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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.
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From "JAFBG"  [XGC]:
Write ten truthful statements.
1. Truth is hard to come by these days; many people would rather embrace comforting lies.
2. Something I learned yesterday: The famous Dutch tulip economic bubble was not as big as reported, nor had as far-ranging consequences.
3. Joe Biden was the legitimate winner of the 2020 US Presidential election.
4. Despite all the hype surrounding the event, Richard Branson didn't actually go into space, which, by international convention, begins at 100 km above sea level. His flight was to about 80km -- significant, but not, by most definitions, space. (To be fair, it is high enough for the US to consider everyone aboard an "astronaut.")
5. Because yesterday was also Bastille Day: it's estimated that a person's consciousness might continue for up to 10 seconds following beheading by, for instance, guillotine.
6. I have never seen the movie Titanic, and hope I never do.
7. Speaking of space, the first human to get there, Yuri Gagarin (who did indeed reach space) achieved orbit in 1961. This was closer in time to the Wright Brothers' first controlled powered aircraft flight (in 1903) than it is to today.
8. While we're on that theme, yes, the Earth is approximately spherical. Not flat.
9. The surface area of the Moon is about 38 million square kilometers, smaller than the size of Asia (44.5 million km2). The Moon, however, has fewer trees.
10. (You know I had to include something about this subject) "A 3,900-year-old Sumerian poem honouring Ninkasi, the patron goddess of brewing, contains the oldest surviving beer recipe, describing the production of beer from bread made from barley."
Of course, I can't leave it at that. I had to look for the poem itself, at least an English translation of it. And behold, I found it.
Presumably, the copyright has expired (it wasn't written by Disney) so here it is in all its glory:
Hymn to Ninkasi
Borne of the flowing water,
Tenderly cared for by the Ninhursag,
Borne of the flowing water,
Tenderly cared for by the Ninhursag,
Having founded your town by the sacred lake,
She finished its great walls for you,
Ninkasi, having founded your town by the sacred lake,
She finished its walls for you,
Your father is Enki, Lord Nidimmud,
Your mother is Ninti, the queen of the sacred lake.
Ninkasi, your father is Enki, Lord Nidimmud,
Your mother is Ninti, the queen of the sacred lake.
You are the one who handles the dough [and] with a big shovel,
Mixing in a pit, the bappir with sweet aromatics,
Ninkasi, you are the one who handles the dough [and] with a big shovel,
Mixing in a pit, the bappir with [date] - honey,
You are the one who bakes the bappir in the big oven,
Puts in order the piles of hulled grains,
Ninkasi, you are the one who bakes the bappir in the big oven,
Puts in order the piles of hulled grains,
You are the one who waters the malt set on the ground,
The noble dogs keep away even the potentates,
Ninkasi, you are the one who waters the malt set on the ground,
The noble dogs keep away even the potentates,
You are the one who soaks the malt in a jar,
The waves rise, the waves fall.
Ninkasi, you are the one who soaks the malt in a jar,
The waves rise, the waves fall.
You are the one who spreads the cooked mash on large reed mats,
Coolness overcomes,
Ninkasi, you are the one who spreads the cooked mash on large reed mats,
Coolness overcomes,
You are the one who holds with both hands the great sweet wort,
Brewing [it] with honey [and] wine
(You the sweet wort to the vessel)
Ninkasi, (...)(You the sweet wort to the vessel)
The filtering vat, which makes a pleasant sound,
You place appropriately on a large collector vat.
Ninkasi, the filtering vat, which makes a pleasant sound,
You place appropriately on a large collector vat.
When you pour out the filtered beer of the collector vat,
It is [like] the onrush of Tigris and Euphrates.
Ninkasi, you are the one who pours out the filtered beer of the collector vat,
It is [like] the onrush of Tigris and Euphrates.
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