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.

1 Like

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.

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').

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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.

December 28, 2022 at 12:03am

#1042274

Triskaidekaphobia

Feeling lucky?

Why is 13 considered unlucky? Explaining the power of its bad reputation

I'm guessing it's because when you run out of fingers and feet to count, you're boned.

According to the Otis Elevator Co., for every building with a floor numbered “13,” six other buildings pretend to not have one, skipping right to 14.

Knowing that the 14th floor is really the 13th could lead one to believe that the 14th is the unlucky floor, if people were logical. Then again, if people were logical, well, you know.

Many Westerners alter their behaviors on Friday the 13th. Of course bad things do sometimes happen on that date, but there’s no evidence they do so disproportionately.

But when they do, they remember it disproportionately. This is a variant of confirmation bias, kind of like Blue Car Syndrome.

As a sociologist specializing in social psychology and group processes, I’m not so interested in individual fears and obsessions. What fascinates me is when millions of people share the same misconception to the extent that it affects behavior on a broad scale. Such is the power of 13.

If enough people believe something, it gives that something a kind of reality. Kind of like when a winter storm is coming, and you're terrified that the store will run out of French toast supplies, so you run out and buy extra bread, milk, and eggs—which of course makes the store run out of French toast supplies.

The source of 13’s bad reputation – “triskaidekaphobia” – is murky and speculative.

And just as with triskaidekaphobia (one of the few words I can never spell correctly unless it's staring me in the face) itself, people will latch on to an explanation and stick with it in the face of any new evidence. Hell, I do it myself; I'm convinced that it's a relic of the Sumerian counting system, upon which we base our clocks. There's a reason 11 and 12 have their own words that don't end in "-teen."

The historical explanation may be as simple as its chance juxtaposition with lucky 12.

Which might make some sort of sense if people were even a little bit scared of 11, but they're generally not.

Joe Nickell ... points out that 12 often represents “completeness”: the number of months in the year, gods on Olympus, signs of the zodiac and apostles of Jesus. Thirteen contrasts with this sense of goodness and perfection.

Except that there's about 13 months in a year, if one goes by lunar months. And the apostles thing is disputable; it's quite likely that the number of apostles was codified at 12 to fit the "compleness" argument, and not vice-versa. Twelve is just a fun and easily divisible number, as has been known since ancient times (again, Sumerians), while 11 and 13 are both primes.

But the truth is, sociocultural processes can associate bad luck with any number. When the conditions are favorable, a rumor or superstition generates its own social reality, snowballing like an urban legend as it rolls down the hill of time.

It's not just bad luck. Lots of numbers, especially smaller ones, have well-known folk associations, irrespective of their usefulness in mathematics. Three, for instance, is the Trinity and the mind/body/spirit beliefs. Four is the number of classical elements, and has associations with the Divine in Western religions; it's also unlucky in parts of Asia (I've stayed in Chinese-owned hotels in the US that don't have a fourth floor, much as many US-owned buildings don't have a 13th). Five is associated with our fingers, and also plays heavily in religious and spiritual symbolism (the pentagram). And so on. The point being that regardless of objective reality, some people believe in all of these associations.

In Japan, 9 is unlucky, probably because it sounds similar to the Japanese word for “suffering.” In Italy, it’s 17. In China, 4 sounds like “death” and is more actively avoided in everyday life than 13 is in Western culture – including a willingness to pay higher fees to avoid it in cellphone numbers. And though 666 is considered lucky in China, many Christians around the world associate it with an evil beast described in the biblical Book of Revelation. There is even a word for an intense fear of 666: hexakosioihexekontahexaphobia.

My spell checker just rattled, sputtered and smoked.

There are many kinds of specific phobias, and people hold them for a variety of psychological reasons.

There are also levels of phobia, I think (I'm no expert). I've admitted before that I'm absolutely terrified of anything touching my eyeballs. I think many people are at least a little freaked out by the concept, but mine is extreme—though I did manage to make it through eye surgery without losing too much of my shit. Thanks, Valium!

Part of 13’s reputation may be connected to a feeling of unfamiliarity, or “felt sense of anomaly,” as it is called in the psychological literature. In everyday life, 13 is less common than 12. There’s no 13th month, 13-inch ruler, or 13 o'clock.

Never read Nineteen Eighty-Four, or joined the military, did you?

People also may assign dark attributes to 13 for the same reason that many believe in “full moon effects.” Beliefs that the full moon influences mental health, crime rates, accidents and other human calamities have been thoroughly debunked. Still, when people are looking to confirm their beliefs, they are prone to infer connections between unrelated factors. For example, having a car accident during a full moon, or on a Friday the 13th, makes the event seem all the more memorable and significant. Once locked in, such beliefs are very hard to shake.

Which is what I was saying up there.

Then there are the potent effects of social influences. It takes a village – or Twitter – to make fears coalesce around a particular harmless number. The emergence of any superstition in a social group – fear of 13, walking under ladders, not stepping on a crack, knocking on wood, etc. – is not unlike the rise of a “meme.”

Goddammit all to hell, that is the actual definition of a meme. Fortunately, the article goes on to explain that, and even credits Dawkins properly.

Also, there are good, practical reasons not to walk under ladders. And I was very, very disappointed when, as a child, my mother's back flat-out refused to break when I'd deliberately stomp on cracks when she pissed me off (that might have been the start of my skepticism about folk "wisdom.")

I'm looking at my black cat right now, though. She must have crossed my path 13 times yesterday, and I don't feel any unluckier than usual.

Anyway, the article ends by reiterating that despite not having any basis in objective reality, superstitions are hard to displace and can have very real consequences. My black cat, for instance: black cats tend to be adopted less frequently than other cats because of the idiot superstitions surrounding them. Well, their loss is my gain; Robin is a good kitty. Here's a poem I wrote about her, with picture:

STATIC

Cat in the Sink

(ASR)
Some ideas are more amusing in theory than in practice.
#2183875 by Robert Waltz

In the end, 13 is just another number, and Friday the 13th is just another day on the completely made-up calendar. There's one next month, in fact. It being January, it's very likely that it will snow somewhere in the US that day. If so, I'll bet people will complain that it happened because it's Friday the 13th. Never mind the snowstorm that paralyzed part of the country on Christmas Eve, which practically no one considers an unlucky day, and besides, a "white Christmas" is supposed to be a good thing—just don't tell that to the unlucky folks in Buffalo.

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