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Pi appears in non-circular formulas mainly because of hidden rotational symmetry and because oscillations are mathematically tied to circular motion. Some appearances come from how constants are defined.
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Zero first marked empty places in written numbers, in Babylon and elsewhere. It became a number with its own arithmetic rules in India, where Brahmagupta set them out in 628 AD.
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In standard real-number mathematics, 0.999… is defined as the limit of 0.9, 0.99, 0.999, and so on, and that limit is 1. They are two names for one number.
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Division is defined as the inverse of multiplication, and nothing multiplied by zero can give a nonzero answer. Assigning any value to 1/0 leads to a contradiction, so mathematicians leave it undefined.
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Yes, by most estimates. Shannon's figure of about 10^120 possible games far exceeds the roughly 10^80 atoms, though the number of positions is much smaller.
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The arithmetic holds: 42 doublings of a 0.1 mm sheet give about 440,000 km, past the Moon. But the fold cannot be done, and a geometric argument says the sheet itself would need to be enormous.
Starter thread · AI-reviewed against its sources
