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

March 25, 2023 at 11:03am

#1046966

Attention Payment

I've been wanting to share this arti- SQUIRREL!

We’ve always been distracted
Worried that technology is ‘breaking your brain’? Fears about attention spans and focus are as old as writing itself

If you suspect that 21st-century technology has broken your brain, it will be reassuring to know that attention spans have never been what they used to be.

No, technology hasn't broken our brains. Endgame capitalism has.

Even the ancient Roman philosopher Seneca the Younger was worried about new technologies degrading his ability to focus. Sometime during the 1st century CE, he complained that ‘The multitude of books is a distraction’.

For the record, this was after the fall of the Library of Alexandria.

By the 12th century, the Chinese philosopher Zhu Xi saw himself living in a new age of distraction thanks to the technology of print: ‘The reason people today read sloppily is that there are a great many printed texts.’

"But Waltz, I thought Gutenberg invented printing in like 1450" No, he mostly just put together movable type and lots of actual moving parts.

And in 14th-century Italy, the scholar and poet Petrarch made even stronger claims about the effects of accumulating books

Depends how big your storage space is.

A torrent of printed texts inspired the Renaissance scholar Erasmus to complain of feeling mobbed by ‘swarms of new books’, while the French theologian Jean Calvin wrote of readers wandering into a ‘confused forest’ of print.

Calvin was easily confused.

We can now worry that the cognitive circuitry of the brain has been ‘rewired’ through interactions with Google Search, smartphones and social media. The rewired mind now delegates tasks previously handled by its in-built memory to external devices.

Yes. This is a good thing. Unless you want to win bar trivia games without cheating.

Writing during the 13th century, the grammarian Geoffrey of Vinsauf had plenty of advice for writers overwhelmed with information. A good writer must not hurry; they must use the ‘measuring line of the mind’ to compose a mental model before rushing into the work of writing: ‘Let not your hand be too swift to grasp the pen … Let the inner compasses of the mind lay out the entire range of the material.’

That's great, Jeff, but we have a deadline to meet.

The article is fairly long, but has lots of interesting historical bits. Unfortunately, my attention span petered out about halfway down the page.

4 Like • Comment

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