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!

about 2 hours 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.

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Adherennium Plotting something

about 4 hours 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.

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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.

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.

August 31, 2021 at 12:04am

#1016405

The Real Decoy

Eons ago, in the Before Time, I wrote a blog entry that mentioned the Decoy Effect. Here it is if you're interested: "Gindex"

. I don't expect anyone to remember this. Hell, I barely remembered it; I had some vague feeling that I'd talked about it before so I did a search and that was the only entry that came up.

I just wanted to verify that I hadn't linked this particular article, which even predates that entry. Apparently, I hadn't. So here it is.

The decoy effect: how you are influenced to choose without really knowing it

Price is the most delicate element of the marketing mix, and much thought goes into setting prices to nudge us towards spending more.

Everyone knows, or at least should know, about the bullshit that is the "sale." Nevertheless, it works. Say you want a widget. It's $100, which is outrageous. So you wait. A couple weeks later, you see: Widget Sale! 50% off! And you buy a widget for $50 plus tax.

Thing is, the widget cost the store $20. So you're still getting boned. Yeah, I know, they gotta make a profit, but really?

There’s one particularly cunning type of pricing strategy that marketers use to get you to switch your choice from one option to a more expensive or profitable one.

It’s called the decoy effect.

Gosh wow, I've never heard of that! Except it's in the headline.

The article goes on to provide a retail example, but it also applies to menu options at a restaurant, as I noted in that long-ago entry.

The decoy effect is defined as the phenomenon whereby consumers change their preference between two options when presented with a third option – the “decoy” – that is “asymmetrically dominated”. It is also referred to as the “attraction effect” or “asymmetric dominance effect”.

You can tell that those phrases weren't marketed to consumers because they're from Latin roots.

What asymmetric domination means is the decoy is priced to make one of the other options much more attractive. It is “dominated” in terms of perceived value (quantity, quality, extra features and so on). The decoy is not intended to sell, just to nudge consumers away from the “competitor” and towards the “target” – usually the more expensive or profitable option.

Not sure why they'd necessarily push people to the more expensive option, unless it's something that confers bragging rights, like a car or a fine scotch. Usually the company only cares that they make more profit. If you can make $20 profit on a $100 woogie, or a $10 profit on a $200 woogie, you want to sell more of the $100 woogies.

The article then describes an actual psych experiment where this was demonstrated. However, the experiment, in my view, suffers from the same bias as almost every psych experiment: their guinea pigs weren't chosen at random from the general population, but from college students.

Still, the results were interesting and changed the way businesses priced things.

The decoy effect is thus a form of “nudging” – defined by Richard Thaler and Cass Sunstein (the pioneers of nudge theory) as “any aspect of the choice architecture that alters people’s behaviour in a predictable way without forbidding any options”. Not all nudging is manipulative, and some argue that even manipulative nudging can be justified if the ends are noble. It has proven useful in social marketing to encourage people to make good decisions such as using less energy, eating healthier or becoming organ donors.

I can't say I know a lot about economics (anyone who can say that is probably lying, to themselves or to others), but I've been told that one of the foundational principles of economics is: people respond to incentives. Those incentives can be financial, or, in the case of my town switching to single-stream recycling some years back, time savings.

Another example provided in the article talks about pricing a magazine subscription. I don't know; when I went through the example, I didn't choose the "forced" option (actually I didn't choose any of the options because I'm a cheap bastard, but I mean hypothetically). This is because, in that hypothetical situation where you can get online only, print only, or online and print, I have no interest whatsoever in print options because I get enough mail as it is, and most of it ends up in the aforementioned recycle bin without having been read.

The next example is more representative, I think, of peoples' daily choices.

Consider the price of drinks at a well-known juice bar: a small (350 ml) size costs $6.10; the medium (450 ml) $7.10; and the large (610 ml) $7.50.

Which would you buy?

If you’re good at doing maths in your head, or committed enough to use a calculator, you might work out that the medium is slightly better value than the small, and the large better value again.

But the pricing of the medium option – $1 more than the small but just 40 cents cheaper than the large – is designed to be asymmetrically dominated, steering you to see the biggest drink as the best value for money.

Putting aside for a moment the fact that any of those prices is way too much to pay for a drink that isn't beer, I have to admit I don't see the point here. Even people who suck at math can figure out that the large is nearly twice the size as the small for a modest increase in price (why everyone does that probably has to do with cup and labor costs), and will tend to get the large. Unless - like with me at a coffee shop perusing the tea prices (because I don't like coffee) - you decide that's just way too much drink, and pay the higher unit cost for a small because that's really all you want.

Still, the decoy effect is real; it's marketing and applied psychology. I'd be willing to bet, though, that there are plenty of people it's lost on. All I'm saying is: be aware of it, and buy what you want anyway; just know that what you "want" is, as always, being influenced by external forces.

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