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:

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.

3 Like

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.

2 Like

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.

1 Like

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.

1 Like

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').

1 Like

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.

3 Like

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

1 Like

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.

2 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.

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