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:

Write_Mikey_Write!

a day ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
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.

August 1, 2019 at 12:13am

#963522

Stupid Is As Stupid Does

Welcome to August. This month, I'm not following prompts. This month, I might not blog every day; I'll be spending most of a week in Vegas, which is three time zones away, and between that, windowless casinos, and the inevitable drinking, I won't even know what day or time of day it is.

I'm looking forward to that. But until then, how about some music - and philosophy?

You look into her eyes, it's more than your heart will allow
In August and everything after
You get a little less than you expected, somehow

http://nautil.us/blog/the-case-for-professors-of-stupidity

The Case for Professors of Stupidity

On this past International Holocaust Remembrance Day, I reread a bit of Bertrand Russell. In 1933, dismayed at the Nazification of Germany, the philosopher wrote “The Triumph of Stupidity,” attributing the rise of Adolf Hitler to the organized fervor of stupid and brutal people—two qualities, he noted, that “usually go together.” He went on to make one of his most famous observations, that the “fundamental cause of the trouble is that in the modern world the stupid are cocksure while the intelligent are full of doubt.”

Yeah... that idea reverberates back even further. Consider Yeats:

Turning and turning in the widening gyre
The falcon cannot hear the falconer;
Things fall apart; the centre cannot hold;
Mere anarchy is loosed upon the world,
The blood-dimmed tide is loosed, and everywhere
The ceremony of innocence is drowned;
The best lack all conviction, while the worst
Are full of passionate intensity.

The linked article goes on to relate Russell's quote to the Dunning-Kruger Effect, which has become fairly well-known - but read the article if you're unfamiliar with the idea.

On a related note, I started calling Bitcoin and other cryptocurrencies "Dunning-Krugerrands." This amuses me far more than it probably should.

Anyway...

But what exactly is stupidity? David Krakauer, the President of the Santa Fe Institute, told interviewer Steve Paulson, for Nautilus, stupidity is not simply the opposite of intelligence. “Stupidity is using a rule where adding more data doesn’t improve your chances of getting [a problem] right,” Krakauer said. “In fact, it makes it more likely you’ll get it wrong.” Intelligence, on the other hand, is using a rule that allows you to solve complex problems with simple, elegant solutions. “Stupidity is a very interesting class of phenomena in human history, and it has to do with rule systems that have made it harder for us to arrive at the truth,” he said.

That... well, as usual, I'm no expert, but it sounds contradictory to say that stupidity is more than just the opposite of intelligence, and then define it as (what seems to me to be) the opposite of intelligence. But hey, I can be pretty stupid.

I'd like to take a moment to talk about what else stupidity isn't. It isn't the same thing as ignorance, though the two can be related. Ignorance is not having all the relevant facts. But - who among us can say we have all the relevant facts in a particular situation? No one, that's who. It always pissed me off when someone used the old saw: "Never assume anything, because it makes an ASS out of U and ME." I mean, what the fuck? You have to make certain assumptions. Always, in every case, without exception, or you're paralyzed with indecision. Granted, though, there's no excuse for not finding out all the facts that you can within any time constraints - ignorance can be fixed, but not willful ignorance.

It's no surprise to me that when I see that bullshit quote, it's almost always associated with sports, a profession known for many things - intelligence not among them. Which is not to say there aren't really smart athletes; it's just the old mind / body dichotomy going on.

Intelligence, then, as I see it, is the ability to make beneficial decisions based on this incomplete information; while stupid decisions are the result of failing to make appropriate assumptions.

But, the article makes a good point at the end (and in the headline): we need to study this more. But given the prevalence of what they're calling "stupidity," I doubt that people will accept the results. After all, even those of us who are aware of Dunning-Kruger often have a higher than warranted opinion of our own competence.

The worst thing about the DKE is the conclusion that we don't even know what we're ignorant about, much of the time. I know there have been several instances in my life that I can look back on and go, "Wow, I was being stupid there." Even though it seemed smart at the time. No, I'm not sharing; besides, my mind tends to block these things from my direct consciousness, unless I'm, I dunno, trying to fall asleep and something from 20 years ago pokes me awake.

I'll end with one last chord: it's easy to assume stupidity for other people, while not attributing one's own failings to it. This happens a lot in politics; Democrats assume Republicans are stupid, and vice-versa. Granted, there are a lot of stupid politicians, but we should at least consider the idea that maybe the other side has good reasons - even if based on different assumptions - for what it says and does.

Except maybe on Twitter.

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