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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Show Recent Comments

10 Most Recent Blog Comments:

tj-Merry Mischief Maker

about 8 hours 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

a day 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

September 19, 2024 at 11:15am

#1076986

Bad Science

After tomorrow, entries will happen at weird and unpredictable times; they won't be my usual stuff, and I might even *gasp* miss a day. But meanwhile, here's another Cracked article that hits a little close to home.

4 Medical Myths We Got Thanks to Bad Reading Comprehension
A bunch of doctors exchanged jokes. People took them seriously and canceled Chinese food

You don’t need to spend your nights carrying out experiments or poring over data. That’s the job of scientists. Then scientists summarize their findings in papers. Then the media summarize those papers in articles. Then other outlets summarize those articles with their own articles. You might not even read those articles, but luckily, those are summarized as headlines.

And that's as good a summary as any of lots of the science reporting I post here.

4. MSG Causing ‘Chinese Restaurant Syndrome’ Was a Joke

Not much worse than having a joke escape into the wild and be taken seriously.

For decades, monosodium glutamate (MSG) has had a reputation for messing with people’s health. Some say it gives you headaches, while other people imagine various more serious long-term effects. In reality, MSG has the same effects on your body as salt does — except, it takes more MSG than salt to get that same effect.

"But MSG really does give me headaches!" Yeah, look up the nocebo effect.

The idea that MSG may be bad originated with one letter written to the New England Journal of Medicine in 1968 by a doctor named Robert Ho Man Kwok.

One reader wrote demanding the original letter’s true identity, reasoning that Robert Ho Man Kwok had to be his troll name. It sounded like “Human Crock,” didn’t it? In reality, that was the guy’s actual name...

I gotta be honest, I thought it was a troll name at first, too.

Soon, newly informed members of the public reported feeling headaches and tingling after eating Chinese food. They didn’t in clinical studies, but they did in the wild, and they blamed MSG.

And this might be the best example ever on why clinical studies should carry more weight than personal anecdote.

Of course, now that "MSG is bad" has permeated the public consciousness, no amount of fact-correcting is going to get people to change their minds.

Some of you might feel baffled or even outraged that a scientific publication like the New England Journal of Medicine should act as a forum for comedy.

No more baffled or outraged that a comedy publication like Cracked should act as a fact-checking source.

Incidentally, I did a whole entry on the MSG bullshit last year: "Umami"

3. Medical Errors Aren’t a Leading Cause of Death

To come up with stats, the JHU analysis averaged results from a few previous studies, and it averaged them badly. One was a study of 12 deaths, while a second was of 14. You shouldn’t use tiny numbers like that to extrapolate a total of 400,000 deaths nationwide.

What have I been saying about small sample sizes?

Yeah, I know I just said to trust studies over personal anecdotes, and I stand by that, but one must also remain skeptical of the study methods. Which this section of the article does, and better than most sources I've seen.

The JHU analysis said 62 percent of all hospital deaths may be from medical errors. Other metanalyses come up a number more like 4 percent. Which is still a lot, but that’s a big difference. It’s the difference between “let’s go to the hospital and get that gaping wound looked at” and “actually, with all the mistakes they do there, we’re better off just staying home and hoping for the best.”

In fairness to everyone involved, "cause of death" can be tricky and complicated, and it's why we have professionals to make those determinations. And then you get the same sort of denial as I talked about above: for instance, when certain people refused to accept any "cause of death" that included COVID, because they held a strong (wrong) inner conviction that COVID couldn't possibly be fatal and because I don't know, some sort of conspiracy about funding and politics.

Still, based entirely on what this section says, it's probably not the third-greatest cause of death. Which I have issues with, anyway, because every time we work on #1 and it gets pushed to #2, then we work on the new #1 and it gets pushed down the list... I guess what I'm saying is that there will always be a leading cause of death, even if we somehow cure cancer and heart disease and keep people from going out where there are snakes and cliffs, and personally, I think we should worry less about that and more about enjoying what life we have, because we're all going to die of something.

2. Men Aren’t More Likely to Abandon a Sick Spouse

No, but it's apparently okay to be sexist if the target is men.

The researchers now did the right thing and issued a formal retraction. “We conclude that there are not gender differences in the relationship between gender, pooled illness onset, and divorce,” they now said, and you’ll today find only the corrected second paper they published, not the original. Some news sites even reported on the retraction. But those articles we linked to before are still up, with no corrections.

And, just like with Wakefield's shameful connection of vaccinations to autism, there are, and always will be, stubborn people with primacy bias or confirmation bias, who will continue to believe the lie in spite of all the retractions.

1. Left-Handers Don’t Die Younger Than Right-Handers

Huh... I'd never heard this nonsense. Or maybe I did and dismissed it because I didn't care, being right-handed.

Many articles that round up all the indignities lefties suffer mention a stat from a 1991 study, which says left-handed people on average die nine years younger than right-handers.

Nah. If I'd seen that, even back then, the alarm bells in my head would have been heard for miles.

Then, when we dig up the study, we see that it was based on just 19 car accidents by left-handed people and tried extending that rate to the population at large.

oh for fucks sake

In conclusion, science is great. Sometimes you get wrong conclusions, though, because people are involved and people are fallible (see also: death from medical mistakes). But it's generally self-correcting in the long run. Which hardly matters as long as there are people who only pay attention to what they want to believe, or to the first headline on any subject, ignoring the retractions and corrections.

But hey, as long as there are bad studies, bad science articles, bad reading comprehension, and idiots, I'll never run out of blog fodder.

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