "be greedy when others are fearful and fearful when others are greedy." Short-term market thinking results in getting in and out of positions, usually at the exact wrong time. For example, if you sold stocks last Monday, you were doing so when the market was crashing - which it was doing because the majority of people in the market were also selling. However, if you used it as an opportunity to buy, you'd be richer today than you were Monday (what it'll do tomorrow is anyone's guess).

Point is, there's a difference between "speculators" and "investors," and the former tend to follow the herd, to their eventual detriment.

So anyway, just linking this because it's an interesting article and made me think about fashion. Who knew it was subject to mathematical modeling? I would have never thought about it like that. But then, I'm proudly contrarian.

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