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 2 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 4 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.

December 26, 2021 at 12:02am

#1023670

Catstory

Today's link is ancient (2007), but short and about cats.

A Brief History of House Cats
It may be that “nobody owns a cat,” but scientists now say the popular pet has lived with people for 12,000 years

It's possible that new science has come out on the subject since then, but who cares when the illustration is a kitty with a ball of yarn?

Speaking of science, that picture reminded me of string theory: The Universe is a big ball of string, and God is a cat.

Cats were first domesticated in the Near East, and some of the study authors speculate that the process began up to 12,000 years ago.

Headline: 12,000 years. Body text: "speculation." And online journalism has only gotten worse since the noughties.

While 12,000 years ago might seem a bold estimate—nearly 3,000 before the date of the Cyprus tomb's cat—it actually is a perfectly logical one, since that is precisely when the first agricultural societies began to flourish in the Middle East's Fertile Crescent.

Sure, I get the logic behind it, but it's still speculative.

Cats, on the other hand, only became useful to people when we began to settle down, till the earth and—crucially—store surplus crops. With grain stores came mice, and when the first wild cats wandered into town, the stage was set for what the Science study authors call "one of the more successful 'biological experiments' ever undertaken." The cats were delighted by the abundance of prey in the storehouses; people were delighted by the pest control.

I still don't know why everyone who writes about cats have to defend their "usefulness." How "useful" are the paintings in the Sistine Chapel? How "useful" is a ballet? Cats are living art, and they don't need to justify their existence.

Of course, sometimes they're living art that pukes on the couch, but still.

Anyway, worth reading the article, but I wanted to get on with my movie review.

One-Sentence Movie Review: Spider-Man: No Way Home

It is not easy to write a review that does this movie justice without spoiling the best surprises; all I can really say is that if you're already a fan you'll love it, and if you're not, you'll have no idea what's going on, but if that's the case you're not going to a comics-adapted movie anyway, and you should probably go see Licorice Pizza instead (which I haven't seen yet but looks artsy as hell).

Rating: 5/5

5 Like • Comment • View 4 comments

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