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:

Wordsmitty ✍️

about 2 hours ago
In response to "Gray Matters"

"who were between age nine and 39," - so the old folks need the rejuvenation formula first to turn back the years.

I wonder if anyone has studied why hair goes gray in certain places and not everywhere. Maybe different kinds of stress?

1 Like

🌖 HuntersMoon

about 5 hours ago
In response to "Gray Matters"

I've always believed "better gray than gone away..."

1 Like

tj-Merry Mischief Maker

a day ago
In response to "A Mammoth Discovery"

This makes think of a book, "Evolution and Genomic Diversity" by Gene Poole.

1 Like

tj-Merry Mischief Maker

a day ago
In response to "Hot Shoulder"

Maybe if we say, "Betelgeuse" three times in a row we can witness it.

1 Like

tj-Merry Mischief Maker

Dec 23, 2024 at 4:55pm
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🎁

Dec 23, 2024 at 12:56pm
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

Dec 23, 2024 at 10:04am
In response to "Trash Pandas"

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

4 Like

Jayngle Bells

Dec 23, 2024 at 10:03am
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

June 16, 2024 at 10:11am

#1072725

Revisited: "A Whole Lotto Nonsense"

I try to do these retrospectives on Sundays, picking an old blog entry at random and checking in with the topic and how my thoughts may have changed. I have a few self-made rules, though, one of them being that any entry from less than one year ago is excluded.

Well, today's randomly selected older entry barely squeaks by that restriction, having been posted just over a year ago: "A Whole Lotto Nonsense"

The primary point of the linked article (which is still available) is to question the scary narratives trotted out every time a lottery jackpot gets big enough to attract the attention of reporters, or what passes for reporters these days. Apart from a few instances of lousy editing on my part, mostly involving punctuation, I don't think I'd change any of my commentary on it—not too surprising, as nothing major has happened in the past year to change my mind about that subject.

I also have another lottery story in the queue, but I have no idea when it might come up. Like lottery winners, I pick one using random numbers when I'm ready to do an entry. Could be tomorrow. Could hang out there for a year (in which case it probably won't go in this blog, because I have fewer than 300 entries left before it's full). So, spoiler alert: the particular article awaiting my public scrutiny will provide a somewhat more nuanced view of windfalls, showcasing some of the pros and cons, and focuses on just one data point (that is, one lottery winner).

Instead of rehashing stuff you can just click on a couple of links to read, today I'll provide an aside about probability and chance.

It's tempting to think that if there's a one in a million chance for something to happen, it will definitely happen if you try a million times, or that once it happens once, it'll take a million tries for it to happen again. You see this sometimes in reporting about floods and such: two "100 year" floods happen in back-to-back years, so some people scoff at the entire concept of 100 year floods (which, to be fair, hydrological statistics are kind of up in the air right now what with climate change, but their problem is generally with the math, not the science).

I'm not going to go into the math here; you can look it up elsewhere; I try not to bore or confuse people too much, and it can be confusing, even to me, and hydrology was something I actually trained in. But what a 100 year event really means is that it has a chance, based mostly on historical data, of 1% (1 in 100) to occur in any given year. The probability of an isolated event, such as a particular result of a die roll or a coin flip, is independent of any previous occurrences of the event. You flip a coin, it lands on "heads." You're tempted to bet that the next flip is "tails." This is the gambler's fallacy; in reality, the next flip has a 50/50 chance of being heads or tails.

Shuffle a deck of cards. Cut it. The chance of the cut card being an ace is 1 in 13. Shuffle again, and cut. What's the chance of it being an ace again? That's right; 1 in 13. On the flip side (pun intended), you could repeat this exercise 13 times and never once cut to an ace. No, I'm not going to calculate what those odds are.

Lottery odds are much smaller than any of these examples, but the idea remains. Part of the reason lotteries are described as a tax on people who are bad at math is because very large and very small numbers can be difficult concepts to deal with. When it comes to that, most of us are bad at math. Hell, it's hard for most of us to conceptualize the difference between 10 million and a billion. Yes, even me. It's not intuitive, like the difference between 1 and 100 is... and those are the same ratio.

All of which is to say: don't let the scare stories about lotteries affect your decision. There are plenty of good, actual reasons not to play the lottery (generally several million of them, as in your chance of winning is one in several million), but fear of bad shit happening to you shouldn't be one of them.

After all, unlike the numbers picked in the lottery drawing, you have some control over your attitude and actions (philosophical issues about free will notwithstanding). In other words, your chances of getting bitten by a shark drop to very close to zero if you never choose to go swimming.

And they were pretty damn low to begin with.

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