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:

Wordsmitty ✍️

about an hour 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 4 hours ago
In response to "Gray Matters"

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

1 Like

tj-Merry Mischief Maker

about 23 hours 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 5, 2024 at 9:47am

#1072179

Putting the Mental in Experimental

Part of the whole point of experiments is to falsify a hypothesis. But sometimes, if your hypothesis is precious to you, and your experiment fails to support it, well, really, your only choices are to lie about it or suppress the results.

Or you could, you know... own up to being wrong, like some of these folks highlighted in Cracked.

5 Experiments That Proved the Exact Opposite of What They Wanted
Turns out Flat Earthers and anti-Semites are no match for science

Did you know that visitors to this site are statistically likely to be sexier than people in general?

Well, obviously I always suspected it.

At least, that’s the hypothesis behind the new study we’re doing.

For some reason, I wasn't contacted about the study.

Sometimes, a study doesn’t prove what it sets out to.

Seriously, though, that's usually a good thing. For starters, it's job security for researchers.

5 The Wallet Experiment

You find a wallet someone dropped. Do you seek to return it, or do you keep the cash and toss the rest?

Toss a perfectly good wallet containing perfectly good identity theft material? No way!

Some wallets contained cash, while others did not. Researchers predicted that people would more likely return wallets that held no money. The researchers also held surveys before the experiment, asking either the general population or hundreds of economists, and all agreed that the cashless wallets would more likely be returned.

Well, sure. That's just common sense.

The experiment ended up offering all kinds of data about how people vary in honesty, but it showed one consistent trend everywhere: People were always more likely to return a wallet that held money.

And this supports my opposition to "common sense."

One possible explanation, that people hoped for a larger reward from the money wallets, did not hold water.

They seem to have overlooked the most likely explanation: Contrary to common sense, people in general don't suck. Most of them are good, or at least neutral. This ties in with my Lone Asshole Theory, that while the majority of people are decent, all it takes is one bad one to ruin your day.

4 The Attempt to Make Cuddlier Hamsters

There's science, and then there's mad science.

In 2022, scientists used CRISPR tech to totally remove a type of receptor from the brains of hamsters. These receptors are called Avpr1as, and they respond to a chemical called arginine-vasopressin, which makes males aggressive. By removing the receptors, the scientists figured we’d render the males passive and cuddly.

Spoiler: it had the opposite effect.

And this is why science fiction should be required reading/viewing for scientists. In this case, specifically, Serenity, from 2005.

3 The German Census of Jewish Shirkers

Today's equivalent might be "the fundamentalist Christian census of drag queen pedophiles."

So, the government had expected to find evidence of Jewish soldiers shirking duty, but they found the opposite.

Bet they didn't like that.

Instead, the census really did debunk those theories, the way the government claimed they wanted but really did not. So, authorities responded by refusing to release the results.

No, they did not like that at all.

Apparently, the other option (flat-out lying about it) didn't occur to them, so... point for prewar Germany?

Nah.

2 Zeeman and His Math Sphere

And we both just lost half our readers.

Not going to quote this one; the article makes it complex enough.

1 The Flat Earthers’ $20,000 Gyroscope Investigation

Speaking of dimensional spheres, if you’re seeking evidence that the world is round, you’ll manage that easily. Just watch the sunrise over the horizon, and boom, you’ve seen the curvature of the Earth.

Except that flat-earthers find ways to convince themselves that this doesn't mean Earth is roughly spherical.

If you’re seeking evidence that the world is flat, however, well, that’s a bit tougher.

Yeah, that would require a trip to North Dakota.

(People think Kansas is flat, but it's got nothing on ND.)

A second experiment used more advanced technology. Host Bob Knodel brought out a $20,000 laser gyroscope, which should keep to the same vertical orientation no matter what goes on with the ground beneath it.

At least they came up with an experiment. Most flat-earthers just spout their nonsense on YouTube or whatever.

What I find ironic about this is that the idea that Earth is round, and the technology behind $20,000 laser gyroscopes, are both products of science. How a person can accept one without the other is questionable.

This, of course, reminded me of "Mad" Mike Hughes, a flat-earther who built a fucking rocket to "prove" his hypothesis. Spoiler: He died in the process. I've written about him before, mostly here: "Bad Advice"

Troublesome data must be rejected. When experiments challenge your world view, you must switch to a different experiment you can trust.

Well, no. But that's certainly how some people get through life.

4 Like • Comment • View 1 comment

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