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🎶

a day 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.

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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.

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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.

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October 24, 2024 at 9:49am

#1078850

A Winner's Tale

I've written before about the bad-news bias toward lottery winners, where the media latches on to a tragedy connected to someone who won the lottery as if to say: "See? You have to work for your money, or bad things will happen." Whereas in reality, the majority of big-ticket lottery winners go on to live decent lives. Sure, they eventually get sick and/or die, but that's going to happen whether you have lots of money or not.

Well, here's a rare article, from Business Insider, that takes an honest look at one guy's lottery win.

I won million in the lottery when I was 21. It changed everything.

The temptation there is to react with envy or sour-grapes. I'm just looking at it as one person's experience, which of course may not be representative of everyone's.

This as-told-to essay is based on a transcribed conversation with Timothy Schultz, who won the Powerball Lottery in 1999.

So, a quarter-century ago. Plenty of time for someone to figure out what to do with all that money.

I started playing the lottery once or twice weekly, buying a single ticket. I visualized winning and told people about it. They said, "Well if anyone's going to win, you're going to win."

Then I did.

Plenty of people "visualize" winning and continue to lose. I keep visualizing a date with Halle Berry, but it hasn't happened yet.

A press conference announced I had won the $28 million Powerball lottery. After that, our phone was inundated with messages. People I knew congratulated me, but there were stacks of letters from strangers, some of whom asked for money.

I'd like to see a study about whether this happens more often to major lottery winners than to other rich people.

I'd always imagined what I'd do if I ever won: pay off debt and put myself through college, but I'd never thought about how it would change my life.

I immediately question how a 21-year-old without student loans can get themselves into debt so deep the lottery has to get involved. I've been in debt, but it took me longer than 21 years to get there.

Suddenly, I'd gone from a gas station attendant to retired at 21. I felt like I was holding a magic wand. Everything was possible, but I also wanted to be financially responsible.

Some would say that one important aspect of financial responsibility is to not play the lottery. Those who are irresponsible before winning a jackpot will generally continue to be irresponsible. There are, of course, exceptions.

Before turning in the ticket, I consulted with wealth professionals to understand how much I could afford to spend and give to others.

Not a bad idea, depending on how much the "wealth professionals" gouge you for the advice.

Before I received the money, I set up a plan with advisors to invest it. We invested conservatively so the returns could last me over a lifetime.

Also a good idea.

Basic, napkin calculation: you win $20M. Assuming that's the actual lump sum payout and not the amortized version, you then want to do immediate things with it like, maybe, set your family up with houses or give some to the local animal shelter or whatever. Maybe take a round-the-world cruise. Whatever floats your boat. Say you set aside half of it just for funsies. That leaves $10M to invest.

Conservative investments tend to return an average of 8% a year. This is the stock market we're talking about, so some years it might be negative, while other years it'll be higher. In 1999, the dot-com crash was still a few months away, but you'd have no way of knowing that.

Because inflation is also a thing, there's a rule of thumb about living on 4% of the investment account balance on an annual basis. Some question that rule, but again, just being quick here. 4% of $10M is $400K. That's more than most people make in the US, even now, let alone in 1999.

Keep in mind that these assumptions maintain, or possibly slightly grow, the balance. It can stay there, working for you, while you sit on the beach very much not-working, until a) someone fucks up the portfolio, b) someone gets irresponsible with it, c) society collapses, or d) hyperinflation takes over.

But as a 21-year-old, the first thing I bought was the latest video game system.

Honestly, I can't blame him one bit. And what's that cost in 1999? $500? That's a rounding error.

I went back to college to study film and broadcast journalism, a dream come true.

Perhaps the best thing about having money at a young age would be the ability to pursue whatever college degree you wanted, without concerning yourself about ROI on tuition. Or maybe that's not everyone.

People were supportive, but some treated me differently. Some tried to get closer to me, which made me feel like a walking, talking ATM.

It may be true that, once you have money and people know it, you always wonder if someone's becoming your friend because you're cool, or because they see a way to get money out of you.

When you win the lottery, people don't view the money as something you've earned. A family member explicitly told me I got something for nothing by winning the lottery and should keep giving them and others money.

Bullshit.

Also, "I want unearned income too!" is massively hypocritical.

It was a steep learning curve navigating the social aspect of winning the lottery.

Technically, it would have been a shallow learning curve, as per an entry I did here a while back, but there's no accounting for English idioms.

People ask all the time, "Does money buy happiness?" Money doesn't necessarily change who you are. It can affect happiness by buying time, providing opportunities, and alleviating stress about debt.

I know I've harped on this before, but "Does money buy happiness?" is the wrong question, with wrong assumptions. But whoever says it doesn't has never enjoyed a really good single-malt scotch.

I wish I had invested in bitcoin a few years ago, but that's my only regret about how I've spent the winnings.

An investor's worst enemy is hindsight. Personally, I'm glad I never got into Dunning-Krugerrands. Don't trust 'em.

These days, I don't buy anything too crazy. Like many people, I live within a budget.

Budgeting is important, rich or poor.

But if I were 21 now and had the option, I would consider claiming the prize anonymously, especially if it was a large prize.

I'd recommend that for lottery winners at any age.

So. Nothing exceptionally bad happened to him, some good things did, but it did change his life and how he looked at things. Basically, life happened.

This is just one person, though, as I noted. But it's nice to see an honest look at the results for that one person, without sensationalism. Even if the article is an ad for his media conglomerate.

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