As far as signed binary numbers are involved for a 32bit machine, the leftmost bit is allocated because that the sign. Once I try to calculation the smallest negative integer value which can be save on computer in a machine, ns follow the following steps:

I entrust the most far-ranging bit together 1 (because the is negative)I ar 1"s right into the remaining bits, for this reason the variety of 1"s I have actually for the magnitude of the number is 31. I transform this binary number come decimal:

$$2^30+ 2^29+...+2^1+2^0=2^31-1$$

With the an adverse sign, the the smallest number is $$1-2^31$$

However the following source says otherwise. It says the smallest worth is $$-2^31$$ on the 3rd page.

You are watching: What is the smallest negative integer

https://www.uio.no/studier/emner/matnat/cg-tower.com/MAT-INF1100/h12/kompendiet/kap4.pdf

What am ns doing wrong here?

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asked Jul 25 "18 at 13:44

Ali KıralAli Kıral
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You are assuming the representation is signed-magnitude, i m sorry is a valid depiction of an adverse numbers. Your answer is exactly for that representation. Most computers use two"s complement, which allows one an ext negative value. The most an unfavorable value is $1$ adhered to by $31$ zeros and also represents $-2^-31$. The motivation is that you have the right to do arithmetic in two"s complement without worrying around whether numbers are positive or an unfavorable and the comes the end right, simplifying the design of the chip.

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answer Jul 25 "18 in ~ 13:55

Ross MillikanRoss Millikan
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The smallest an adverse number is a $1$ adhered to by $31$ zeros i m sorry is understood as $-2^31.$ since twos" match is basically arithmetic modulo $2^32,$ it would certainly be equally logical to interpret it as $2^31.$ The negative value is preferred so that the an adverse integers are precisely those v a $1$ as the most far-reaching bit.

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answer Jul 25 "18 in ~ 13:51

saulspatzsaulspatz
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Your calculation in itself is correct. However, over there are an ext efficient methods to represent an unfavorable integers than sign & magnitude, i m sorry is the an approach you use. Such methods include "two"s complement", which is in fact defined in the connect you provided. When they create that the smallest an adverse integer we have the right to represent is $-2^31$ they refer to the much more efficient methods, and not the naive method. I imply you proceed to read the section about two"s complement and also maybe then it will certainly all make an ext sense.

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answer Jul 25 "18 in ~ 13:55

Dean GurvitzDean Gurvitz
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In the two"s match representation,

\beginalign000&\to0\\001&\to1\\010&\to2\\011&\to3\\100&\to-4\\101&\to-3\\110&\to-2\\111&\to-1\endalign

In this representation, the values are in increasing order (except because that the $3/-4$ jump), therefore that practically the same adder have the right to be provided for unsigned and signed additions/subtractions.

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answered Jul 25 "18 in ~ 14:01
user65203user65203
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