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# 23 integer arithmetic 59 twos complement addition 4

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Unformatted text preview: s, however, we might wish to determine whether overﬂow has occurred. For example, suppose we compute × Ü +u Ý , and we wish to Û determine whether × equals Ü · Ý . We claim that overﬂow has occurred if and only if × Ü (or equivalently × Ý.) To see this, observe that Ü · Ý Ü, and hence if × did not overﬂow, we will surely have × Ü. On the other hand, if × did overﬂow, we have × Ü · Ý ¾Û . Given that Ý ¾Û , we have Ý ¾Û ¼, and hence × Ü · Ý ¾Û Ü. In our earlier example, we saw that +u ½¾ . We can see that overﬂow occurred, since . Modular addition forms a mathematical structure known as an Abelian group, named after the Danish mathematician Niels Henrik Abel (1802–1829). That is, it is commutative (that’s where the “Abelian” part comes in) and associative. It has an identity element 0, and every element has an additive inverse. Let us consider the set of Û-bit unsigned numbers with addition operation +u . For every value Ü, there must Û be some value -u Ü such that -u Ü +u Ü ¼....
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