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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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.

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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.

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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').

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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.

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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

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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

September 9, 2021 at 12:03am

#1017070

Fight The Power

Here's an article from 2017, but one that seems more relevant now than in the Before Time.

The Five Universal Laws of Human Stupidity
We underestimate the stupid, and we do so at our own peril.

Normally, I'd link to the original source, but it's paywalled. This is a reprint.

Stupid people, Carlo M. Cipolla explained, share several identifying traits: they are abundant, they are irrational, and they cause problems for others without apparent benefit to themselves, thereby lowering society’s total well-being.

I would be remiss if I didn't note that intelligence isn't the last word on human worth. Half the world's human population possesses below-average intelligence (more or less), and it's possible to be stupid and nice, just as it's possible to be smart and an asshole. A university professor like Cipolla might have had some bias in the matter, though.

The only way a society can avoid being crushed by the burden of its idiots is if the non-stupid work even harder to offset the losses of their stupid brethren.

Wait, you want me to be smart and work hard? Let's not go too far, okay?

Of course, everyone does stupid things from time to time. I think it's a matter of proportion.

Let’s take a look at Cipolla’s five basic laws of human stupidity:

Oh, good. Having "The Five Universal Laws" in the headline and not stating them in the body would be... well... stupid.

Law 1: Always and inevitably everyone underestimates the number of stupid individuals in circulation.

No matter how many idiots you suspect yourself surrounded by, Cipolla wrote, you are invariably lowballing the total.

Yeah, best to assume everyone's an idiot until they show otherwise, and then still be prepared to revise one's opinion of them upon further observation.

But like I said... it's about half the population (depending on what's meant by "average").

Law 2: The probability that a certain person be stupid is independent of any other characteristic of that person.

Cipolla posits stupidity is a variable that remains constant across all populations. Every category one can imagine—gender, race, nationality, education level, income—possesses a fixed percentage of stupid people.

This may seem counterintuitive. Okay, it does seem counterintuitive. But the whole purpose of reasoning, and of science, is to check our intuition, which is usually less reliable than logic.

Law 3. A stupid person is a person who causes losses to another person or to a group of persons while himself deriving no gain and even possibly incurring losses.

Cipolla called this one the Golden Law of stupidity. A stupid person, according to the economist, is one who causes problems for others without any clear benefit to himself.

I'm just going to come right out and say it: anyone who trusts horse dewormer over a proven (yes, proven) preventative vaccine is, beyond any shadow of a doubt, in this group.

But to me, the important thing to know is the spectrum this economics professor proposed to map humanity:

This law also introduces three other phenotypes that Cipolla says co-exist alongside stupidity. First there is the intelligent person, whose actions benefit both himself and others. Then there is the bandit, who benefits himself at others’ expense. And lastly there is the helpless person, whose actions enrich others at his own expense.

You'll have to click on the link to see the graphic.

Law 4: Non-stupid people always underestimate the damaging power of stupid individuals. In particular non-stupid people constantly forget that at all times and places and under any circumstances to deal and/or associate with stupid people always turns out to be a costly mistake.

We underestimate the stupid, and we do so at our own peril.

There's a quote attributed to Robert A. Heinlein that's relevant and I've kept it in mind for lo these many years: "Never underestimate the power of human stupidity."

Law 5: A stupid person is the most dangerous type of person.

If "stupid" is defined as above, then yeah, I can believe that.

Declining societies have the same percentage of stupid people as successful ones. But they also have high percentages of helpless people and, Cipolla writes, “an alarming proliferation of the bandits with overtones of stupidity.”

“Such change in the composition of the non-stupid population inevitably strengthens the destructive power of the [stupid] fraction and makes decline a certainty,” Cipolla concludes. “And the country goes to Hell.”

So, based on this, the only way to keep the stupid in check is to actively increase one's position on the article's graph, up and to the right. Can it be done? I don't know. I'm too lazy to try, which I guess puts me in the "stupid" quadrant.

I think it takes an economics professor to think of things in this way. I have to wonder what his thoughts on Rational Market Theory

were, considering the percentage of people who act irrationally. It was in vogue around the time he wrote the above. But I can't be arsed to look it up.

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