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by Dylan16807
922 days ago
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But if you look at limits you get "0" and "diverges". And decimal "...999999" is an infinity, which should immediately set off red flags and tell you that you need to be extra careful when analyzing it. In computers your series of 1s is not infinite, there's a modulus that steps in. And this analysis depends on the modulus being an exact power of the base. But you could make a system that's decimal but has a modulus of 999853, for example, and then "-1" would be 999852. |
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That isn't quite correct. The series of 1s really is conceptually infinite. That's why we have sign extension. The analysis (of the sum of all natural powers of 2) will work for any modulus that is an integral power of 2, including a modulus where the integer to which 2 is raised is infinitely large. Such an infinite modulus will still be evenly divided by a finite power of 2 -- as well as by itself -- and so it will disappear whenever you're working in any finite modulus that is a power of 2 -- or when you are working modulo 2^ℕ. The modulus of 2^ℕ will prevent any distinct finite integers from falling into the same equivalence class.
This is what enables you to have an infinite series of leading 1s, or leading patterns, without problems.