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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Brandiwyn🎶

a day 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

February 1, 2023 at 12:01am

#1043972

Like the Plague

The hated month of February begins. Appropriately enough, the random number generator pulled up a plague article.

Scientists say they've solved a 700-year-old mystery: Where and when Black Death began

And no, they didn't pinpoint the exact flea.

As the deadliest pandemic in recorded history – it killed an estimated 50 million people in Europe and the Mediterranean between 1346 and 1353 — it's a question that has plagued scientists and historians for nearly 700 years.

...really, NPR? You're going to do that pun? Who do you think you are, the BBC?

Incidentally, other sources put the number of victims higher than 50 million. That would be a significant number of people today; at the time, it was likely over 10% of the world's human population.

Now, researchers say they've found the genetic ancestor of the Black Death, which still infects thousands of people each year.

Which is not what the headline implied.

New research, published this month in the journal Nature, provides biological evidence that places the ancestral origins of Black Death in Central Asia, in what is now modern-day Kyrgyzstan.

See what lack of vowels will do?

Oh, and by "this month" they mean when the article was published, back in June of last year.

What's more, the researchers find that the strain from this region "gave rise to the majority of [modern plague] strains circulating in the world today," says Phil Slavin, co-author on the paper and a historian at the University of Stirling in Scotland.

One could ask where that strain came from in turn.

The article goes on to explain the various strands of evidence that led them to this conclusion, and it was apparently a multidisciplinary effort. That's the part I find interesting.

Does this mean that the mystery of the origin of Black Death has been solved?

"I would be very cautious about stretching it that far," says Hendrik Poinar, evolutionary geneticist and director of the McMaster University Ancient DNA Center in Ontario, Canada, who was not involved in the study. "Pinpointing a date and a specific site for emergence is a nebulous thing to do."

I'm just including this because I didn't want anyone to get the impression it's settled science, as the headline might make one believe.

When the Black Death swept across Eurasia, no one had the slightest clue what DNA was. Or a cell. Or hell, even the notion that organisms too small to see could be what caused the plague. No, they thought it was God's punishment, or witches' curses, or a comet or some shit like that.

I'm just throwing that out there because I continue to see romanticization of the past. While today sucks, the past sucked worse.

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