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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.
March 25, 2023 at 11:03am March 25, 2023 at 11:03am
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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. |
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