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Why is a negative number a pseudo-positive number in ones' complement?

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Problem Detail: 

I'm studying ones' complement representation and I read in a book that a negative number in ones' complement is a pseudo-positive of value $R^n - 1 - |x|$.

However there is no demostration and I can't find it anywhere. If someone could tell me the reason I will be very thankful.

Asked By : Joseph

Answered By : David Richerby

One's complement stores negative numbers as their bitwise negation.

Suppose we're working in $b$ bits. A positive integer $x$ is represented in binary by writing it as $x = \sum_{i=0}^b a_i2^i$, where each $a_i\in\{0,1\}$. The bitwise negation of $x$, then, is $$\overline{x} = \sum_{i=0}^b (1-a_i)2^i = \sum_{i=0}^b 2^i - \sum_{i=0}^b a_i2^i = 2^b - 1 - x\,.$$

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