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🎶

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

September 21, 2021 at 12:01am

#1017797

We Got The Beat

Hey look - it's a link about writing.

‘I don’t care’: text shows modern poetry began much earlier than believed
Academic finds that lines widely reproduced in the eastern Roman empire are ‘stressed’ in a way that laid the foundations for what we recognise as poetry

"I don't care?" Wow, Generation X is older than I thought!

For Taylor Swift, the “haters gonna hate”, but she’ll just “shake it off”. Now research by a Cambridge academic into a little-known ancient Greek text bearing much the same sentiment – “They say / What they like / Let them say it / I don’t care” – is set to cast a new light on the history of poetry and song.

Good thing that copyright long ago expired, huh?

The anonymous text, which concludes with the lines “Go on, love me / It does you good”, was popular across the eastern Roman empire in the second century, and has been found inscribed on 20 gemstones and as a graffito in Cartagena, Spain.

I mean, seriously, how many of us have seen poems like that? Or, well... written them?

In my defense, I was 14.

Before the emergence of stressed poetry, poetry was quantitative – based on syllable length.

Honestly, this is a distinction I never could wrap my head around -- hence why most of my forays into "poetry" are doggerel or parody.

“We’ve known for a long time that there was popular poetry in ancient Greek, but a lot of what survives takes a similar form to traditional high poetics. This poem, on the other hand, points to a distinct and thriving culture, primarily oral, which fortunately for us in this case also found its way on to a number of gemstones,” said Whitmarsh.

This kind of reminds me of children's rhymes. Not the widely-known ones, but the ones that rarely, if at all, get written down, and are passed down to younger children exclusively from older children, who continue to pass it down in an attempt to look cool. Since they're not written down, they usually transform like a generational game of Telephone (what do they call that these days, anyway? "Reply All?")

This poem pushes back the earliest appearance of stressed poetry by at least 300 years,” he said. “It has this sort of magnetic rhythm to it, four beats to the bar, a stress on the first beat, and weaker stress on the third beat, which is rock’n’roll and pop music as well.”

I once heard a theory that rhythm -- whether drumbeats or poetry -- was connected to heartbeat. No idea if that's the actual case or not, but it makes sense.

I also find it fascinating that, even in classical antiquity, there was a distinction between "low art" and "high art," and this seems to be one of the few instances of "low art" that survived. I suppose it's possible that a lot of modern music, especially folk and the like, owe its existence to those half-forgotten words and rhythms. A message across time. We may be facing different challenges now, but we're not so different than those working-class Greeks of old, are we?

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