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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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').

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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.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

August 19, 2021 at 12:02am

#1015838

Curiosity

Scientists have studied cats. This is not news. (Mews?) But these scientists studied why cats like to settle into boxes.

What cats’ love of boxes and squares can tell us about their visual perception
"Vision has evolved to answer questions having to do with boundaries and contours."

Some things about cat behavior might make sense if you think about it. Like how their vision is adapted for lower light levels, and they're often active at night. But the box thing? It's not like cats and boxes evolved in parallel. There were cats. And then there were humans. And then the humans made boxes, and cats lived happily ever after.

It is a truth universally acknowledged—at least by those of the feline persuasion—that an empty box on the floor must be in want of a cat.

Gotta love articles that start out parodying Jane Austen.

This behavior is generally attributed to the fact that cats feel safer when squeezed into small spaces, but it might also be able to tell us something about feline visual perception.

Okay, that's not so obvious.

That's the rationale behind a new study in the journal Applied Animal Behaviour Science with a colorful title: "If I fits I sits: A citizen science investigation into illusory contour susceptibility in domestic cats (Felis silvers catus)."

That title. THAT TITLE.

I am compelled, however to point out that the species nomenclature there is both wrong and outdated. From Wikipedia:

In 2003, the International Commission on Zoological Nomenclature ruled that the domestic cat is a distinct species, namely Felis catus. In 2007, it was considered a subspecies, F. silvestris catus, of the European wildcat (F. silvestris) following results of phylogenetic research. In 2017, the IUCN Cat Classification Taskforce followed the recommendation of the ICZN in regarding the domestic cat as a distinct species, Felis catus.

If you click on the link in the article to the actual paper, you'll see they spelled "silvestris" correctly (it's from "woodland" not "silver"), but the date is this year so they should have left that part out entirely.

This is apropos of nothing, really, but when I see a mistake like that I immediately wonder what other errors they're propagating. But I can't be arsed to confirm and cross-reference everything. I mean, look, it's a study on why cats like boxes; it's not, like, the latest epidemiological research or some shit. Just bask in the knowledge that someone managed to get a scientific paper published about the symbiosis between boxes and cats.

Now, I'm finding it difficult to find other sentences in the article that make sense out of context, and there really aren't any; the text builds upon itself. So I'll just leave it at this, but I definitely encourage you to click on the link. The text is pretty short, but it explains everything well; still, if you can't be arsed to read it, it's worth going there just for all the pictures of good kitties.

Why do we study such things when there might not be any direct rewards to reap from them? Well, first of all, you never know what line of research might yield something useful; for instance, I can see how this might have AI applications. And second, not everything has to be useful; sometimes we're just satisfying our own curiosity.

And in this case, fortunately, curiosity didn't kill the cat.

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