Why are 11, 111, 1111, ... equivalent to -1 in two's …
https://electronics.stackexchange.com/questions/164413/why-are-11-111-1111-equivalent-to-1-in-twos-complement
Modular ArithmeticUnsigned ArithmeticTwo's Complement Arithmetic This proof is actually quite easy. Theorem: an n-bit binary number that is all 1s has value v=2n−1v=2n−1. Proof:By adding the place values of all bits, we have: v=2n−1+2n−2+…+21+20v=2n−1+2n−2+…+21+20. Note that: 2v−v=(2n+2n−1+…+22+21)−(2n−1+2n−2+…+21+20)2v−v=(2n+2n−1+…+22+21)−(2n−1+2n−2+… Reviews: 2
This proof is actually quite easy. Theorem: an n-bit binary number that is all 1s has value v=2n−1v=2n−1. Proof:By adding the place values of all bits, we have: v=2n−1+2n−2+…+21+20v=2n−1+2n−2+…+21+20. Note that: 2v−v=(2n+2n−1+…+22+21)−(2n−1+2n−2+…+21+20)2v−v=(2n+2n−1+…+22+21)−(2n−1+2n−2+…
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