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:

Turkey DrumStik

about 9 hours ago
In response to "Barley There"

And approximately 15 seconds later, someone's going to screech about genetically engineered crops.

That would be me...screeching for more genetically engineered crops.

Write_Mikey_Write!

Nov 14, 2024 at 2:04pm
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

Nov 14, 2024 at 12:10pm
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

October 26, 2018 at 12:53am

#944204

Crow-Magnon

https://www.bbc.com/news/science-environment-45968935

Humans like to restrict what they eat.

I'm not just talking about dieting to lose weight. I mean, like, kosher or halal, or vegan or vegetarian or pescatarian or lacto-ovo-whateverthefuck. Or sticking to organic only, or "natural" only (that rant is for another time). It's as if they have too many choices and have to narrow them down to stay relatively sane, or maybe just to maintain group identity.

I don't like to limit myself, though I'm not the adventurous type who tries disgusting foods from other cultures. My own culture's disgusting food is good enough for me. Offer me some balut in the Philippines, and I'll politely decline. That rotten fish thing they eat in Norway? Nope. But other than the very subjective "don't eat disgusting food" restriction, I have developed one simple rule about eating:

I refuse to eat anything that's smarter than I am.

Now, not to brag or anything, but I'm incredibly smart. Okay, I'm bragging, fine. Point is, I'm smarter than the vast majority of humans. (That doesn't mean I'll eat them, though. Probably.) But after seeing how cuttlefish act and organize themselves in the ocean, I decided they're smarter than I am. So, no cuttlefish. Well, it's not like there are a lot of places that serve cuttlefish, but chances are that calamari in the Italian place down the street is actually cuttlefish, not squid. Not that it would matter; squid are pretty damn smart too. Also octopodes - I think if they lived on land we'd end up enslaved by them.

Dolphins are smarter, certainly, though I haven't encountered actual dolphin steaks anywhere. Cats, definitely. Not dogs - except maybe border collies; those suckers can solve partial differential equations. I think that's actually what they're doing while they're herding sheep, just to keep themselves from being bored. Not just counting sheep, but integrating them over time. Doesn't matter - I don't eat dogs for other reasons, such as "carnivores taste terrible." Other dogs are merely trainable, which doesn't necessarily correlate with intelligence; after all, if I could train you to sit and play dead and fetch a ball on command, would that be proof of your intelligence? I don't think so.

Anyway, to bring the subject back to the link I provided above, it seems that even though I've decided not to eat anything smarter than I am, still, somehow, I regularly end up eating crow.

2 Like • Comment

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