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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10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 21 hours ago
In response to "Trash Pandas"

Being an avid camper, keeping items out of the reach of raccoons can be challenging. We've had them figure out how to open coolers, open chests, and even gnaw through rope used to suspend a cooler out of their reach.

2 Like

🐕GeminiGem🎁

a day ago
In response to "Trash Pandas"

I wrote one of my Danger Dog stories based on a childhood camping trip that involved a raccoon stealing grapes from our cooler "Danger Dog's Camping Capers"

I had always heard that cockroaches will be the only thing that survives a nuclear war/apocolypse, but maybe it will actually be the raccoons.

4 Like

🌖 HuntersMoon

a day ago
In response to "Trash Pandas"

Just wait until we team up with the cats...

4 Like

Jayngle Bells

a day ago
In response to "Trash Pandas"

The back of my property is sheltered by buildings and cedar trees, so I live on a "racoon highway." Tinfoil works until one takes it for the team and moves it. The only scent that somewhat deters then is ammonia - but they alter their path within the already preferred walkway, not avoid it completely. They generally leave the garbage bin alone, but that's because of the bears, not because of anything we do 😂

I've had more wildlife problems in the city than I ever did in the country. The first time I saw a deer up close? In the middle of my street directly in front of my house. Probably looking for the trash panda party. Lol

4 Like

Annette

Dec 21, 2024 at 11:59am
In response to "Neutral Zone"

It made me happy that I was able to like it. I take the small victories.

1 Like

tj-Merry Mischief Maker

Dec 20, 2024 at 6:06pm
In response to "To Err Is Human"

Right from the beginning I find myself asking, "How many times can anything go extinct?" On a side note, the other day while shopping I saw a neanderthal in the produce section. He was trying to fit in with modern clothes, but it was pretty obvious he was a neanderthal.

1 Like

Annette

Dec 20, 2024 at 1:00pm
In response to "To Err Is Human"

Comments work.

"Like" button sends me this message:

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2 Like

Annette

Dec 20, 2024 at 12:58pm
In response to "To Err Is Human"

Can I leave this comment?

Wordsmitty ✍️

Dec 19, 2024 at 9:03pm
In response to "Five Seconds"

Way, way back when I was little, it was a three second rule that I remember. Sometime since it was upped to five seconds because of all the chemical preservatives that are pumped into everything now.

1 Like

tj-Merry Mischief Maker

Dec 19, 2024 at 3:43pm
In response to "Five Seconds"

I also ate carrots right out of the garden growing up, I also picked and ate fruits and berries growing in the wild without washing. I do remember food dropped on the floor was either rinsed off and then eaten, or if it was something not easily rinsed, like buttered toast buttered side down, we gave it to the dog.

In our house, there's not a five second rule; unless it's something dogs can't eat, it's whichever one gets it first.

2 Like

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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