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!

about 8 hours 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

about 10 hours 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.

March 1, 2024 at 8:39am

#1065309

They Ain't Here

The truth is out there... but you don't believe it. From New York, not to be confused with The New Yorker:

No, Aliens Haven’t Visited the Earth
Why are so many smart people insisting otherwise?

Just to get this out of the way: Smart people can still draw bogus conclusions. They can be fooled. They can be exceptionally good at fooling. I wouldn't give a smart person's opinion more weight, unless it's in a field they're credentialed in, any more than I'd trust a rich person's opinion over that of a poor person.

Last month, Sean Kirkpatrick, the head of the Pentagon office responsible for investigating unexplained aerial events, stepped down.

The article's from the end of January, so I guess that happened in December?

He said he was tired of being harassed and accused of hiding evidence, and he lamented an erosion in “our capacity for rational, evidence-based critical thinking.”

I feel that.

He may have been pushed over the edge by a pair of events from the past summer. In June of last year, Avi Loeb, an astronomer at Harvard, announced that he had found some tiny blobs of metal by dragging a magnetic sled over the bottom of the Pacific near Papua New Guinea. He claimed that these blobs were metallic droplets that had melted off an interstellar object that might have been “a technological gadget with artificial intelligence” — the product of beings from another star system.

That's the same numbskull who claimed that the interstellar wanderer called 'Oumuamua

was a product of technology.

Look, most of the time, I see Avi Loeb's name, Harvard gets mentioned. I think that's an attempt to give him some credentials. But it has the opposite effect on me: instead of being more inclined to believe Loeb's crackpot "it-must-be-smart-aliens" conclusions, I become more inclined to dismiss anything that comes out of Harvard.

In other words, that guy's such a disgrace to that prestigious university that his association with it (along with a few other Harvard-associated people recently) decreases its prestige.

Oh, and sure, random blobs of metal could have come from an interstellar visitor. I don't consider the odds of that to be very high. Claim like that, though, you're going to have to rule out terrestrial origins, first.

Thoughtful, sensible-seeming, non-crankish people at Harvard, at The New Yorker, at the New York Times, and at the Pentagon seemed to be drifting ever closer to the conclusion that alien spaceships had visited Earth.

One of those things is not like the others...

To be clear, we should investigate these claims, within reasonable budgetary limits. Though any debunking that gets done sure won't convince the UFO nuts. On the other hand, show me definitive evidence, and I'll be convinced.

Yet even after more than 70 years of claimed sightings, there was simply no good evidence. In an age of ubiquitous cameras and fancy scopes, there was no footage that wasn’t blurry and jumpy and taken from far away. There was just this guy Grusch telling the world that the government had a “crash-retrieval and reverse-engineering program” for flying saucers that was totally supersecret and that only people in the program knew about the program. Grusch said he had learned about it while serving on a UAP task force at the Pentagon. He interviewed more than 40 people, and they told him wild things. He said he couldn’t reveal the names of the people he interviewed. He shared no firsthand information and showed no photos.

And that, folks, isn't evidence. It's crackpottery, albeit perpetrated by someone who would be in a position to know. It makes me wonder what he was paid to distract us from. Yes, I have my own conspiracy leanings, but they involve human activity, not purported alien.

Now, the article goes on for a while, and it's absolutely worth a look, at least a skim. I won't comment on it further, though, except to say that our collective fascination with the subject says nothing about aliens and everything about humans.

2 Like • Comment

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