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 14 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 16 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.

April 1, 2020 at 12:12am

#979863

April Fools

I considered doing an April Fools' entry but, deciding there's already enough foolishness in the world (and then some), decided against it.

Instead, let's look at a particular brand of fool: the conspiracy theorist.

https://aeon.co/essays/the-intellectual-character-of-conspiracy-theorists

Indeed, peculiar theories about all manner of things are now widespread. There are conspiracy theories about the spread of AIDS, the 1969 Moon landings, UFOs, and the assassination of JFK.

This article is from 2015. I'm sure you've heard the conspiracy theories regarding COVID-19. If not, I don't suggest looking them up.

Sometimes, conspiracy theories turn out to be right – Watergate really was a conspiracy – but mostly they are bunkum.

And Watergate was exposed. In general, the more people there are supposedly involved in a conspiracy, the more vanishingly improbable it is to be true.

A 2009 Harris Poll found that between one‑fifth and one‑quarter of Americans believe in reincarnation, astrology and the existence of witches.

I assure you, witches exist. I've met many. Most of them are actually quite nice, if a little catty in larger groups, and way too fond of New Age music.

The article is fairly long, but worth a read if you're at all interested in why it's so frustrating to argue with conspiracy theorists. Like most internet arguments, it skates perilously closely to epistemology: the study of how we know what we know.

Thing is, I think it would be just as wrong to dismiss conspiracy "theories" out of hand as it would be to uncritically accept them. People in power, for example parts of the US government, have certainly done shady things and covered up mistakes. This, for example.

It's legitimately difficult, sometimes, to separate truth from opinion, opinion from fiction, and facts from speculation.

We see a lot of this on April Fools' Day in particular. The internet is replete with sites that celebrate what I like to call Comedy Christmas by posting something vaguely plausible as fact -- and some people, inevitably, fall for it. This practice, of course, preceded the internet - like this report from 1957

- but the Web has made it easier to get your AFD prank to huge numbers of people.

And yeah, I've been pranked. Chances are you have too. Come on, yes, you have. The only alternative is to accept nothing as fact, and that's just as dangerous as being criminally gullible. Just as foolish, even.

No, the best thing to do is to keep an open mind, but be skeptical at the same time. It's not easy, and I certainly fail at it a lot. We're all fools, but sometimes the fool has the real truth, while others are too blind to see it. The peril lies in believing you're immune to foolishness.

2 Like • Comment

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