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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Annette in France

about 9 hours ago
In response to "It's Hot in The South, Once Again!"

The quote is too small and low contrast to be read.

Elycia Lee ☮

about 19 hours ago
In response to "There Is A Such Thing As Coincidence"

I loveeee the photo! You guys look good!

Coincidences are great! So happy you met!

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Wordsmitty ✍️

a day ago
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That's definitely a wonderful picture that shows the whole story. When enlarged, it's a little fuzzy as if after quite a few bottles of wine.

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Annette in France

a day ago
In response to "Two Castles and a Cave"

After being in Tours for an afternoon/evening and only, my favorite part of my time there was to see you.
Old Tours with its medieval houses is charming to stroll through, but France has many charming old towns.

The most impressive cave system that I went to this vacation is under the Chateau de Brézé. It is where the Troglodytes lived. Their caves didn't only house people, they even had underground horse stables.

Interestingly, Chateau de Brézé is also a wine castle complete with an attached tasting room. Even the gift shop sells mostly wine along some postcards.

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Elisa: Middle Aged Stik

a day ago
In response to "Two Castles and a Cave"

Oddly enough, Tours is a sister city to Minneapolis. That still isn't enough of a reason for me to visit France, but it's still a nice bit of trivia.

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Kåre Enga in Montana

Sep 25, 2024 at 7:53pm
In response to "There Is A Such Thing As Coincidence"

That's awesome. Enjoy your visit of the Loire.

TGV = tren à grande vitesse = HST. I only took it once, Paris > Luxembourg. Everything was a blur.

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Annette in France

Sep 25, 2024 at 3:20pm
In response to "There Is A Such Thing As Coincidence"

That was a total coincidence. I didn't even plan to go to Tours at all. I was there because of the way I planned out a travel leg. To run into you on the other side of the Atlantic in a random town was pure luck.
It was fun to catch up with you.

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Brandiwyn🎶

Sep 25, 2024 at 2:49pm
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Love it when stars align!

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Wordsmitty ✍️

Sep 24, 2024 at 5:15pm
In response to "All Out of Louvre"

I agree. You should always check the "Five Must Dos in ..." before you go, so you know what to avoid from crowds.

Smurfs are universal.

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Elisa: Middle Aged Stik

Sep 24, 2024 at 11:30am
In response to "All Out of Louvre"

That street art is definitely a step up.

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August 16, 2019 at 1:01am

#964273

Losers

https://theoutline.com/post/7737/abolish-state-lotteries

It’s time to get rid of the lottery
States should not rely on a scam to fund much-needed services.

Despite the one-in-292-million odds of winning the multi-state Powerball jackpot (you have a greater chance of dying from a falling coconut, which is one in 250 million), Americans spent $71.8 billion on lottery tickets in 2017. The bulk of this revenue was generated by the largest consumers of lottery tickets, who also happen to be the poorest Americans.

I've heard it called a tax on people who can't do math. Part of me doesn't have a problem with that.

Also, somehow I doubt that 1 in 250 million chance of dying from a falling coconut. The chance of a coconut hitting me on the noggin on any given day is effectively zero, as the nearest coconut palm is about 500 miles from me. On the rare occasions when I go to tropical paradises, I specifically avoid hanging out under coconut-bearing trees, because duh. According to some interpretations of quantum physics, there's a nonzero chance that a fully-formed coconut could appear in the air above my head, at sufficient height to kill me upon falling, but that chance is way less than 1 in 250 million.

The Cornell study also found that people who made less than $30,000 a year were more likely to play the lottery for money (as opposed to those who play purely for entertainment), meaning that poor lottery players play as a legitimate strategy for financial stability.

As someone who gambles for entertainment on occasion, I believe that there are way more entertaining things than dealing with bored, minimum-wage convenience store clerks and waiting hours to days for confirmation that you lost. Blackjack dealers, for example, are fun to talk to; like bartenders, they have to be nice so they'll get tips. But hey, whatever twipples your nipples.

A 2006 survey found that one in five Americans believe that winning the lottery “represents the most practical way for them to accumulate several hundred thousand dollars.” This number jumps to one in three Americans for those with incomes below $25,000.

I imagine that, for some people, it really is the most practical way for them to accumulate several hundred thousand dollars, in that any other means of doing that is slightly more impractical. Which is not to say that it's actually "practical," only that while the chance of winning, say, $500K is astronomical, the chance of a minimum-wage worker saving that much money is indistinguishable from zero.

“This is California. And if a dream is going to come true anywhere... it’s going to come true here.”

Sure, if that dream involves banning grocery bags, plastic straws and e-cigs, or declaring that every consumer product in the world causes cancer.

As Jonathan Cohen, a Ph.D. candidate at the University of Virginia studying Amerian lotteries, told Bloomberg last year: “I don’t think it’s a coincidence that state lotteries started emerging in the 1970s and 1980s when rates of social mobility in the traditional economy stagnated and then declined.”

Shout out to the old alma mater... but it might be helpful to know what Mr. Cohen is actually a Ph.D. candidate for. Economics? Statistics? Anthropology? Art history?

The lottery is a narrative of false potential, pacifying the ever more precarious masses by dangling in front of us a better, shinier life. Instead of asking how the system is rigged against you, you ask how you can win.

This just in: marketing is about highlighting the positives about something, and minimizing the negatives. Why do you think the side-effects of prescription drugs are in Ferengi print at the bottom, or, in spoken ads, presented by someone who went to Auctioneer School? "Fukitol can make you irresistible to members of your preferred gender, help to cure your ingrown toenail, and increase your IQ! Possible side effects include intestinal bleeding, growth of a third arm, brain damage, brittle bones, fatal constipation, and loss of sexual function." But I've never seen a state lottery that doesn't have the chances of winning noted prominently. The information is available.

The outcome of the state lottery becomes a deflection of responsibility: it directs players’ frustration away from the state for its failures — to provide sufficient welfare, to fund its public school system without relying on those in poverty, to provide a livable minimum wage — and transfigures the state into a potential fairy godmother.

This is amusing to me because the mascot for the Virginia lottery used to be an incarnation of Lady Luck, and she looked a lot like a Fairy Godmother. But still, okay, you're a state government and you need to raise funds. What's going to play better with the constituency: institute a lottery, giving everyone who plays a chance, however tiny, of hitting it big? Or selling them on a tax increase?

In case I'm not clear, I'm not buying this author's argument. There might be other arguments more compelling; I don't know. Other authors have tried to kick lotteries by highlighting the people whose lives were ruined after winning. News flash: some of the people who play the lottery tend to lack the financial discipline necessary to handle any windfall sensibly.

Would people be better off socking away the money they'd otherwise spend on lottery tickets? Yeah, probably. But many of them wouldn't save it; they'd just spend it on something just as frivolous, and without even the minuscule chance of a return. I've known people who, if the lottery suddenly stopped, would instead just spend the money on cheap-ass "beer," thus still pissing it away.

So there are two facets to this that I can think of.

1) Do we really want to live in a world where everyone is protected from themselves? I know I don't. Sometimes, making non-optimal decisions can lead to enlightenment. But

2) Should our governments be involved in facilitating - as opposed to merely ignoring - those non-optimal decisions? See also: state-owned liquor stores like we have here in Virginia.

I used to think that the way to deal with this sort of thing was through education. A proper grounding in basic finance, and maybe even some decent teaching about probabilities, I thought, would go a long way to addressing the "poor people play the lottery" issue. But some people are immune to facts. I mean, just look around.

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