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:

Brandiwyn🎶

about 22 hours ago
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.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 4, 2024 at 8:17am

#1079484

It's Hard Not To Scare

This Cracked article might have been better to post before Halloween, but remember, tomorrow is Election Day here. Scary stuff is still in our future.

Scientific Studies That Sound Like the Beginning of a Horror Movie
Two words: Spider goats

I have often looked at articles about what some scientists are doing and think: "You fools! Have you never seen a horror movie?"

Animal testing is an unfortunate but necessary part of certain scientific fields — that is, until the scientists decide to go all Dr. Moreau on some rats. Then it’s just weird. It’s like they’ve never seen a single monster movie.

See? I'm not the only one. (Sometimes, "horror" gets replaced with "science fiction," but usually, horror is still a subgenre.)

5 The Spider Goat

Spider silk is super useful for a lot of different things, but there’s a reason you don’t see a lot of spider farms.

I saw a video recently on how silkworms do their thing. Silkworms are larval moths, and generally unpleasant to look at, but they lack the visceral horror of arachnids. Still, spiders are cool... from a distance.

In 2012, Utah State University geneticists rectified that problem by splicing spider DNA into goat embryos, who eventually grew up to lactate spider silk.

Friendly neighborhood spider-goat.

4 The Man Mice

In 2013, scientists at the University of Rochester implanted human glial brain cells into the brains of newborn mice, who became much smarter and learned faster than other mice as a result.

"What are we going to do tomorrow night, Brain?"

Also, anyone who has ever seen a horror movie should have implored them to stop.

3 Acid Elephant

In the ‘60s, experimenting with LSD was all the rage, in both the scientific and “hanging out in your friend’s cousin’s basement” senses of the word. By 1962, scientists at the University of Oklahoma had run out of ideas until one of them asked, “What if we gave an elephant a thousand doses?”

Which makes me wonder: if we see pink elephants, what do elephants see? Gray humans?

2 Magnetized Cockroaches

“Can you turn a cockroach into a magnet?” is a question you only ask after you’ve been seriously scientifically jaded...

...or on a serious acid trip.

1 Zombie Dogs

Believe it or not, we do know how to bring once-living creatures back from the dead.

Thus shifting the definition of "dead." Used to be: heart stopped = death, but then CPR came along, and now doctors have to pretty much guess when the point of no return occurs.

Cornish hoped to try his method on humans, specifically a recently executed prisoner, but the government forbade it not out of any fear of a zombie apocalypse but because they weren’t sure how double jeopardy laws applied to a revived corpse.

Seems like an important legal loophole to fix.

Add to this mix the penchant of certain scientists to revive millennia-old bacteria found in ice cores, and you definitely have the makings of a horror movie. But still not as scary as tomorrow's election.

4 Like • Comment • View 3 comments

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